mathematical models in the manufacturing of glass: c.i.m.e. summer school, montecatini terme, italy...

Lecture Notes in Mathematics 2010

Editors:J.-M. Morel, CachanF. Takens, GroningenB. Teissier, Paris

Fondazione C.I.M.E., Firenze

C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International MathematicalSummer Centre. Conceived in the early fifties, it was born in 1954 in Florence, Italy, and welcomedby the world mathematical community: it continues successfully, year for year, to this day.

Many mathematicians from all over the world have been involved in a way or another in C.I.M.E.’sactivities over the years. The main purpose and mode of functioning of the Centre may be summarisedas follows: every year, during the summer, sessions on different themes from pure and applied math-ematics are offered by application to mathematicians from all countries. A Session is generally basedon three or four main courses given by specialists of international renown, plus a certain number ofseminars, and is held in an attractive rural location in Italy.

The aim of a C.I.M.E. session is to bring to the attention of younger researchers the origins, devel-opment, and perspectives of some very active branch of mathematical research. The topics of thecourses are generally of international resonance. The full immersion atmosphere of the courses and thedaily exchange among participants are thus an initiation to international collaboration in mathematicalresearch.

C.I.M.E. Director C.I.M.E. SecretaryPietro ZECCA Elvira MASCOLODipartimento di Energetica “S. Stecco” Dipartimento di Matematica “U. Dini”Universita di Firenze Universita di FirenzeVia S. Marta, 3 viale G.B. Morgagni 67/A50139 Florence 50134 FlorenceItaly Italye-mail: [email protected] e-mail: [email protected]

For more information see CIME’s homepage: http://www.cime.unifi.it

CIME activity is carried out with the collaboration and financial support of:– EMS - European Mathematical Society

Angiolo Farina ·Axel Klar ·Robert M.M. MattheijAndro Mikelic ·Norbert Siedow

Mathematical Modelsin the Manufacturingof Glass

C.I.M.E. Summer School,Montecatini Terme, Italy 2008

Editor:Antonio Fasano

ABC

EditorAntonio FasanoUniversita degli Studi di Firenze,Dipartimento di Matematica “Ulisse Dini”Viale Morgagni 67/AI-50134 [email protected]

ISBN: 978-3-642-15966-4 e-ISBN: 978-3-642-15967-1DOI: 10.1007/978-3-642-15967-1Springer Heidelberg Dordrecht London New York

Lecture Notes in Mathematics ISSN print edition: 0075-8434ISSN electronic edition: 1617-9692

Mathematics Subject Classification (2010): 76D05, 76D07, 76B10, 76B45, 76M10, 80A20, 80A23,35K05, 35K60, 35R35

c© Springer-Verlag Berlin Heidelberg 2011This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.

The use of general descriptive names, registered names, trademarks, etc. in this publication does not im-ply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.

Cover design: SPi Publisher Services

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Authors: see List of Contributors

Preface

The EMS-CIME Course on Mathematical Models in the Manufacturing of Glass,Polymers and Textiles was held in Montecatini Terme (Italy) from September 8to September 19, 2008. The course was co-directed by John Ockendon (OCIAM,Oxford, UK) and myself. The following topics were treated:

(1) Nonisothermal flows and fibres drawing (Angiolo Farina and Antonio Fasano,Univ. Firenze, Italy, Andro Mikelic, Univ. Lyon, France) (*)

(2) The mathematics of glass sheets and fibres (Peter Howell, OCIAM, Oxford,UK)

(3) Radiative heat transfer in glass industry: modelling, simulation and optimi-sation (Axel Klar and Norbert Siedow, ITWM – Fraunhofer, KaiserslauternGermany) (*)

(4) Modelling and simulation of glass forming processes (Robert Mattheij,TU Eindhoven, The Netherlands) (*)

(5) Injection moulding (Hilary Ockendon, OCIAM, Oxford, UK)(6) Fibre assembly modelling (Hilary Ockendon, OCIAM, Oxford, UK)(7) The mathematics of the windscreen sag process (John Ockendon, OCIAM,

Oxford, UK)

The focus was largely on glass manufacturing processes, with some digression topolymers and textile fibres in a context very close to the area of glass manufacturing.This volume collects the lecture notes of the courses marked with (*), all devoted toproblems in glass industry. It is regrettable that the other lecturers could not providea chapter, because the subjects they illustrated were extremely interesting.

John Ockendon presented a fascinating and quite difficult problem: the produc-tion of a windscreen by the natural bending under gravity of a still soft glass layerclumped at the boundary. The audience was very excited by his colourful expla-nation of the underlying mechanics, making use of any deformable object he hadat hand.

Hilary Ockendon posed stimulating questions about injection moulding and the“flow” of fibres in a fluffy tuft subject to traction. We had exciting afternoon sessionsdiscussing such problems.

v

vi Preface

Peter Howell gave a series of lectures on the manufacturing of glass sheets andfibres which provided an excellent complementary view of the subjects treated byFarina, Fasano, Mikelic.

Indeed he addressed different problems in the same area (e.g.: how to get a fibreof a desired cross section), each with a different mathematical approach.

Fortunately most of the material not included in this volume is retrievable on theCIME web site, either in the form of slides or of excerpts from books.

Altogether the Course presented a remarkable review of quite advanced techno-logical problems in the glass industry and of the mathematics involved. It was quiteamazing to realize that such a seemingly small research area is on the contrary ex-tremely rich and it calls for an impressively large variety of mathematical methods.

Despite the fact that the volume is not collecting all the material presented at theCourse, it deals with a number of problems which are very typical in the field ofglass manufacturing and it can certainly be useful not only to applied mathemati-cians, but also to physicists and engineers, who can find in it an overview of themost advanced models and methods.

The Chapter by J.A.W.M. Groot, R.M.M. Mattheij, and K.Y. Laevsky illustratesthe various processes of glass forming, starting from the basic physical information,developing the mathematical models for each process, and analyzing the proceduresof numerical computations.

Then we have two Chapters on radiative heat transfer in glass. The first one is byM. Frank and A. Klar, treating in detail the physics of radiation in glass and variousapproximated methods to model it, with an eye to numerical complexity. This is aquite substantial piece of work, due to the extension and the intrinsic difficulty of theproblem. It is followed by the contribution of N. Siedow, who, after continuing theinvestigation of numerical methods for heat transfer problems including radiationand convection, passes to a question of great importance: the measurement of glasstemperature from the observation of the spectrum of emitted optical radiation. Fromthe mathematical point of view this is formulated as an inverse problem, which istypically ill posed.

The way of circumventing this difficulty is explained in detail and examples areprovided.

The last Chapter, by A. Farina, A. Fasano, A. Mikelic, deals with the industrialprocess of glass fibre drawing, which goes through several stages having differentthermal and mechanical characterizations, and analyzes in general non-isothermalmotions of viscous fluids which are mechanically incompressible and thermally ex-pansible.

I must abstain from commenting the scientific level of the present volume, sinceI am among the contributors, but at least I wish to express my deep gratitude to theAuthors for their valuable work. Finally, also on behalf of John Ockendon, I wish tothank EMS and CIME for having made this Course possible. A particular thank tothe Secretary of CIME, Prof. Elvira Mascolo, who took care of so many details.

Antonio Fasano

Contents

Mathematical Modelling of Glass Forming Processes . . . . . . . . . . . . . . . . . . . . . . . . . 1J.A.W.M. Groot, Robert M.M. Mattheij, and K.Y. Laevsky1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1 Glass Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Process Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Geometry, Problem Domains and Boundaries . . . . . . . . . . . . . . . . . . . 82.2 Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Parison Press Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Slender-Geometry Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Motion of the Plunger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Blow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Glass-Air Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Direct Press Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Radiative Heat Transfer and Applications for Glass ProductionProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Martin Frank and Axel Klar1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 Radiative Heat Transfer Equations for Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.1 Fundamental Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

vii

viii Contents

2.2 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.3 The Transfer Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.4 Overall Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.5 Boundary Conditions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.6 Summary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 Direct Numerical Methods .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.1 Ordinates and Space Discretizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2 Linear System Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3 Preconditioning Techniques.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.4 A Fast Multilevel Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 Higher-Order Diffusion Approximations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.1 Asymptotic Analysis and Derivation of the SPN

Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.2 Boundary Conditions for SPN Approximations .. . . . . . . . . . . . . . . . . 100

5 Moment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.1 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Minimum Entropy Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.3 Flux-Limited Diffusion and Entropy Minimization . . . . . . . . . . . . . 1085.4 Partial Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.5 Partial Moment PN Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.6 Partial Moment Entropy Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Frequency-Averaged Moment Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.1 Entropy Minimization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 Inversion of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7 Numerical Comparisons.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.2 Grey Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.3 Grey Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.4 Multigroup Transport.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.5 Multigroup Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.6 Adaptive methods for the Simulation of 2-d and

3-d Cooling Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Radiative Heat Transfer and Applications for Glass ProductionProcesses II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135Norbert Siedow1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1352 Models for Fast Radiative Heat Transfer Simulation . . . . . . . . . . . . . . . . . . . . . . 137

2.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373 Indirect Temperature Measurement of Hot Glasses. . . . . . . . . . . . . . . . . . . . . . . . 148

3.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Contents ix

3.2 The Basic Equation of Spectral RemoteTemperature Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

3.3 Some Basics of Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.4 Spectral Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.5 Reconstruction of Initial Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.6 Conclusions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Non-Isothermal Flow of Molten Glass: MathematicalChallenges and Industrial Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173Angiolo Farina, Antonio Fasano, and Andro Mikelic1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1732 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

2.1 Definitions and Basic Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1762.2 Fluids Physical Properties and Constitutive Equations . . . . . . . . . . 1772.3 The General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1812.4 Scaling and Dimensionless Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 183

3 Study of the Stationary Non-Isothermal Molten Glass Flow in a Die . . . . 1873.1 Existence and Uniqueness Result for the Stationary Problem.. . 1893.2 Oberbeck–Boussinesq Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

4 Modelling the Viscous Jet at the Exit of the Die . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.1 Definition of L and Jet’s Profile at the End of Stage (c) . . . . . . . . . 201

5 Terminal Phase of the Fiber Drawing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2075.1 Derivation of the Model of Matovich–Pearson

for the Thermal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.2 Solvability of the Boundary Value Problems for

the Stationary Effective Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Contributors

Angiolo Farina Universita degli Studi di Firenze, Dipartimento di Matematica“Ulisse Dini”, Viale Morgagni 67/A, I-50134 Firenze, Italy [email protected]

Antonio Fasano Universita degli Studi di Firenze, Dipartimento di Matematica“Ulisse Dini”, Viale Morgagni 67/A, I-50134 Firenze, Italy [email protected]

Martin Frank University of Kaiserslautern, Erwin-Schrodinger-Strasse,67663 Kaiserslautern, Germany [email protected]

J.A.W.M. Groot Department of Mathematics and Computer Science, EindhovenUniversity of Technology, PO Box 513, 5600 MB Eindhoven, The [email protected]

Axel Klar University of Kaiserslautern, Erwin-Schrodinger-Strasse, 67663Kaiserslautern, Germany and Fraunhofer ITWM, Fraunhofer Platz 1, 67663Kaiserslautern, Germany [email protected]

K.Y. Laevsky [email protected]

Robert M.M. Mattheij Department of Mathematics and Computer Science,Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven,The Netherlands [email protected]

Andro Mikelic Universite de Lyon, Lyon, F-69003, FRANCE; Universite Lyon 1,Institut Camille Jordan, UMR 5208 CNRS, Bat. Braconnier, 43, Bd du onzenovembre 1918 69622 Villeurbanne Cedex, [email protected]

Norbert Siedow Fraunhofer-Institut fur Techno- und WirtschaftsmathematikKaiserslautern, Germany [email protected]

xi

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

Mathematical Modelling of Glass FormingProcesses

J.A.W.M. Groot, Robert M.M. Mattheij, and K.Y. Laevsky

Abstract An important process in glass manufacture is the forming of the product.The forming process takes place at high rate, involves extreme temperatures andis characterised by large deformations. The process can be modelled as a coupledthermodynamical/mechanical problem including the interaction between glass, airand equipment. In this paper a general mathematical model for glass forming isderived, which is specified for different forming processes, in particular pressing andblowing. The model should be able to correctly represent the flow of the glass andthe energy exchange during the process. Various modelling aspects are discussedfor each process, while several key issues, such as the motion of the plunger andthe evolution of the glass-air interfaces, are examined thoroughly. Finally, someexamples of process simulations for existing simulation tools are provided.

Nomenclature

Br Brinkman numberFr Froude numberNu Nusselt numberPe Peclet numberRe Reynolds numberFe [N] External force on plunger

J.A.W.M. Groot (�)Department of Mathematics and Computer Science, Eindhoven University of Technology,PO Box 513, 5600 MB Eindhoven, The Netherlandse-mail: [email protected]

R.M.M. MattheijDepartment of Mathematics and Computer Science, Eindhoven University of Technology,PO Box 513, 5600 MB Eindhoven, The Netherlandse-mail: [email protected]

K.Y. Laevskye-mail: [email protected]

A. Fasano (ed.), Mathematical Models in the Manufacturing of Glass,Lecture Notes in Mathematics 2010, DOI 10.1007/978-3-642-15967-1 1,c© Springer-Verlag Berlin Heidelberg 2011

1

[email protected]

[email protected]

[email protected]

2 J.A.W.M. Groot et al.

Fg [N] Force of glass on plungerL [m] Characteristic lengthT [K] TemperatureTg [K] Glass temperatureTm [K] Mould temperatureV [m s−1] Characteristic flow velocityVp [m s−1] Plunger velocitycp [J kg−1 K−1] Specific heatg [m s−2] Gravitational accelerationp [Pa] Pressurerp [m] Radius of plungert [s] Timezp [m] Vertical plunger positionα [W m−2 K−1] Heat transfer coefficientβ [N m−3 s] Friction coefficientλ [W m−1 K−1] Effective conductivityμ [kg m−1 s−1] Dynamic viscosityρ [kg m−3] Densityn Unit normalt Unit tangentu [m s−1] Flow velocityuw [m s−1] Wall velocityx [m] PositionI Unit tensorE [s−1] Strain rate tensorT [Pa] Stress tensor

1 Introduction

Nowadays glass has a wide range of uses. By nature glass has some specialcharacteristics, including shock-resistance, soundproofing, transparency and reflect-ing properties, which makes it particularly suitable for a wide range of applications,such as windows, television screens, bottles, drinking glasses, lenses, fibre opticcables, sound barriers and many other applications. It is therefore not surprising thatglass manufacture is an extensive branch of industry.

1.1 Glass Forming

The manufacture of a glass product can be subdivided into several processes. Belowa common series of glass manufacturing processes are described in the order of theirapplication.

Mathematical Modelling of Glass Forming Processes 3

1.1.1 Melting

In industry the vast majority of glass products is manufactured by melting rawmaterials and recycled glass in tank furnaces at an elevated temperature [33, 61].Examples of raw materials include silica, boric oxide, phosphoric oxide, soda andlead oxide. The temperature of the molten glass in the furnace usually ranges from1,200 to 1,600 ◦C. A slow formation of the liquid is required to avoid bubbleforming [61].

1.1.2 Forming

The glass melt is cut into uniform gobs, which are gathered in a forming machine. Inthe forming machine the molten gobs are successively forced into the desired shape.The forming technique used depends on the type of product. Forming techniquesinclude pressing, blowing and combinations of both, and are discussed further on.During the formation the glass slightly cools down to below 1,200 ◦C. After theformation the glass objects are rapidly cooled down as to take a solid form.

1.1.3 Annealing

Development of stresses during the formation of glass may lead to static fatigueof the product, or even to dimensional changes due to relaxation or optical refrac-tion. The process of reduction and removal of stresses due to relaxation is calledannealing [61].

In an annealing process the glass objects are positioned in a so-called Anneal-ing Lehr, where they are reheated to a uniform temperature region, and then againgradually cooled down. The rate of cooling is determined by the allowable finalpermanent stresses and property variations throughout the glass [61].

1.1.4 Surface Treatment

An exterior surface treatment is applied to reduce surface defects. Flaws in the glasssurface are removed by chemical etching or polishing. Subsequent flaw formationmay be prevented by applying a lubricating coating to the glass surface. Crackgrowth is prevented by chemical tempering (ion exchange strengthening), thermaltempering or formation of a compressive coating. For more information about flawremoval and strengthening of the glass surface the reader is referred to [61].

The process step of interest in this paper is the actual glass formation. Belowthree widely used forming techniques are discussed. See [22, 61, 73] for furtherdetails on glass forming.


Fig. 1 Pressing machine

1.1.5 Press Process

Commercial glass pressing is a continuous process, where relatively flat products(e.g. lenses, TV screens) are manufactured by pressing a gob that comes directlyfrom the melt [61]. This process is usually referred to as the press process. In thispaper also the term direct press is used to distinguish the process from the parisonpress, which is explained further on.

The direct press is depicted in Fig. 1. Initially, the glass gob is positioned in thecentre of a mould. Over the mould a plunger is situated. In order to enclose thespace between the mould and the plunger, so that the glass cannot flow out duringthe process, a ring is positioned on top of the mould. During the direct press theplunger moves down through the ring so that the gob is pressed into the desiredshape.

1.1.6 Press-Blow Process

A hollow glass object is formed by inflating a glass preform with pressurised air.This is called the final blow. The preform is also called parison.

In a press-blow process first a preform is constructed by a press stage, to avoidconfusion with the direct press here referred to as the parison press. Figure 2 shows aschematic drawing of a press-blow process. In the press stage a glass gob is droppeddown into a mould, called the blank mould, and then pressed from below by aplunger (see Fig. 2a). Once the gob is inside the blank mould, the baffle (upper partof the mould) closes and the plunger moves gradually up. When the plunger is at itshighest position, the ring closes itself around the plunger, so that the mould-plungerconstruction is closed from below. Finally, when the plunger is lowered, the ringis decoupled from the blank mould and the glass preform is carried by means of arobotic arm to another mould for the blow stage (final blow). In the blow stage thepreform is usually first left to sag due to gravity for a short period. Then pressurised


gob

a

mould

plunger

baffle

parison press

preform

b

final blow

Fig. 2 Schematic drawing of a press-blow process

ring

glass

a

mould

settle blow

preform

b

counter blow

Fig. 3 Schematic drawing of the first blow stage of a blow–blow process

air is blown inside to force the glass in a mould shape (see Fig. 2b). It is importantto know the right shape of the preform beforehand for an appropriate distribution ofglass over the mould wall.

1.1.7 Blow–Blow Process

A blow-blow process is based on the same principle as a press-blow process, buthere the preform is produced by a blow stage. By means of a blow stage a hol-low preform can be formed, which is required for the production of narrow-mouthcontainers. In practice the glass gob is blown twice to create the preform. Figure 3


shows a schematic drawing of the first blow stage of a blow-blow process. First inthe settle blow the glass gob is blown from above to form the neck of the container(see Fig. 3a), then in the counter blow from below to form the preform (see Fig. 3b).After the counter blow the preform is carried to the mould for the second blow stage,the final blow. The final blow is basically the same as in the press-blow process, butthe preform is typically different.

The temperature of the material of the forming machine is typically around500 ◦C. Because of the high temperature of the gob, the surface temperature of thematerial will increase. To keep the temperature of the material within acceptablebounds the mould and plunger are thermally stabilised by means of water-cooledchannels.

1.2 Process Simulation

In the recent past glass forming techniques were still based on empirical knowledgeand hand on experience. It was difficult to gain a clear insight into formingprocesses. Experiments with glass forming machines were in general rather ex-pensive and time consuming, whereas the majority had to be performed in closedconstructions under complicated circumstances, such as high temperatures.

Over the last few decades numerical process simulation models have becomeincreasingly important in understanding, controlling and optimising the process [33,71]. Since measurements are often complicated, simulation models may offer a goodalternative or can be used for comparison of results.

The growing interest of glass industry for computer simulation models has been amotivation for a fair number of publications on this subject. The earliest papers dealtwith computer simulation of glass container blowing [12, 52]. The first publicationin which both stages of the press-blow process were modelled was presented in [74],although the model did not include aspects such as the drop of the gob into the blankmould and the transfer of the preform to the mould for the final blow [28].

Shortly afterward also the first PhD theses on the modelling of glass formingprocesses appeared. A numerical model for blowing was published in [54]. Notmuch later a simplified mathematical model for pressing was presented in [27].Both models assumed axial symmetry of the forming process.

Subsequent publications reported the development of more advanced models.A fully three-dimensional model for pressing TV panels was addressed in [35].A complete model for the three-dimensional simulation of TV panel forming andconditioning, including gob forming, pressing cooling and annealing was devel-oped by TNO [7]. A simulation model for the complete press-blow process, fromgob forming until the final blow, was presented in [28]. The model also includedeffects of viscoelasticity and surface tension. Finally, an extensive work on themathematical modelling of glass manufacturing was published in the book entitled‘Mathematical Simulation in Glass Technology’ [33].


More recent papers focus on optimisation of glass forming processes [25]. Forinstance a numerical optimisation method to find the optimum tool geometry in amodel for glass pressing was introduced in [42]. Optimisation methods have alsobeen developed to estimate heat transfer coefficients [9] or the initial temperaturedistribution in glass forming simulation models [41]. An engineering approach tofind the optimum preform shape for glass blowing was addressed in [39]. Theiralgorithm attempts to optimise the geometry of the blank mould in the blow–blowprocess, given the mould shape at the end of the second blow stage. More recently,an optimisation algorithm for predictive control over a class of rheological formingprocesses was presented in [4].

By far most papers on modelling glass forming processes use FEM (FiniteElement Methods) for the numerical simulation. Exceptions are [75], in which aFinite Volume Method is used, and [14], in which Boundary Element Methods areused. FEM in models for forming processes usually go together with re-meshingtechniques, sometimes combined with an Eulerian formulation or a Lagrangianmethod. In [7] remeshing was completely avoided by using an arbitrary Euler–Lagrangian approach to compute the mesh for the changing computational domaindue to the motion of the plunger and by using a Pseudo-Concentration Method totrack the glass-air interfaces. In [20] a Level Set Method was used to track the glass-air interfaces in a FEM based model for glass blowing.

In this paper a general mathematical model for the aforementioned formingprocesses is derived. Subsequently, the model is specified for different forming pro-cesses, thereby discussing diverse modelling aspects. The paper focusses on themost relevant aspects of the forming process, rather than supplying a model that isas complete as possible. As discussed previously, a considerable amount of work onmodelling glass forming processes has been done in the recent past. For complete-ness, comparison or reviewing various references to different simulation modelsare included. Finally, some examples of process simulations for existing simulationtools are provided. Most of the work in this paper has been done by CASA1 (Centrefor Analysis, Scientific computing and Applications).

1.3 Outline

The paper is structured as follows. First in Sect. 2 the physical aspects of glass form-ing are described and a general mathematical model for glass forming is derived.Then the mathematical model is specified for the parison press in Sect. 3, for boththe counter blow and the final blow in Sect. 4 and for the direct press in Sect. 5. Ineach of these sections also a computer simulation model for the forming process inquestion is described and some results are presented.

1 A research group in the Department of Mathematics and Computer Science of Eindhoven Uni-versity of Technology.


2 Mathematical Model

This section presents a mathematical model for glass forming in general. Thesection has the following structure. First Sect. 2.1 defines the physical domains intowhich a glass forming machine can be subdivided and in which the boundary valueproblems for glass forming are defined. Then Sect. 2.2 describes the physical aspectsof glass forming, sets up the resulting balance laws and derives the correspondingboundary value problems. Sect. 2.2.1 is concerned with the thermodynamics andSect. 2.2.2 with the mechanics.

2.1 Geometry, Problem Domains and Boundaries

In order to formulate a mathematical model for glass forming, the forming machineis subdivided into subdomains. First the space enclosed by the equipment is subdi-vided into a glass domain and an air domain. Then separate domains for componentsof the equipment (e.g. mould, plunger) are considered. Subdomains of the equip-ment are of interest when modelling the heat exchange between glass, air andequipment. On the other hand, for less advanced heat transfer modelling it can be as-sumed that the equipment has constant temperature, so that the mathematical modelcan be restricted to the glass and air domain. In this case the equipment domains aredisregarded.

Figure 4 illustrates the domain decomposition of 2D axi-symmetrical formingmachines for the direct press, parison press, counter blow and final blow, respec-tively. The entire open domain of the forming machine, consisting of equipment,glass melt and air, is denoted by Σ . The ‘flow’ domain Ω consists of the open glassdomain Ωg, the open air domain Ωa and the glass-air interface(s) Γi. For blowingΩ := Ωg∪Ωa∪Γi is fixed, while for pressing Ω changes in time. Furthermore, Ωg

and Ωa are variable in time for any forming process. The boundaries of the domainsare:

Γb : Baffle boundary Γi : Glass-air interface Γm : Mould boundaryΓo : Outer boundary Γp : Plunger boundary Γr : Ring boundaryΓs : Symmetry axis

Note that not necessarily all boundaries exist for each forming machine. Domain Σis enclosed by Γo∪Γs. Domain Ω is enclosed by ∂Ω := Ω ∩

(Γe∪Γo∪Γs

), with

Γe := Γb∪Γm∪Γp∪Γr (equipment boundary). (1)

In addition, define ∂Ωa := Ω a∩∂Ω and ∂Ωg := Ω g∩∂Ω . Finally, the boundariesfor the glass domain and the air domain are distinguished:

Γa,e = ∂Ωa∩Γe, Γg,e = ∂Ωg∩Γe, Γa,o = ∂Ωa∩Γo, Γg,o = ∂Ωg∩Γo.


Γi

r

z mould

a

plunger

ring

glass airΩg Ωa

tn Γp

Γr

Γm

Γo

Γs

Γo Γo

direct press

Ω

ΩΓ

Γ

Γ Γ

Γ

Γ

Γ

Γ

plunger

mould

baffle

gglass

a air

s

b

o

om

i

p

g,o

t

n

z

r

b

parison press

c

Ω

Ω

Γ

Γ

Γ

Γ

Γ

Γ

ΓΓ

a,o

g,o

s

g

glass

a

air

m

baffleb

o

o

n

t

i

Γi

mould

r

z

counter blow

d

Ω

ΩΩ

Γ ΓΓ

Γ

Γ

Γ

Γ

n

t

a

air

a

air

gglass

om

i

i

s

o,a o,g

mould

r

z

final blow

Fig. 4 2D axi-symmetrical problem domains of glass forming models

Remark 2.1. In pressing sometimes only the forces acting on the glass are of inter-est. If the density and viscosity of air are negligible compared to the density andthe viscosity of glass, the mathematical model may be restricted to the glass domainΩg. In this case the interface Γi is replaced by Γo.

2.2 Balance Laws

In this section the physical aspects of glass forming are described and the resultingbalance laws are formulated in the form of boundary value problems. Subsequentlya dimensional analysis of the balance laws is applied. Section 2.2.1 is concernedwith the thermodynamics and Sect. 2.2.2 with the mechanics. In subsequent sectionsthese balance laws are specified for different forming processes.


2.2.1 Thermodynamics

Glass forming involves high temperatures within a typical range of 800–1,400 ◦C.Temperature variations within this range may cause significant changes in the me-chanical properties of the glass.

• The range of viscosity for varying temperature is relatively large for glass: itamounts from 10 Pa s at the melting temperature (about 1,500 ◦C) to 1020 Pa sat room temperature. The viscosity increases rapidly as a glass melt is cooled, sothat the glass will retain its shape after the forming process. Typical values forthe viscosity in glass forming processes lie between 102 and 105Pa s [61, 73].The temperature dependence for the viscosity of glass within the forming tem-perature range is given by the VFT-relation, due to Vogel, Fulcher and Tamman[61, 73]:

μ(T )\[Pas] = exp(−A + B/(T−TL)

). (2)

Quantities A [−], B [ ◦C] and TL [ ◦C] represent the Lakatos coefficients, whichdepend on the composition of the glass melt. Figure 5 shows how strongly theviscosity depends on temperature for soda-lime-silica glass with Lakatos coeffi-cients A = 3.551, B = 8575 ◦C and TL = 259 ◦C [49].

• The following density-temperature relation can be deduced

ρ(T ) = ρ0

(1−αV

(T −Tref

)), (3)

where

– αV [ ◦C−1] is the volumetric thermal expansion coefficient– Tref [ ◦C] is a reference temperature– ρ0 [kg m−3] is the density at the reference temperature

The volumetric thermal expansion coefficient is often assumed constant; formolten glass it is typically ranged from 5·10−5 to 8·10−5 ◦C−1. The density

Fig. 5 VFT-relation

800 900 1000 1100 12004

6

8

10

12

14

Temperature (oC)

log

visc

osity


of molten glass is of the order 2,300–2,500 kg m−3 and is 5–8% lower than atroom temperature. Thus it is quite reasonable to assume incompressibility formolten glass [73].

Clearly, the mechanics of glass forming is related to the heat transfer in the glass.The heat transfer in glass, air and equipment is described by the heat equation forincompressible continua:

ρcp(∂T

∂ t+ u·∇T

)︸︷︷︸

advection

=− ∇·q︸︷︷︸conductionradiation

+ 2μ(E:∇⊗u

)︸︷︷︸

dissipation

, in◦Σ , (4)

where the temperature distribution T [K] is unknown. Here◦Σ denotes domain

Σ minus the interfaces between the continua. On the interfaces between differentcontinua a steady state temperature transition is imposed,

[[λ n·∇T

]]= 0. (5)

The heat flux q [W m−2] is the result of the contribution of both conduction andradiation,

q = −λ · ∇T, (6)

where λ is the effective conductivity [W m−1K−1], given by

λ = λc + λr. (7)

Here λc is the thermal conductivity and λr is the radiative conductivity. The ther-mal conductivity measures 1.0 W m−1K−1 at room temperature for soda-lime glassand increases with approximately 0.1 W m−1K−1 per 100 K. In this paper it isassumed to be constant. The calculation of the radiative conductivity λr is oftena complicated process. However, for non-transparent glasses it can be simplified bythe Rosseland approximation [16, 51, 73]

λr(T ) =163

n2σT 3

α, (8)

where

• σ is the Stefan Boltzmann radiation constant [W m−2K−4]• n is the average refractive index [−]• α is the absorption coefficient [m−1]

The radiative conductivity λr in the sense of (8) is called the Rosseland param-eter. Relation (8) cannot be applied for highly transparent glasses, since in thiscase not all radiation is absorbed by the glass melt [73]. A more simple approachis to omit the radiative term, which is often reasonable for clear glass [34, 37].


For more information on heat transfer in glass by radiation the reader is referredto [33, 37, 38, 43, 73]. The specific heat cp is in general slightly temperature de-pendent. In [3] an increase in the specific heat of less than 10% in a temperaturerange of 900–1,300 K for soda-lime-silica glass is reported. For various specificheat capacity models the reader is referred to [3, 60].

In order to analyse the energy exchange problem quantitatively the heat equationis written in dimensionless form. Define a typical: velocity V , length scale L, vis-cosity μ , specific heat cp, effective conductivity λ , glass temperature Tg and mouldtemperature Tm. Then introduce the dimensionless variables

t∗ :=VtL

, x∗ :=xL

, u∗ :=uV

, T ∗ :=T −Tm

Tg−Tm,

c∗p :=cp

cp, λ ∗ :=

λλ

, μ∗ =μμ

, (9)

For convenience all dimensionless variables, spaces and operators with respectto the dimensionless variables are denoted with superscript ∗. Substitution of thedimensionless variables (9) in the heat equation and splitting up the effectiveconductivity into (7) lead to the dimensionless form,

Pe(∂T∗

∂ t∗+ u∗ ·∇∗T∗)= ∇∗ ·(λ ∗∇∗T∗

)+ 2μ∗Br

(E∗:∇∗ ⊗u∗

), in

◦Σ ∗, (10)

where

Pe =ρ cpVL

λ, (11)

Br =μV 2

λ(Tg−Tm

) (12)

are the Peclet number and the Brinkman number, respectively. The Peclet numberrepresents the ratio of the advection rate to the diffusion rate. On the other hand, theBrinkman number relates the dissipation rate to the conduction rate. The dimension-less numbers are useful for assessing the order of magnitude of the different termsin (10).

The energy BCs follow from symmetry and heat exchange with the surroundings:(λ ∇T

)·n = 0, on Γs(λ ∇T

)·n = α(T −T∞

), on Γo,

where T∞ is the temperature of the surroundings. The heat transfer coefficient α[W m−2K−1] can differ for separate equipment domains, such as the mould and theplunger. Let α be a typical value for the heat transfer coefficient, then the Nusseltnumber is defined by

Nu =αL

λ. (13)


The dimensionless boundary conditions become

(λ ∗∇T ∗

)·n = 0, on Γs(λ ∗∇T ∗

)·n = Nu α∗(T∗ −T∗∞

), on Γo,

(14)

with α∗ = α/α . On the other hand, if the heat transfer in the equipment domain isnot of interest, the boundary condition on Γo in (14) is imposed on Γe, with T∞ thesurface temperature of the mould.

2.2.2 Mechanics

A balance law for the mechanics of the glass melt is formulated. In general, glasscan be treated as an isotropic viscoelastic Maxwell material [3,53,54,61], that is thestrain rate tensor can be split up into an elastic and a viscous part:

E = Ee + Ev, (15)

where the elastic and viscous strain rate tensors, Ee and Ev respectively, are givenby [54]

Ee =1−2ν

E

(αV

∂T∂ t

+13

tr(T))

+1 + ν

Edev(T) (16)

Ev =1

2μdev(T). (17)

Here αV [ ◦C−1] is the volumetric thermal expansion coefficient, E [Pa] is theYoung’s modulus, ν [-] is the Poisson’s ratio and T [Pa s−1] denotes the stress ratetensor. However, at relatively low viscosities the relation between shear stress andviscosity becomes approximately linear. For example, for soda lime silica glassesthe viscosity as a function of the strain rate and the temperature becomes [6, 63]

μ(E,T ) =μ0(T )

1 + 3.5 ·10−6E μ0.760 (T )

, (18)

where μ0 is the Newtonian viscosity. Consequently, the motion of glass is dominatedby viscous flow and the influence of elastic effects can be neglected [3, 53, 61].Moreover, as verified in Sect. 2.2.1, glass is practically incompressible in theforming temperature range, from which it follows that

tr(E) = 0. (19)

It can be concluded that glass in the forming temperature range behaves as an in-compressible Newtonian fluid [3, 53, 73], i.e.

T =−pI+ 2μE, (20)


with

E =12

(∇⊗u+(∇⊗u)T ). (21)

For simplicity the pressurised air is considered as an incompressible, viscousfluid with uniform viscosity. Thus the motion of glass melt and pressurised air isdescribed by the Navier–Stokes equations for incompressible fluids. These involvethe momentum equations,

ρ(∂u

∂ t+ u·(∇⊗u

))︸︷︷︸

inertia

= −∇p︸︷︷︸pressure

+ ρg︸︷︷︸gravity

+2∇·(μE)

︸︷︷︸viscosity

, in Ω \Γi, (22)

and the continuity equation, which follows directly from (19),

∇·u = 0, inΩ \Γi. (23)

The unknowns are the flow velocity u [m s−1] and the pressure p [Pa].Flow problem (22)–(23) is coupled to the energy problem (10) in two ways:

firstly the viscosity is temperature dependent and secondly the heat transfer is partlydescribed by convection and diffusion.

In order to apply a quantitative analysis the Navier–Stokes equations are writtenin dimensionless form. First a dimensionless pressure is defined by

p∗ :=LpμV

. (24)

The gravity force can be written as−ρgez, where ez is the unit vector in z-direction.Substitution of the dimensionless variables (9) and (24) into the Navier–Stokesequations (22)–(23) and division by the order of magnitude of the diffusionterm, μV

L2 , lead to the dimensionless Navier–Stokes equations

Re

(∂u∗

∂ t∗+ u∗ ·(∇∗ ⊗u∗

))= −∇∗p∗ − Re

Frez + 2∇∗·(μ∗E∗

), in Ω ∗ \Γ ∗i ,

∇∗ ·u∗ = 0, in Ω ∗ \Γ ∗i , (25)

where

Re =ρVL

μ, (26)

Fr =V 2

gL(27)

are the Reynolds number and the Froude number, respectively. The Reynoldsnumber measures the ratio of inertial forces to viscous forces, while the froudenumber measures the ratio of inertial forces to gravitational forces.


The jump conditions between two immiscible viscous fluids are the continuity ofthe flow velocity,

[[u]] = 0, (28)

as well as the continuity of its tangential derivative,

[[(∇⊗u

)· t]] = 0, (29)

and a dynamic jump condition stating the balance of stress across the fluid interface[31, 45],

[[Tn]] = 0. (30)

Remark 2.2. If the influence of surface tension is taken into account, the dynamicjump condition is

[[Tn]] =−γκn, (31)

where n points in the air domain. Here γ[N m−1] denotes the surface tension andκ [m−1] denotes the curvature. The influence of surface tension highly depends onthe glass composition [61]. In this paper the influence of surface tension is simplydisregarded, although for some glasses a dimensional analysis can point out that thesurface tension term is not negligible.

Boundary conditions for the flow problem can be determined as follows. On Γs

symmetry conditions are imposed. On Γg,e a suitable slip condition for the glassshould be adopted. The air can escape through small cavities in (part of) the mouldwall. This aspect can be modelled by allowing air to flow freely through the mouldwall. Thus free-stress conditions are proposed on this part of the equipment bound-

ary, which is referred to as Γ (1)a,e . A free-slip condition is prescribed on the remaining

part, Γ (2)a,e . On Γo the normal stress should be equal to the external pressure.

A commonly used boundary condition to describe fluid flow at an impenetrablewall [13, 19, 29, 50] is Navier’s slip condition:

(Tn+ β (u−uw)

)· t = 0, (32)

where β is the friction coefficient [N m−3s] and uw is the velocity of the wall[m s−1]. A similar condition can be obtained by using the Tresca model [16].Introduce a dimensionless friction coefficient,

β ∗ :=Lβμ

, (33)

then the dimensionless Navier’s slip condition reads:

(T∗n+ β ∗(u∗ −u∗w)

)· t = 0. (34)

The order of magnitude of the friction coefficient depends on many parameters, suchas the type of glass, temperature, pressure or presence of a lubricant [15, 18, 50].


Remark 2.3. For β ∗ → ∞ Navier’s slip condition together with the boundarycondition for an impenetrable wall, (u∗ −u∗w)·n = 0, can be reformulated as a noslip condition:

u∗ = u∗w. (35)

In summary, the boundary conditions for the flow problem can be formulated as:

u∗·n = 0, T∗n· t = 0, on Γs,

(u∗ −u∗w)·n = 0,(T∗n+ β ∗(u∗ −u∗w)

)· t = 0, on Γg,e,

T∗n·n = 0, T∗n· t = 0, on Γ (1)a,e ,

(u∗ −u∗w)·n = 0, T∗n· t = 0, on Γ (2)a,e ,

T∗n·n = p0, T∗n· t = 0, on Γo,(36)

where p0 is the external pressure.

3 Parison Press Model

This section presents a mathematical model for the parison press. Section 3.1 speci-fies the mathematical model described in Sect. 2 for the parison press. By restrictingthe analysis to a narrow channel between the plunger and the mould, an analyticalapproximation of the flow can be derived. Section 3.2 explains this concept, knownas the slender geometry approximation [50]. Section 3.3 describes the motion ofthe plunger by an ordinary differential equation. It appears that the motion of theplunger is coupled to the flow, which considerably complicates the parison pressmodel. Section 3.4 presents a numerical simulation model for the parison press.The motion of the free boundaries is emphasised. Finally, Sect. 3.5 shows some ex-amples of parison press simulations. The simulation tool used for these results ispresented in [34].

3.1 Mathematical Model

In Sect. 2.2 the balance laws for glass forming were formulated. In this section thesebalance laws are further specified for the parison press process. Typical values forthe parison press are:

Glass density : ρg = 2.5 ·103 kg m−3

Glass viscosity : μg = 104 kg m−1s−1

Gravitational acceleration : g = 9.8 m s−2

Flow velocity : V = 10−1 m s−1

Length scale of the parison : L = 10−2 mGlass temperature : Tg = 1,000 ◦CMould temperature : Tm = 500 ◦CSpecific heat of glass : cp = 1.5 ·103 J kg−1K−1

Effective conductivity of glass : λ = 5 W m−1K−1,


As a result the following dimensionless numbers are found:

Peglass≈ 7.6 ·102, Brglass≈ 4.0 ·10−3, Reglass≈ 2.5 ·10−4, Fr≈ 1.0 ·10−1. (37)

Apparently, the heat transport in glass is dominated by thermal advection. As a resultthe heat equation (10) simplifies to

dT ∗

dt∗=

∂T ∗

∂ t∗+ u∗ ·∇∗T ∗ = 0, inΩ ∗g , (38)

Thus the temperature remains constant along streamlines. Consequently, if the initialtemperature distribution in the glass is (approximately) uniform, the glass viscositycan be considered constant. From the small Reynolds number for glass it can beconcluded that the inertia forces can be neglected with respect to the viscous forces.Furthermore,

Reglass

Fr≈ 2.5 ·10−3,

which means that also the contribution of gravitational forces is rather small. Inconclusion, the glass flow can be described by the Stokes flow equations:

∇∗ ·T∗ = 0, ∇∗ ·u∗ = 0, inΩ ∗g , (39)

where T∗ is the dimensionless stress tensor, which satisfies (20) in terms of the di-mensionless variables. The air domain is ignored for the following reasons. Firstly,the force of the plunger acts directly on the glass domain. Secondly, the density andviscosity of air are negligible compared to those of glass, so that air hardly formsany obstacle for the glass flow. Thirdly, the simplification of the heat equation toconvection equation (38) gives reason to restrict the energy exchange problem tothe glass domain, or all together ignore the heat transfer in case of an uniform ini-tial glass temperature. Subsequently, the glass-air interfaces are treated as an outerboundary of the flow domain, Γo, on which free-stress conditions are imposed. Notethat although the problem can be considered as a free-boundary problem, the geom-etry is constrained by the mould and the plunger.

Remark 3.1. Close to the equipment wall extreme temperature variations occur overa small length scale. Therefore, the conductive heat flux close to the equipment wallshould, strictly speaking, not be disregarded [50,64]. Moreover, the viscosity in thisregion may increase by several orders, so that the fluid friction may not be negligibleand the influence of heat generation by dissipation should be taken into account foroptimal accuracy [50]. Although the reader should take notice of these boundarylayer effects, in this paper it is simply assumed that they are small enough to beignored. For more advanced heat modelling during the parison press the reader isreferred to [23, 38, 64].


The boundary conditions for the flow problem (2.2.2) can be specified for theparison press:

u∗·n = 0, T∗n· t = 0, on Γs,

(u∗ −Vpez)·n = 0,(T∗n+ β ∗p(u∗ −Vpez)

)· t = 0, on Γg,p,

u∗·n = 0,(T∗n+ β ∗mu∗

)· t = 0, on Γg,m,

T∗n·n = 0, T∗n· t = 0, on Γo.(40)

In the remainder of this section it is assumed that the glass gob initially has anuniform temperature distribution, so that with (38) it follows that μ = μ is constant.

3.2 Slender-Geometry Approximation

In the model for glass pressing the analysis can be restricted to the flow in a narrowchannel between plunger and mould. In other words, the analysis focusses on theflow in the slender geometry around the plunger, while the flow between the plungertop and the baffle (Fig. 4b) is considered practically stagnant. For a more completeanalysis the reader is referred to [50]. In the slender-geometry approximation ofthe flow two typical length scales � and L are considered, with � L, where � is thelength scale for the width of the channel and L is the length scale for the length of thechannel. Thus variations in r-direction are scaled by � and variations in z-directionare scaled by L. By means of this scaling the following dimensionless variables canbe defined:

t∗ :=VtL

, ε :=�

L, r∗ :=

rεL

, z∗ :=zL

,

u∗r :=ur

εV, u∗z :=

uz

V, p∗ :=

ε2LpμV

. (41)

In the remainder of this section all variables, spaces and operators are dimension-less and the superscript ∗ is ignored. Substitution of the dimensionless variables(41) into the Navier–Stokes equations (22)–(23) leads to the dimensionless 2D axi-symmetrical Navier–Stokes equations

ε3Re�

(∂ur

∂ t+ur

∂ur

∂ r+uz

∂ur

∂z

)=−∂ p

∂ r+ε2 ∂

∂ r

(1r

∂∂ r

(rur))

+ε4 ∂ 2ur

∂ z2 , inΩg,

εRe�

(∂uz

∂ t+ur

∂uz

∂ r+uz

∂uz

∂z

)=−∂ p

∂ z+

∂∂ r

(1r

∂∂ r

(ruz))

+ε2 ∂ 2uz

∂ z2 +εRe�

Fr, inΩg,

1r

∂∂ r

(rur)+

∂uz

∂ z= 0, inΩg, (42)


where Re� is the Reynolds number with respect to length scale �, i.e.

Re� =ρV�

μ= εReL. (43)

Thus, for small ε , (42) can be simplified to

∂ p∂ r

= O(ε2), inΩg,

∂ p∂ z

=∂∂ r

(1r

∂∂ r

(ruz))

+ O(ε2), inΩg,

1r

∂∂ r

(rur)+

∂uz

∂ z= 0, inΩg. (44)

If the O(ε2) terms are neglected, system of (44) can be recognised as Reynolds’ 2Daxial-symmetrical lubrication flow [19, 50].

The equipment boundary conditions in the slender-geometry approximation canbe simplified accordingly [50]. Define the plunger surface and the mould surface by:

r = rp(z− zp(t)

), r = rm(z), (45)

respectively, where z = zp(t) is the top of the plunger at time t (see Fig. 6). Considerthe plunger position zp ≡ zp(t) at a fixed time t. Then the outward unit normal np

and the counterclockwise unit tangent tp on the plunger wall are given by

np =εr′pez− er√

1 + ε2r′2p, tp =

−ez− εr′per√1 + ε2r′2p

. (46)

Fig. 6 Geometry ofthe plunger

glass

plungermould

rp

rm

zpr

z

p

o

s

m


Analogously, the outward unit normal nm and the counterclockwise unit tangent tm

on the plunger wall on the mould wall are given by

nm =−εr′mez + er√

1 + ε2r′2m, tm =

ez + εr′mer√1 + ε2r′2m

. (47)

By substituting (46)–(47) into Navier’s slip condition (32) and scaling the frictioncoefficient β by a factor ε the following boundary conditions are obtained

βp

((uz−Vp

)+ ε2urr

′p

)=

(1− ε2r′2p

)(∂uz

∂ r+ ε2 ∂ur

∂ z

)−2ε2r′p

(∂uz

∂ z− ∂ur

∂ r

)√

1 + ε2r′2p,

on r = rp(z− zp) (48)

−βm

(uz + ε2urr

′m

)=

(1− ε2r′2m

)(∂uz

∂ r+ ε2 ∂ur

∂ z

)−2ε2r′m

(∂uz

∂ z− ∂ur

∂ r

)√

1 + ε2r′2m,

on r = rm(z), (49)

where Vp is the velocity of the plunger and βp, βm are the friction coefficients cor-responding to the plunger and the mould, respectively. For small ε the Navier’s slipcondition on the plunger and mould surfaces can be written as

βp(uz−Vp

)=

∂uz

∂ r+ O(ε2), on r = rp(z− zp), (50)

−βmuz =∂uz

∂ r+ O(ε2), on r = rm(z). (51)

The componentwise boundary condition for the impenetrable wall is

ur = (uz−Vp)r′p, on r = rp(z− zp), (52)

ur = uzr′m, on r = rm(z). (53)

Note that for βm,p→∞ the error in boundary condition (50)–(51) due to the slender-geometry approximation vanishes.

Following [40,50] the analytical solution to system of (44) with set of boundaryconditions (50)–(53) on the equipment boundary and free-stress conditions on theother boundaries can be obtained. Neglecting the O(ε2) terms system of equations(44) becomes

∂ p∂ r

= 0, inΩg, (54)

∂ p∂ z

=∂∂ r

(1r

∂∂ r

(ruz))

, inΩg, (55)


1r

∂∂ r

(rur)+

∂uz

∂ z= 0, inΩg. (56)

Firstly, from (54) it follows that p is a function of z only. Secondly, from (55) andthe boundary conditions it follows that

uz(r,z) =14

dpdz

(z)(

r2 +ψm(z)χp(r,z)−ψp(z)χm(r,z)

χm(r,z)− χp(r,z)

)+Vp

χm(r,z)χm(r,z)− χp(r,z)

, (57)

with

χp(r,z) = log(rrp(z−zp)

)− 1βprp(z−zp)

,ψp(z)=rp(z−zp)(

rp(z−zp)− 2βp

),

χm(r,z) = log(rrm(z)

)+

1βmrm(z)

,ψm(z) = rm(z)(

rm(z)+2

βm

). (58)

Thirdly, from (56) and the boundary conditions it follows that

ur(r,z) =1r

(rmrm

′uz(rm,z)+∫ rm

rs

∂uz

∂ z(s,z)ds

)=

1r

ddz

∫ rm

rs uz(s,z)ds. (59)

Finally, to find the pressure gradient, Gauss’ divergence theorem is applied to thecontinuity equation,

0 =∫

Ωg

∇·udΩ =∫

Γg,o

u·ndΓ +∫

Γp

u·ndΓ +∫

Ωg∩Γs

u·ndΓ +∫

Γm

u·ndΓ . (60)

The fluxes through the symmetry axis and the mould wall are zero. Since thecontinuity equation also holds in the plunger domain Σp, it holds that

∫

Γp

u·ndΓ =−∫

Σp∩Γo

u·ndΓ = πVpr2p. (61)

As a result, ∫

Γg,o

u·ndΓ = 2π∫ rm

rp

ruzdr =−πVpr2p. (62)

Substitution of (57) into (62) yields

[r2

(18

dpdz

(r2− ψm(2χp + 1)−ψp(2χm + 1)

χm− χp

)+Vp

χm + 12

χm− χp

)]r=rm

r=rp

=−Vpr2p.

(63)By solving (63) the following solution for the pressure gradient is found:


0.005 0.01 0.015 0.02

0.06

a

0.08

0.1

0.12

0.14

r [m]

z [m

]

mould

plun

ger

plunger and mould boundaries

b

0.04 0.06 0.08 0.1 0.12 0.14 0.166

7

8

9

10

11x 104

z [m]

−dp

/dz

[Pa/

m]

pressure [m]

r-component of flow velocity [m/s] z-component of flow velocity [m/s]

Fig. 7 Analytical solution of slender-geometry approximation for zp = 0.5

dpdz

(z) = 4Vpψp(z)−ψm(z)(

ψp(z)−ψm(z))2− (χp− χm

)(z)

(ωp(z)−ωm(z)

) , (64)

with

ωp(z) = ψp(z)2− 4rp(z− zp)2

β 2p

, ωm(z) = ψm(z)2− 4rm(z)2

β 2m

. (65)

Figure 7–8 plot the solution at zp = 0.05m and zp = 0.01m, respectively, forε = 0.1 and Vp = −0.1ms−1. The (dimensionless) geometries of the plunger andmould have been taken from [50]:

rp(z) =−0.1√

5z, rm(z) = 0.8√

5z.

For simplicity no-slip boundary conditions at the mould and plunger wall have beenused. In [50] it is reported that the results are in good agreement with the numericalsolution obtained by using FEM. For more results the reader is referred to [40, 50].


0.005 0.01 0.015 0.02

0.02

0.04

0.06

0.08

0.1

r [m]

z [m

]

a

mould

plun

ger

plunger and mould boundaries

b

0 0.02 0.04 0.06 0.08 0.1 0.120

1

2

3

4

5

6x 105

z [m]

−dp

/dz

[Pa/

m]

pressure [m]

r-component of flow velocity [m/s] z-component of flow velocity [m/s]

Fig. 8 Analytical solution of slender-geometry approximation for zp = 0.1

3.3 Motion of the Plunger

In the previous section it was assumed that the plunger moves with a constant flowvelocity. However, in practice the plunger is pushed by a piston. This means thatthe flow velocity is the result of an external force applied to the plunger. In thissection it can be seen that the plunger velocity is coupled to the glass flow, whichconsiderably complicates the parison press model.

The press process is initiated by applying an external force Fe to the plunger. Thiscauses the plunger to move with velocity Vp(t). This plunger velocity is the resultof the total force F on the plunger, which is the sum of the external force Fe and theforce of the glass on the plunger Fg:

dVp

dt(t) =

F(t)mp

=Fe + Fg(t)

mp, (66)

where mp is the mass of the plunger [34, 50]. The force Fg is determined by themechanical forces of the glass acting on the plunger and hence depends on the


glass flow. The glass flow at its turn is caused by the plunger motion. Therefore,differential equation (66) is fully coupled to the flow problem. Clearly, if a constantexternal force is applied to the plunger, the plunger moves until the force of the glasson the plunger is equal in magnitude to the external force.

Below the equation for the motion of the plunger (66) is examined morethoroughly. For a complete analysis the reader is referred to [34, 50]. First theforce of the glass on the plunger is analysed. At every time t the force can be fullydescribed by the stress tensor (20) integrated over the plunger surface Γp,

Fg =∫

Γp

(Tn)·ezdΓ . (67)

Using the definition of the plunger surface (45) the surface element dΓ can bewritten as

dΓ = 2π√

1 + r′2p rpdz, on Γp. (68)

By means of the definition of the stress tensor (20) and the expressions for the unitnormal and tangent (46) and the surface element (68), the expression for the forceof the glass on the plunger becomes

Fg = 2π∫ z0

zp

((p−2μ

∂uz

∂ z

)r′p + μ

(∂ur

∂ z+

∂uz

∂ r

))rpdz. (69)

where z = z0(t) is the bottom glass level at time t. Substitution of the dimensionlessvariables (41) into (69) yields

Fg = 2πμVL∫ z∗0

z∗p

((p∗ −2ε2 ∂u∗z

∂ z∗)

r∗p′+(

ε2 ∂ur

∂ z∗+

∂u∗z∂ r∗

))r∗pdz∗. (70)

Considering expression (70) it makes sense to define the dimensionless force of theglass on the plunger as

F∗g :=Fg

2πμVL. (71)

For small ε the dimensionless force can be simplified to

F∗g =:∫ z∗0

z∗p

(p∗r∗p

′+∂u∗z∂ r∗

)r∗pdz∗+ O(ε2). (72)

The flow velocity and pressure in (72) are implicit functionals of the plungervelocity Vp. However, the slender-geometry approximation in Sect. 3.2 can be usedto find approximate solutions for u∗z and p∗. By substituting (57) and (65) into(70)–(71) the dimensionless force can be written as

F∗g (t∗) = V ∗p (t∗)I∗(t∗), (73)


with an assumed error of O(ε2), where the dimensionless function I∗(t∗) onlydepends on the geometry and the friction coefficient β ∗ [34, 50]. Then the equationfor the motion of the plunger (66) in dimensionless form becomes

dV ∗pdt∗

(t∗) =2πμL2

mpV

(V ∗p (t∗)I∗(t∗)+ F∗e

)+ O(ε2), (74)

where the dimensionless external force F∗e is defined in the same way as F∗g in (71).Typical values in the slender-geometry approximation are:

Viscosity : μg = 104 kg m−1s−1


Length scale of the parison : L = 10−1 mMass of the plunger : mp = 1 kg

As a result the dimensionless coefficient in (74) is typically

2πμL2

mpV≈ 104.

This value is rather large, which indicates that (74) is a stiffness equation. Accordingto [34] this phenomenon can also be observed if (74) is solved numerically usingthe Euler forward scheme for time integration. This means that one would have toresort to an implicit time integration scheme in order to solve (74). Unfortunately,a fully implicit scheme is practically impossible, since the plunger velocity is notknown explicitly. See [34] for further details on this stiffness phenomenon.

The stiffness phenomenon previously described indicates that the coupling ofthe equation for the plunger motion (66) to the boundary conditions on the plungerfor the Stokes flow problem is undesirable. Therefore, in the following the plungervelocity Vp(t) is decoupled from the parameter Vp in the boundary conditions forthe flow problem. For a more detailed analysis the reader is referred to [34]. Thefollowing lemma is used:

Lemma 3.2. Let (uυ , pυ) be the family of solutions of Stokes flow problem(39)–(40) with plunger velocity Vp = υ . Define (u, p)υ := (uυ , pυ). Then

(u, p

)k1υ1+k2υ2

=(k1uυ1 + k2uυ2 , p0 + k1(pυ1 − p0)+ k2(pυ2 − p0)

)

for arbitrary constants k1,k2.

The lemma can be proven by direct substitution of the solution(k1uυ1 +k2uυ2 , p0 +

k1(pυ1 − p0)+ k2(pυ2 − p0))

into the Stokes flow problem with plunger velocityVp = k1υ1 + k2υ2 [34]. In the remainder of this section all variables, spaces andoperators are dimensionless and the superscript ∗ is ignored. From Lemma 3.2 itimmediately follows that

(u, p

)υ =

(υu1, p0 + υ(p1− p0)

), (75)


and as a result,Fg,υ = Fg,0 + υ

(Fg,1−Fg,0

). (76)

Substitution of (76) with υ = Vp(t) into the dimensionless form of (66) yields

dVp

dt(t) =

2πμL2

mpV

(Vp(t)

(Fg,1(t)+ Fg,0(t)

)+ Fg,0(t)+ Fe

). (77)

The force Fg,1(t) can be calculated by solving the Stokes flow problem in Ωg(t)with plunger velocity Vp = 1. Thus the equation for the motion of the plunger isdecoupled from the boundary conditions for the Stokes flow problem. Note that theStokes flow problem is still coupled to differential equation (77) as the geometry ofthe glass domain is determined by the plunger velocity.

The time dependency of the plunger velocity seems a bit awkward as the Stokesflow problem is not explicitly time dependent; the flow merely changes in timethrough the changing geometry. It seems more convenient to define the force of theglass on the plunger and the plunger velocity as functions of the plunger positionz := zp:

Fg := Fg(z), Vp := Vp(z). (78)

As a result also the motion of the plunger can be described by the plunger position.By the chain rule of differentiation,

dVp

dt(t) =

dVp

dz(z)Vp(z). (79)

As for the initial condition, let z = 0 and Vp = V0 at t = 0. Then the motion of theplunger involves the following initial value problem:

⎧⎨⎩

12

dV 2p

dz(z) =

2πμL2

mpV

(Vp(z)

(Fg,1(z)+ Fg,0(z)

)+ Fg,0(z)+ Fe

)

Vp(0) = V0

(80)

Since the plunger velocity does not need to be known to determine the glass domainat given plunger position z, an implicit time integration scheme can be used to solve(80), thus overcoming the stiffness problem [34].

3.4 Simulation Model

A simulation tool for the parison press process was designed [34]. The tool is able tocompute and visualise the velocity field, the pressure and also the temperature in theparison during the press stage. The input parameters for the simulations include the2D axi-symmetrical parison geometry, i.e. the description of the mould and plungersurfaces, the initial positions of the plunger and the glass domain, as well as the


Fig. 9 Mesh for presssimulation of a jarparison [34]

physical properties of the glass. The geometry of the initial computational domainis defined by that of the glass gob (see Sect. 2.1). At the beginning of each time stepthe computational domain is discretised by means of FEM. A typical mesh for theinitial glass domain in a press simulation of the parison for a jar is depicted in Fig. 9.The mesh distribution depends on the geometries of the mould and the plunger. Forexample, the mesh in the ring domain requires a relatively small scale in comparisonwith the mould domain and the plunger domain.

The implicit Euler method is used to solve the initial value problem for themotion of the plunger (80):⎧⎪⎪⎨⎪⎪⎩

12

V k+1p

2−V kp

2

zk+1− zk =2πμL2

mpV

(V k+1

p

(Fg,1(zk+1)+ Fg,0(zk+1)

)+ Fg,0(zk+1)+ Fe

)

V 0p = V0

(81)The plunger position and hence the geometry of the glass domain can be updatedusing the approximation

zk+1 = zk + Δ tkVp(zk), tk+1 = tk + Δ tk, (82)

where Δ tk denotes the kth time step.The motion of the free boundaries Γo is described by the ordinary differential

equationdxdt

= u(x), fort ∈ [0, tend). (83)

Let xi, i = 1, . . . ,N be the nodes on a free boundary Γf ⊂Γo and let xki be an approx-

imation of x(tk) (see Fig. 10). Then the new position of the ith node can be obtainedby the explicit scheme

xk+1i = xk

i + Δ t uki , (84)


Fig. 10 Time integration onthe free boundary [34]

�

��

��

��

��

��

�

��

�

��

��

�

Γp Γm

Γf

��

Ωg(tk)

��

Ωg(t k+1)

�

xki

�

xk+1i

vki�

��

��

xk+1i

��

Fig. 11 Clip algorithm

�

��

��

��

��

��

��

��

��

�

��

�

Γp Γm

Γf

��

Ωtk

��

Ωtk+1

�

xki

��xk+1i

with uki := u(xk

i ). A particular question is how to deal with this moving boundary.Depending on the velocity a situation may be encountered where the obtained posi-tion xk+1

i lies outside Ωg (see Fig. 10).In this situation one of the strategies described below may be used. For simplicity

only explicit integration is considered.One approach to deal with the moving boundary is to decrease the time step:

xk+1i = xk

i + αki Δ t uk

i , αki ∈ (0,1], (85)

where xk+1i is the new position of the ith node obtained with the decreased time

step. The kth time step can be defined by Δ t ki := mini αk

i Δ t, such that the nodesare situated inside the glass domain at time t k+1 := t k + Δ t k

i , as depicted in Fig. 10.Unfortunately, this algorithm introduces a variable time step that turns out to be tooirregular in practice and can be excessively small. In order to have consistency inthe topology of the computational domain the time step should be constant [34].

An alternative is illustrated in Fig. 11; the so-called clip algorithm leads the nodesalong a discrete ‘solution curve’ until they reach the boundary, thereby clipping thetrajectories on the boundary. Thus node i, which would originally leave the physicaldomain at time tk+1, is clipped on the boundary by (85), but with αk

i = 1 for a


Fig. 12 Modified clipalgorithm

�

��

��

��

��

��

�

xki

��

vki

��

��

��

��

��

vki

��

��

��

��vki ·n

��

��vk

i · t�

��

��

��

��

non-clipped node; the time step only differs for clipped nodes. Regrettably, also thisalgorithm has an evident drawback: the clipping influences the mass conservationproperty of the glass domain [34].

The clip algorithm can be modified to enforce better mass conservation [34]. Tothis end alter the velocities at the nodes that would otherwise end up outside theglass domain, such that their normal component stays the same, i.e.

uki ·n = uk

i ·n, (86)

while the tangential component uki · t is obtained by rotating the velocity vector, such

that xki ends up on the equipment boundary (see Fig. 12). This can be formulated as

uki := αk

i Rki uk

i , (87)

where αki is the scaling parameter and Rk

i is the 2×2 rotation matrix,

Rki :=

⎛⎝

cosγki −sinγk

i

sinγki cosγk

i

⎞⎠ . (88)

The net outflow for the modified velocity field remains zero and hence the algorithmshould give better mass conservation, i.e.

∫

Γf

u·n dΓ =∫

Γf

u·n dΓ = 0. (89)


An approximate position of a node i that would be outside the glass domain at timetk+1 is obtained using the modified velocity field,

xk+1i = xk

i + Δ t uki . (90)

Instead of (84) also an implicit time integration scheme can be used:

xk+1i = xk

i + Δ t uk+1i . (91)

In order to use (91) it is necessary to compute the flow velocity in Ωg(tk+1).However, even though the plunger position is known in the sense of (82), the freeboundary of Ωg(tk+1) is still unknown. This difficulty can be overcome by employ-ing an algorithm that iterates on xk+1

i . Unfortunately, this straightforward approachrequires each iteration the solution of a Stokes flow problem, which is computation-ally too costly. In [34] a numerical tool is introduced that overcomes the essentialdifficulty of the implicitness of the scheme by using the fact that the flow velocity isautonomous. In [69] this matter is examined into more detail.

3.5 Results

In this section simulations of the pressing of a jar and a bottle parison are shown.This includes the visualisation of the velocity and pressure fields, the tracking of thefree boundaries of the glass flow domain and the computation of the motion of theplunger. For more results of the simulation tool used the reader is referred to [34].Furthermore, see [28] for results of a different parison press model.

First consider the simulation of the pressing of a jar parison. The simulationstarts at the moment the gob of glass has entered the mould and the baffle has closed.Figure 13 visualises the flow velocity field in the glass domain during pressing. Herefull slip of glass at the equipment boundary is assumed, that is β = 0. The plungermoves upward forcing the glass to fill the space between the mould and the plunger.When the glass hits the mould the pressure increases. Figure 14 depicts the flowvelocity of the glass in the neck part ring of the jar in the final stage of the parisonpress. Figure 15 shows the pressure field. It is important to know the pressure of theglass onto the mould during the process, so that a similar pressure can be appliedfrom the outside in order to keep the separate parts of the mould together [34].

Next consider the simulation of the pressing of a bottle parison. The initial posi-tion of the top of the plunger in the simulation is close to the attachment of the ringto the mould. When the glass gob is dropped into the mould it almost reaches thering before the plunger starts moving (see Fig. 16a). Figure 16 visualises the flowvelocity of the glass during pressing. Figure 17 depicts the flow velocity in the bot-tle neck in the final stage of the parison press. Figure 18 shows the pressure field.The results are similar to the jar parison simulation. Again full slip of glass at theequipment boundary is assumed.


Fig. 13 Pressing of a jarparison: flow velocity[m/s] [34]

Fig. 14 Pressing of a jar parison (neckring part): flow velocity [34]

4 Blow Model

This section presents a mathematical model for blowing. Section 4.1 specifies themathematical model described in Sect. 2 for the counter blow and the final blow. Inaddition to the flow of glass, also the flow of air is modelled. A particular question is


Fig. 15 Pressing of a jarparison: pressure [bar] [34]

how to deal with the glass-air interfaces. Section 4.2 discusses several techniques.Subsequently, Sect. 4.3 explains how a variational formulation is used to combinethe physical problems for glass and air as well as the jump conditions on the inter-faces into one statement. Section 4.4 presents a numerical simulation model for theblow–blow process. Finally, Sect. 4.5 shows some examples of process simulations.The simulation tool used for the results is presented in [21].


In Sect. 2.2 the balance laws for glass forming were formulated. In this section thesebalance laws are further specified for glass blowing. Since the final blow stage startswith the preform obtained in either the parison press stage of the counter blow stage,the orders of magnitude of most physical parameters for these forming processesare typically the same. The temperature of the glass melt in the final blow is usuallyslightly lower than in the preceding stage, but this does not lead to a significantdifference in the order of magnitude of the physical parameters. Therefore, the sametypical values for both the counter blow and the final blow are considered:






Fig. 16 Pressing of a bottleparison: flow velocity[m/s] [34]

Fig. 17 Pressing of a bottle parison (neckring part): flow velocity [34]




Fig. 18 Pressing of a bottleparison: pressure [bar] [34]

The main difference from the physical parameters in the parison press is that thetime duration of a blow stage is typically much larger, hence the flow velocity ismuch smaller. The dimensionless numbers corresponding to either blow stage are:

Peglass ≈ 76, Brglass ≈ 4.0 ·10−5, Reglass ≈ 2.5 ·10−5, Fr≈ 1.0 ·10−3. (92)

The Peclet number for glass is moderately large, while the Brinkman number isnegligibly small. Thus the heat transfer is dominated by advection, convection andradiation:

(∂T ∗

∂ t∗+ u∗ ·∇∗T ∗

)= ∇∗ ·(λ ∗∇∗T ∗

), (93)

with

λ ∗ :=λ

ρ cpVL, (94)

rather than its definition (9) in Sect. 2.2.1. The Reynolds number for glass is suffi-ciently small to neglect the inertia terms in the Navier–Stokes equations (25). Theorder of magnitude of the gravity term is given by

Reglass

Fr≈ 2.5 ·10−2,


which is not extremely small. Therefore, the glass flow is described by the Stokesflow equations:

∇∗ ·T∗ = g∗, ∇∗ ·u∗ = 0, (95)

where T∗ is the dimensionless stress tensor, which satisfies (20) in terms of thedimensionless variables, and g∗ is the dimensionless gravity force, given by

g∗ =ReFr

ez. (96)

Subsequently, the air domain is considered. Note that the arguments to ignore theair domain in the parison press model do not apply to the blow model. Firstly, theinflow pressure is applied at the mould entrance and not directly on the glass, so thatin this case the transport phenomena in air are of higher interest for the blow model.Secondly, because the influence of the heat flux cannot be ignored, the energy ex-change between the glass and its surroundings should be taken into account. For hotpressurised air the following typical values are considered2:

Initial air temperature : T0 = 750 ◦C,Specific heat of air : cp = 103 J kg−1K−1,Thermal conductivity of air : λ = 10−1 W m−1K−1.Air density : ρ = 1 kg m−3,Air viscosity : μ = 10−4 kg m−1s−1.

The resulting dimensionless numbers are:

Peair ≈ 102, Brair ≈ 4 ·10−8, Reair ≈ 1. (97)

It can be concluded that the heat transfer in air is described by (93), while the flowof air is described by the full Navier–Stokes equations (25). This means that themodel for the air flow is more complicated model than for the glass flow, whilethe motion of glass is most interesting. Therefore, air is replaced by a fictitiousfluid with the same physical properties as air, but with a much higher viscosity,e.g. μa = 1. Then the Reynolds number of the fictitious fluid Re ≈ 10−4 is smallenough to reasonably neglect the influence of the inertia forces. On the other hand,the viscosity of the fictitious fluid is still much smaller than the viscosity of glass, sothat the pressure drop in the air domain is negligible compared to the pressure dropin the glass domain [2]. Thus the flow of the ficitious fluid can be described by theStokes flow equations (95).

Remark 4.1. If the glass temperature and the pressure in air can be reasonably as-sumed to be uniform, the calculations can be restricted to the glass domain and theglass-air interfaces can be treated as free boundaries with a prescribed pressure. Inthis case it can be recommended to use Boundary Element Methods to solve theflow problem [13, 14].

2 The true orders of magnitude may be slightly different from their rough estimates in the table,but this will not affect the final results.


The boundary conditions for the flow problem are given in Sect. 2.2, but can befurther specified for glass blowing. Free-stress conditions are imposed for air at themould wall. If no lubricate is used, usually a no-slip condition is imposed for glassat the mould wall. For modelling of the slip condition in the presence of a lubricatethe reader is referred to [15,16,36]. As a result the boundary conditions for the flowproblem become:

u∗ ·n = 0, T∗n· t = 0, on Γs,

u∗ ·n = 0, u∗ · t = 0, on Γg,e,

T∗n·n = 0, T∗n· t = 0, on Γa,e,

T∗n·n = p0, T∗n· t = 0, on Γo,

(98)

with

p0 ={

pin on Γa,o

0 on Γg,o,(99)

where pin is the pressure at which air is blown into the mould. Alternatively, onemay prefer to introduce the boundary condition

u∗ ·n = 0, T∗n· t = 0, on Γg,o. (100)

This boundary condition avoids outflow of glass through the mould entrance duringblowing. However, this involves the definition of separate boundaries Γa,o and Γg,o,rather than the single boundary Γo. Moreover, these boundaries can change in time.In the next section it can be seen that this is not always convenient, particularly if asingle fixed mesh is used for the discretisation of the flow domain. Instead Γg,o canbe conceived as the boundary between the glass and the ring, thereby imposing ano-slip condition,

u∗ = 0, on Γg,o. (101)

Note that in this case Γa,o and Γg,o remain fixed.The boundary conditions for the energy exchange problem are defined on the

boundary of the flow domain:(λ ∗∇T ∗

)·n = 0, on Γs∪Γe,(λ ∗∇T ∗

)·n = Nu α∗(T∗ −T∗∞

), on Γo,

(102)

4.2 Glass-Air Interfaces

A two-phase fluid flow problem is considered, involving the flow of both glass andair. The flow domain Ω is described by the geometry of the mould and hence fixed.On the other hand, the air domain Ωa and the glass domain Ωg are separated by mov-ing interfaces Γi, as depicted in Fig. 4c–d, and therefore change in time. In order tomodel the two-phase fluid flow problem, the glass-air interfaces have to captured.There are different numerical techniques to deal with the moving interfaces in


two-phase fluid flow problems. They can be classified in two main categories [20]:interface-tracking techniques (ITT) and interface-capturing techniques (ICT).

Interface-tracking techniques (ITT) attempt to find the moving interfaces explic-itly. ITT involve separate discretisations of domains Ωa and Ωg; the meshes of bothdomains are updated as the flow evolves by following the flow velocity on the in-terfaces (see e.g. Sect. 3.4). The major challenge of ITT is the mesh update. Theprocedure of updating the mesh can become increasingly computationally expen-sive as the mesh size decreases or the mesh has to be updated more frequently. See[12,28,52] for examples of ITT in mathematical modelling of glass blow processes.

Interface-capturing techniques (ICT) are based on an implicit formulation of theinterfaces by means of interface functions, which allow ICT to function on a fixedmesh for domain Ω . An interface function marks the location of the correspondinginterface by a given level set. Two widely used ICT are Volume-Of-Fluid (VOF)Methods [26, 56] and Level Set Methods [1, 8, 58, 68].

In VOF methods the interface function denotes the fraction of volume withineach element of either fluid. VOF methods are conservative and can deal with topo-logical changes of the interface. However, they are often rather inaccurate; highorder of accuracy is hard to achieve because of the discontinuity of the interfacefunction [46]. Furthermore, they can suffer from small remnants of mixed-fluidzones (‘flotsam and jetsam’) [44, 48]. Still VOF methods are attractive because oftheir rigorous conservation properties.

In Level Set Methods the interface is generally represented by the zero contourof the interface function. Level Set Methods automatically deal with topologicalchanges and it is in general easy to obtain high order of accuracy [46]. In addition,properties of the interfaces, such as the normal and the curvature, are straightforwardto calculate. Also Level Set Methods generalise easily to three dimensions [65].A drawback of Level Set Methods is that they are not conservative. Poor massconservation of Level Set Methods for incompressible two-phase fluid flow prob-lems is addressed in [17, 46, 72]. A major concern in Level Set Methods is there-initialisation of the interface function in order to avoid numerical problems. Forexamples of Level Set Methods in mathematical modelling of glass blow processesthe reader is referred to [2, 20].

Of the aforementioned methods Level Set Methods seem to be most attractivefor this application. The interfaces are accurately captured, topological changes arenaturally dealt with, a generalisation to three dimensions is relatively easy and com-plicated re-meshing algorithms are avoided. In order to compensate for the mass lossor gain coupled Level Set and VOF Methods have been developed [48,62,67]. How-ever, [20] reports a change in mass of less than 1% during the glass blow processsimulations, which can be further improved by using higher order time integrationschemes or by taking smaller time steps. This indicates that in this case Level SetMethods are suited to be used as ICT.

The basic idea of Level Set Methods is to embed the moving interfaces as the zerolevel set of the interface function φ , the so-called level set function (see Fig. 19):

φ(x,t)

= 0, x ∈ Γi(t). (103)


Fig. 19 Level set functionfor a glass preform

airairq < 0

q = 0

q < 0

glassq > 0

The equation of motion of the interfaces follows by the chain rule,

∂φ∂ t

(x,t)+ u·∇φ

(x,t)

= 0, x ∈ Γi(t) (104)

Here the flow velocity is obtained from the Stokes flow problem (95). Initially,the level set function is defined as the signed Euclidean distance function to theinterfaces Γi(0). If furthermore the level set equation (104) is extended to the flowdomain the corresponding level set problem becomes

⎧⎪⎪⎨⎪⎪⎩

∂φ∂ t

+ u·∇φ = 0, in Ω ,

∂φ∂ t

(x,0) :={−d

(x,Γi(0)

), x ∈Ωa(0)

d(x,Γi(0)

), x ∈Ωg(0),

(105)

whered(x,Γ

)= inf

y∈Γ‖x−y‖2 . (106)

Note that since two interfaces are involved φ is not everywhere differentiable. Thisproblem can be overcome by defining two level set functions φ1 and φ2, one for eachinterface, and subsequently solving the two corresponding level set problems. Thenthe level set function for both interfaces is φ = min{φ1,φ2}.

One of the difficulties encountered in Level Set Methods is maintaining the de-sired shape of the level set function. The flow velocity does not preserve the signeddistance property, but may instead considerably distort and stretch the shape ofthe function, which eventually leads to additional numerical difficulties [10, 58].To avoid this the evolution of the level set function is stopped at a certain pointin time to rebuild the signed distance function. This process is referred to as re-initialisation. There are several ways to accomplish this. One approach is to solvethe partial differential equation [47, 58, 68]


∂ φ∂τ

+ sign(φ0)(‖∇φ‖2 −1

)= 0, in Ω, (107)

with φ (x,0) = φ0 := φ(x,t), to steady state. If properly implemented the function φconverges rapidly to the signed distance function around the interface [47,68]. How-ever, the technique does not properly preserve the location of the interface [58, 66].This problem was fixed in [66] by adding a constraint to (107) that enforces massconservation within the grid cells. This re-initialisation technique has appeared tobe quite successful in practice. Another approach is to solve the Eikonal equation,

‖∇φ‖2 = 1, (108)

given φ = 0 on Γi, using Fast Marching Methods (FMM) [11, 57, 58]. FMM buildthe solution outward starting from a narrow band around the interface and subse-quently marching along the grid points. Depending on the implementation FMMcan be extremely computationally efficient. A Fast Marching Method is discussedin Sect. 4.4.

4.3 Variational Formulation

The variational formulation combines the physical problems for glass and air as wellas the jump conditions on the interfaces into one statement. This results in a singleproblem formulation for the entire flow domain, while Level Set Methods can beused to deal with the interfaces. Moreover, the variational formulation is used forthe discretisation by FEM in the simulation model.

First consider a variational formulation of Stokes flow problem (95), (98). Definevector spaces

Q := L2(Ω), (109)

U :={

u ∈ H1(Ω)d∣∣∣ u·n = 0 onΓs, u = 0 onΓg,e

}, (110)

where L2(Ω) is the set of Lebesgue 2-integrable functions over Ω and H1(Ω) isthe first order Hilbert space over Ω . Superscript d denotes the dimension of theflow. Then the variational formulation is to find (u, p) ∈ U ×Q, such that for all(v,q) ∈U×Q, {

a(v,u)+ b(v, p) = c(v),b(u,q) = 0.

(111)

where a : U×U �→R and b : U×Q �→R are bilinear forms and c : U �→R is a linearform, defined by

a(v,u) =∫

Ωμ(∇⊗v

):((

∇⊗u)T + ∇⊗u

)dΩ , (112)


b(v,q) = −∫

Ωq∇·v dΩ , (113)

c(v) = −∫

Ωg·v dΩ +

∫

Γo

p0n·v dΓ . (114)

It is verified that the variational formulation indeed follows from problem (95), (98)with corresponding jump conditions (28), (30) on Γi. First note that

a(v,u)−b(v, p) =∫

Ω

(∇⊗v

):(

μ((

∇⊗u)T + ∇⊗u

)− pI

)dΩ

=∫

Ω

(∇⊗v

):T dΩ . (115)

By means of (115) the first equation in (111) can be written as

∫

Ω

((∇⊗v

):T+ g·v)dΩ =∫

Γo

p0n·v dΓ . (116)

The integral on the left side of (116) can be split up into

∫

Ω

((∇⊗v

):T+ g·v)dΩ =∫

Ωg

((∇⊗v

):T+ g·v) dΩ

+∫

Ωa

((∇⊗v

):T+ g·v)dΩ . (117)

Successive application of identity

∇· (Tv) = v· (∇·T)+(∇⊗v

):T, (118)

Gauss’ divergence theorem and the boundary conditions yields∫

Ωg

((∇⊗v

):T+ g·v)dΩ =∫

Ωg

∇· (Tv)dΩ

=∫

∂Ωg

n·Tv dΓ

=∫

Γi,g

Tn·v dΓ , (119)

since p0 = 0 on Γg,o, and∫

Ωa

((∇⊗v

):T+ g·v)dΩ =∫

Γi,a

Tn·v dΓ +∫

Γa,o

p0n·v dΓ . (120)

Here Γi,g and Γi,a denote the interfaces construed as part of the boundary of Ωg

and Ωa, respectively. As a result,∫

Γi

[[Tn·v]]dΓ = 0, (121)


for all v ∈U , which implies condition (30). Analogously, from the second equationin (111) it follows that ∫

Γi

[[u·∇q]]dΓ = 0, (122)

for all q ∈ Q, which implies condition (28).In addition, consider a variational formulation of energy exchange problem (93),

(4.1). Define vector spaces

V :={

T ∈ H1(Ω)× [0,∞)∣∣ T = T0 on∂Ω , T (x,0) = T0(x) forx ∈Ω

}, (123)

W :={

ω ∈ H1(Ω)× [0,∞)∣∣ ω = 0 on∂Ω , lim

t→∞ω(x, t) = 0 forx ∈Ω

}. (124)

Then the variational formulation is to find T ∈V , such that for all ω ∈W ,∫ ∞

0

∫

Ω

(T

∂ω∂ t

+(T u−λ ∇T ) ·∇ω)

dΩdt

=∫ ∞

0

∫

Γo

αω (T∞−T )dΓ dt +∫

ΩT0ω0 dΩ , (125)

where ω0(x) = ω(x,0) for x ∈ Ω . It is verified that the variational formulation in-deed follows from problem (93)–(4.1) with corresponding jump condition (5) on Γi.To this end split up the integral on the left side of (125) into

∫ ∞

0

∫

Ω

(T

∂ω∂ t

+(Tu−λ ∇T ) ·∇ω)

dΩdt

=∫ ∞

0

∫

Ωg

(T

∂ω∂ t

+(T u−λ ∇T) ·∇ω)

dΩdt

+∫ ∞

0

∫

Ωa

(T

∂ω∂ t

+(T u−λ ∇T ) ·∇ω)

dΩdt. (126)

Successive application of the product rule for differentiation, Gauss’ divergencetheorem and the boundary conditions yields

∫ ∞

0

∫

Ωg

(T

∂ω∂ t

+(T u−λ ∇T) ·∇ω)

dΩdt

=∫ ∞

0

∫

Ωg

(∂∂ t

(T ω

)+ ∇·(ωT u−λ ω∇T

))dΩdt

=∫ ∞

0

∫

Γg

ω(T u−λ ∇T

)·n dΩdt

−∫ ∞

0

∫

Γg

ωT u·n dΓ dt +∫

Ωg

T0ω0 dΩ

=∫ ∞

0

∫

Γg,o

αω (T∞−T )dΓ dt

−∫ ∞

0

∫

Γi,g

λ ω∇T ·n dΓ dt +∫

Ωg

T0ω0 dΩ , (127)


and

∫ ∞

0

∫

Ωa

(T

∂ω∂ t

+(T u−λ ∇T ) ·∇ω)

dΩdt

=∫ ∞

0

∫

Γa,o

αω (T∞−T) dΓ dt

−∫ ∞

0

∫

Γi,a

λ ω∇T ·n dΓ dt +∫

Ωa

T0ω0dΩ . (128)

Therefore, ∫

Γi

[[λ ω∇T ·n]]dΓ = 0, (129)

for all ω ∈W , which implies condition (5).


A simulation model for the blow–blow process is described in [21]. The modelis able to compute and visualise the preform or container shape, the thickness ofthe final product as well as the stress and thermal deformation the glass and mouldundergo during the process. Input parameters for the simulations are the gob volumeor the glass preform, the mould shape, the temperature distribution of the glass anda prescribed inlet air pressure, as well as the physical properties of the glass. The2D axi-symmetrical geometry of the initial computational domain is defined by thatof the flow domain (see Sect. 2.1). The model does not take the heat transfer inthe mould into account. See [23, 28] for a complete description of the heat transferbetween the glass and the mould. FEM are used to solve the problems. TriangularMini-elements are used for the discretisation of the flow problem. Typical meshesfor the moulds for the parison and the bottle are depicted in Fig. 20.

A streamline-upwind Petrov-Galerkin (SUPG) method is used to obtain a sta-bilised formulation of the energy equation [5, 20, 30]. The upwind parameter isdefined by

ξ =hξ2

u·∇ϕi

‖u‖2, inei. (130)

Here h is the width of the element in flow direction, ϕi is the ith basis function, ei

is the ith element and ξ is a tuning parameter. In the classical case ξ = 1 and in themodified case [20, 59]

ξ =12

√Δ t2 +

(h‖u‖2

)+

14

(h

u·Au

)2(u·∇ϕi

), inei, (131)

for some matrix A.


Fig. 20 Typical meshesfor blow–blow simulationof a bottle

parison bottle

Interface

a b

Interface

c

InterfaceA

B

C

d

proximate: correct value by assumptiondistant: not yet consideredtrial: value might not be correctaccepted: value updated by FMM

Fig. 21 Fast marching methods

A triangulated Fast Marching Method is used as a re-initialisation algorithm tomaintain the level set function as a signed distance function [2, 20, 32, 57]. Themethod assumes that the level set values at the nodes within a narrow band aroundthe interface are the correct signed distance values and then builds a signed distancefunction outward by marching along the nodes. Figure 21 shows the algorithmicrepresentation of the Fast Marching Method for a two-dimensional mesh. The curverepresents a glass-air interface. The algorithm is initialised by computing initialvalues for distance function d in all nodes adjacent to the interface φ−1(0). A secondorder accurate initialisation procedure is discussed in [11]. The nodes adjacent tothe interface, to which the initial values are assigned, are tagged as proximate and


are colour marked as orange in Fig. 21a. The proximate nodes are the first nodesto be added to the set of accepted nodes, which contains all nodes of which thecorresponding values are accepted as a distance value. The remaining nodes areinitially tagged as distant, which are colour marked as black in Fig. 21, and are not(yet) of interest. The nodes adjacent to the set of accepted nodes are candidates to beadded to either the set of accepted nodes or the set of distant nodes and are the trialnodes, which are colour marked as green in Fig. 21b. These nodes have a trial valueassigned to them that might not yet be the correct distance value. When the valuesof all trial nodes have been updated, the trial nodes are removed from the set of trialnodes and added to the set of accepted nodes (see Fig. 21c). If multiple values areassigned to a trial node, the smallest one holds. This procedure repeats itself untilthe set of accepted nodes contains all grid points.

An element in Fig. 21c with one trial node C with two accepted nodes A and B isconsidered. If the mesh is structured as in Fig. 21 and the angle between edges ABand AC is right, then the distance value dC at node C is the solution of the quadraticequation (

dC−dA)2 +

(dB−dA

)2 =(dC−dB

)2. (132)

Suppose the mesh is not structured, but the triangles have different edges.Figure 22 shows a triangle with angles α,β ,γ and edge lengths a,b,c. The interfaceis approximated by a line l such that the distance from nodes A and B to l is equalto the approximate distance to the interface, that is dA and dB, respectively. SupposedB ≤ dA. Then the angle δ between AB and l is determined by sin(δ ) = dA−dB

c . Asa result distance value dC can be computed by

dC = asin(δ + β )+ dB. (133)

It should be verified that the shortest distance to l from C intersects the triangle.Therefore the following requirement has to be satisfied:

0≤ acos(δ + β )

cos(δ )≤ c. (134)

If requirement (134) is not fulfilled, the update is performed by taking

dC = min(dA + b,dB + a

). (135)

Fig. 22 Triangulated fastmarching method

dBdA

A

C

B

ba

c

a

bd

l

dA-dB

g


Finally, it is also possible to consider triangles with only one accepted node. Iffor example only A is accepted, then the values at B and C can be computed asdB = dA + c and dC = dA + b.

4.5 Results

In this section simulations of the blow–blow process for a bottle are shown. Theouter surface of the preform in the final blow is used as the mould shape for thecounter blow. The propagation of the interfaces and the temperature distributionduring the blow stages are visualised. For more results of the simulation model usedthe reader is referred to [20, 21]. Furthermore, see [12, 28, 53, 55] for results ofdifferent simulation models.

First consider the simulation of the counter blow. The initial temperature of themould and the air in the counter blow are 500 ◦C. The gob has an uniform initialtemperature distribution. The inlet air pressure is 138 kPa. Figure 23 visualises theevolution of the glass domain (red) and the air domain (blue) and Fig. 24 shows thetemperature distribution.

Next consider the simulation of the final blow. First the preform is left to sag dueto gravity for 0.3 s. Then pressurised air is blown into the mould with an inlet airpressure of 3 kPa. The pressure should be much lower than in the counter blow, be-cause the preform is relatively thin and therefore easily breaks. Figure 25 visualisesthe evolution of the glass domain (red) and the air domain (blue) and Fig. 26 showsthe temperature distribution.

In [21] a change in glass volume of less than 1.5% during the blow–blow simula-tion is reported. The volume conservation can be further improved by decreasing thetime step, improving the mesh quality or by using higher order time discretisationschemes [20, 21].

Fig. 23 Evolution of theglass domain during thecounter blow [21]. Glass isred, air is blue t=0.0s 0.25s 0.75s 1.75s


Fig. 24 Temperature distribution during the counter blow [21]

Fig. 25 Evolutionof the glass domain duringthe final blow [21]. Glassis red, air is blue

t=0.0s 0.3s 0.725s 1.025s

5 Direct Press Model

This section presents a mathematical model for the direct press. Since physicalproblems in the direct press model are described in glass, air and equipment, itis more complicated than the parison press model described in Sect. 3 and the blow


Fig. 26 Temperature distribution during the final blow [21]

model described in Sect. 4. Section 3.1 specifies the mathematical model describedin Sect. 2 for the direct press. Features from both the blow model and the parisonpress model are combined in the direct press model. Section 3.4 presents a numer-ical simulation model for the direct press. An essential difference with the parisonpress is the way the mesh is updated. Finally, Sect. 3.5 shows some examples of two-dimensional TV panel press simulations. The simulation tool used for these resultsis presented in [24].


In Sect. 2.2 the balance laws for glass forming were formulated. In this section thesebalance laws are further specified for direct pressing. Essentially, the typical valuesfor most physical parameters for direct pressing are the same as for parison pressing;the main difference is the larger time duration of the direct press and hence thesmaller velocity:








The corresponding dimensionless numbers are:

Peglass ≈ 76, Brglass ≈ 4.0 ·10−5, Reglass ≈ 2.5 ·10−5, Fr≈ 1.0 ·10−3. (136)

In conclusion, the heat transfer in glass is dominated by convection and radiation,

(∂T ∗

∂ t∗+ u∗ ·∇∗T ∗

)= ∇∗ ·(λ ∗∇∗T ∗

), (137)

where λ ∗ is defined by (94), and the glass flow is described by the Stokes flowequations,

∇∗ ·T∗ = g∗, ∇∗ ·u∗ = 0, (138)

where T∗ is the dimensionless stress tensor and g∗ is the dimensionless gravityforce, given by (96). As the glass is always in contact with both air and equipmentand the influence of radiation and convection cannot be disregarded, it is interestingto model the heat transfer in the whole domain Σ . Consider the typical values forthe physical properties of air in Sect. 4.1. Then the following dimensionless numberscan be found:

Peair ≈ 102, Brair ≈ 4 ·10−8, Reair ≈ 1. (139)

Since the Peclet number for air is not extremely large, heat transfer by conductionand radiation should be taken into account. Heat transfer by dissipation can be ne-glected. The flow of air is described by the full Navier–Stokes flow equations, butthese can eventually be simplified to the Stokes flow equations by replacing air by afictitious fluid (see Sect. 4.4). In the equipment there is no flow, so the heat exchangeis described by:

∂T ∗

∂ t∗= ∇∗ ·(λ ∗∇∗T ∗

), (140)

Separate domains for the mould, plunger and ring should be defined. The mould andplunger can again consist of parts with different material properties, for which sepa-rate subdomains should be defined. For example, if water-cooled channels are usedto thermally stabilise the temperature of the plunger and the mould, these shouldalso be taken into account. In this paper the mould, the plunger and the ring areconsidered as entities with constant physical properties.

The boundary conditions given in Sect. 2.2 are specified for the direct press. Free-slip conditions are imposed for air at the equipment boundary, except for the furthestend of the ring wall, where free flow of air is allowed. In order to distinguish these

boundary conditions the ring boundary Γr is subdivided into two parts Γ (1)r and Γ (1)

r ,as it is illustrated in Fig. 27, denoting the inner boundaries of the lower part and theupper part of the ring, respectively. Finally, the boundary conditions for the flowproblem can be formulated as:


Γi

r

z mould

plunger

ring

glass airΩg Ωa

tn Γp

Γr

Γm

Γo

Γs

ΓoΓo

ringΓr

(2)Γo Γo

mould

plunger

Γr(1)

ΓmΓp

Fig. 27 Problem domains and boundaries of direct press model near the ring

u∗·n = 0, T∗n· t = 0, on Γs,

(u∗ −Vpez)·n = 0,(T∗n+ β ∗p(u∗ −Vpez)

)· t = 0, on Γg,p,

u∗·n = 0,(T∗n+ β ∗mu∗

)· t = 0, on Γg,m,

u∗·n = 0,(T∗n+ β ∗r u∗

)· t = 0, on Γg,r,

u∗·n = 0, T∗n· t = 0, on Γa,m∪Γa,p∪Γ (1)a,r ,

T∗n·n = 0, T∗n· t = 0, on Γ (2)a,r .

(141)

Alternatively, one may prefer a small normal velocity of air on the impermeableequipment boundary to ensure that air does not get trapped during the simulation.The boundary conditions for the energy exchange problem are given by (14).

Essentially, a slender-geometry approximation such as in (3.2) is also possible forthe direct press model as the glass becomes fairly long and thin during pressing (e.g.television screens, lenses). Unfortunately, in this case the influence of convectionand radiation is too large to assume the temperature, and hence the viscosity, willremain constant for constant initial temperature is constant. As a result an analyticalsolution to the slender-geometry approximation cannot be found as easily as forthe parison press. The slender-geometry approximation for the direct press is notconsidered in this paper.


Obviously, the direct press model is much more complicated than the parison pressmodel described in Sect. 3 and the blow model described in Sect. 4. It involves acoupled system of physical problems consisting of a Stokes flow problem, an energyexchange problem and a moving interface problem. In addition, the computationaldomain has a rather complicated geometry, which is changing in time, and is subdi-vided into several subdomains, each with different physical properties.

A simulation model for the two-dimensional direct press process is describedin [24]. The model is able to compute and visualise the velocity field and thepressure in glass and air, as well as the stress and thermal deformation the glass and


y

z x

a

t = 0.0s

y

z x

b

t = 0.5s

y

z x

c

t = 1.0sxz

y

d

t = 1.5s

Fig. 28 Typical mesh for press simulation of a TV panel

mould undergo during the process. Furthermore, it can reproduce the temperaturedistribution in glass, air and equipment. A complete three-dimensional simulationtool with many additional features, such as several advanced radiation models andan annealing process simulation model, is described in [7]. Input parameters forthe simulations are the gob volume, the equipment geometry, the initial tempera-ture distribution and the physical properties of the glass and the equipment. Eachtime step the computational domain is discretised by means of FEM. TriangularMini-elements are used for the discretisation of the flow problem. Figure 28 showsthe deformation of a typical mesh in the press model from t = 0 s to 1.5 s, wheret = 1.5 s is the end time of the process. The mesh for the lowest plunger position inFig. 28d is provided as input for the press model. Then the elements in the flow do-main are stretched in vertical direction by raising the plunger to its initial position toobtain the initial mesh (see Fig. 28a). In each subsequent time step a mesh is gener-ated that fits the plunger position at that time following an arbitrary Euler–Lagrangeapproach, that is each mesh consists of the same numbers of elements along eachdimension of the domain. Thus the mesh of the flow domain is compressed by theplunger, until the mesh for the lowest plunger position is regained. In this way timeconsuming remeshing is avoided, although remeshing is still necessary if the qualityof the elements can become at issue as they are stretched or compressed [24].

Some modelling aspects of the direct press have already been discussed inSects. 3 and 4.

• The motion of the plunger is modelled analogously to the parison press model(see Sects. 3.3, 3.4).


• The evolution of the glass-air interface can be modelled analogously to the blowmodel (see Sect. 4.2). The direct press model described in [7, 24] uses a Pseudo-Concentration Method [70] to track the interface.

• Air is replaced by a fictitious fluid (see Sect. 4.4).• The energy equation is stabilised by means of a SUPG method (see Sect. 4.4).

5.3 Results

In this section two-dimensional simulations of the pressing of a TV panel are shown.This includes the visualisation of the flow velocity and temperature distribution,the tracking of the glass-air interfaces and the computation of the motion of the

t = 0.0s t = 0.25s

t = 0.5s t = 0.75s

t = 1.0s t = 1.5s

Fig. 29 Concentration of glass during the pressing of a TV panel


t = 0.0s t = 0.5s

t = 1.0s t = 1.5s

Fig. 30 Temperature distribution during the pressing of a TV panel

plunger. For more results of the simulation tool used the reader is referred to [7,24].Moreover, see [16, 35, 76] for results of different direct press models.

Figure 29 visualises the (pseudo-)concentration of glass during pressing. In thefigures glass is red and air is blue. The transition layer of the concentration betweenglass and air becomes thinner and smoother as the mesh is refined. A no-slip con-dition for glass at the equipment wall is used, which explains the strongly curvedinterface.

Figure 30 visualises the temperature distribution during pressing. The initialtemperature of the glass is 1,000 ◦C and the initial temperature of the air and theequipment is 500 ◦C. The glass gradually heats up the equipment at the contact sur-face. Apart from a thin boundary layer of roughly 0.2 mm, the temperature of theequipment does not become higher than 550◦C within 1.5s. By contrast, the glassrapidly heats up the air.

In order to verify the accuracy of numerical solutions, the volume conservationof the glass is examined. Figure 31 depicts the ratio of the glass volume to theinitial glass volume as a function of time. It can be observed that the glass volumeinitially makes a relatively strong jump. Possibly, the solver has some difficultiesin accurately meeting the initial conditions. At t = 0.5s a loss volume of less than1% is observed, which is a good result. The maximum change of volume during thesimulation was 1.11%. The volume change can be further reduced by decreasing thetime step, improving the mesh quality or by using higher order time discretisationschemes. Good volume conservation is also reported in [24].


Fig. 31 Glass volume conservation

References

1. Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. J. Comp.Phys. 118, 269–277 (1995)

2. Allaart-Bruin, S.M.A., Linden, B.J.V.D., Mattheij, R.M.M.: Modelling the glass press-blowprocess. In: Di Bucchianico, A., Mattheij, R.M.M., Peletier, M.A. (eds.) Progress in Indus-trial Mathematics at ECMI 2004 (Proceedings 13th European Conference on Mathematicsfor Industry, Eindhoven, The Netherlands, June 21–25, 2004). Mathematics in Industry/TheEuropean Consortium for Mathematics in Industry, vol. 8, pp. 351–355. Springer, Berlin (2006)

3. Babcock, C.L.: Silicate Glass Technology Methods. Wiley, New York (1977)4. Bernard, T., Ebrahimi Moghaddam, E.: Nonlinear model predictive control of a glass forming

process based on a finite element model. In: Proceedings of the 2006 IEEE InternationalConference on Control Applications (2006)

5. Brooks, A.N., Hughes, T.J.R.: Stream-line upwind/petrov-galerkin formulation for convectiondominated flows with particular emphasis on the incompressible navier-stokes equations. Com-put. Methods Appl. Mech. Eng. 32, 199–259 (1982)

6. Brown, M.: A review of research in numerical simulation for the glass-pressing process. Proc.IMechE Part B J. Eng. Manufact. 221(9), 1377–1386 (2007)

7. Op den Camp, O., Hegen, D., Haagh, G., Limpens, M.: Tv panel production: Simulation of theforming process. Ceram. Eng. Sci. Proc. 24(1), 1–19 (2003)

8. Chang, Y.C., Hou, T.Y., Merriman, B., Osher, S.: A level set formulation of eulerian interfacecapturing methods for incompressible fluid flows. J. Comp. Phys. 124, 449–464 (1996)

9. Choi, J., Ha, D., Kim, J., Grandhi, R.V.: Inverse design of glass forming process simulationusing an optimization technique and distributed computing. J. Mater. Process. Technol. 148,342–352 (2004)

10. Chopp, D.L.: Computing minimal surfaces via level set curvature flow. J. Comp. Phys. 106(1),77–91 (1993)

11. Chopp, D.L.: Some improvements on the fast marching method. SIAM J. Sci. Comput. 23(1),230–244 (2001)

12. Cormeau, A., Cormeau, I., Roose, J.: Numerical simulation of glass-blowing. In: Pittman,J.F.T., Zienkiewicz, O.C., Wood, R.D., Alexander, J.M. (eds.) Numerical Analysis of FormingProcesses, pp. 219–237. Wiley, New York (1984)

13. Dijkstra, W.: Condition numbers in the boundary element method: Shape and solvability. Ph.D.thesis, Eindhoven University of Technology (2008)

14. Dijkstra, W., Mattheij, R.M.M.: Numerical modelling of the blowing phase in the productionof glass containers. Electron. J. Bound. Elem. 6, 1–23 (2008)

15. Dowling, W.C., Fairbanks, H.V., Koehler, W.A.: A study of the effect of lubricants on moltenglass to heated metals. J. Am. Ceram. Soc. 33, 269–273 (1950)


16. Dussere, G., Schmidt, F., Dour, G., Bernhart, G.: Thermo-mechanical stresses in cast steel diesduring glass pressing process. J. Mater. Technol. 162–163, 484–491 (2005)

17. Enright, D., Fedkiw, R., Ferziger, J., Mitchell, I.: A hybrid particel level set method for im-proved interface capturing. J. Comput. Phys. 183, 83–116 (2002)

18. Falipou, M., Siclorof, F., Donnet, C.: New method for measuring the friction between hotviscous glass and metals. Glass Sci. Technol. 72(3), 59–66 (1999)

19. Fowler, A.C.: Mathematical Models in the Applied Sciences. Cambridge University Press,Cambridge (1997)

20. Giannopapa, C.G.: Development of a computer simulation model for blowing glass containers.J. Manuf. Sci. Eng. 130 (2008)

21. Giannopapa, C.G., Groot, J.A.W.M.: A computer simulation model for the blow-blow formingprocess of glass containers. In: 2007 ASME Pressure Vessels and Piping Conf. and 8th Int.Conf. on CREEP and Fatigue at Elevated Temp. (2007)

22. Giegerich, W., Trier, W. (eds.): Glass Machines: Construction and Operation of Machines forthe Forming of Hot Glass. Springer, Berlin (1969)

23. Gregoire, S., Cesar de Sa, J.M.A., Moreau, P., Lochegnies, D.: Modelling of heat transfer atglass/mould interface in press and blow forming processes. Comput. Struct. 85, 1194–1205(2007)

24. Groot, J.A.W.M.: Analysis of the solver performance for stokes flow problems in glass formingprocess simulation models. Master’s thesis, Eindhoven University of Technology (2007)

25. Groot, J.A.W.M., Giannopapa, C.G., Mattheij, R.M.M.: Development of a numerical optimisa-tion method for blowing glass parison shapes. In: Proceedings of PVP 2008: ASME PressureVessels and Piping Division Conference (July 27–31, 2008)

26. Hirth, C.W., Nichols, B.D.: Volume of fluid (vof) method for the dynamics of free boundaries.J. Comp. Phys. 39, 201–225 (1981)

27. Humpherys, C.E.: Mathematical modelling of glass flow during a pressing operation. Ph.D.thesis, University of Sheffield (1991)

28. Hyre, M.: Numerical simulation of glass forming and conditioning. J. Am. Ceram. Soc. 85(5),1047–1056 (2002)

29. John, V., Liakos, A.: Time-dependent flow across a step: The slip with friction boundary con-dition. Int. J. Numer. Meth. Fluids 50, 713–731 (2006)

30. Johnson, C., Navert, U., Pitkaranta, J.: Finite element methods for linear hyperbolic problems.Comput. Methods Appl. Mech. Eng. 45, 285–312 (1984)

31. Kang, M., Fedkiw, R.P., Liu, X.D.: A boundary condition capturing method for multiphaseincompressible flow. J. Sci. Comput. 15(3), 323–360 (2000)

32. Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci.USA 95, 8431–8435 (1998)

33. Krause, D., Loch, H. (eds.): Mathematical Simulation in Glass Technology. Springer, Berlin(2002)

34. Laevsky, K.: Pressing of glass in bottle and jar manufacturing: Numerical analysis and compu-tation. Ph.D. thesis, Eindhoven University of Technology (2003)

35. Li, G., Jinn, J.T., Wu, W.T., Oh, S.I.: Recent development and applications of three-dimensionalfinite element modeling in bulk forming processes. J. Mater. Process. Technol. 113(2001),40–45 (2001)

36. Li, L.X., Peng, D.S., Liu, J.A., Liu, Z.Q., Jiang, Y.: An experimental study on the lubricationbehavior of a5 glass lubricant by means of the ring compression test. J. Mater. Process. Technol.102, 138–142 (2000)

37. Linden, B.J.V.D.: Radiative heat transfer in glass: The algebraic ray trace method. Ph.D. thesis,Eindhoven University of Technology (2002)

38. Linden, B.J.V.D., Mattheij, R.M.M.: A new method for solving radiative heat problems inglass. Int. J. Forming Process. 2, 41–61 (1999)

39. Lochegnies, D., Moreau, P., Guilbaut, R.: A reverse engineering approach to the design of theblank mould for the glass blow and blow process. Glass Technol. 46(2), 116–120 (2005)

40. Mattheij, R.M.M., Rienstra, S.W., Thije Boonkamp, J.H.M.T.: Partial Differential Equations:Modeling, Analysis, Computation. SIAM, Philadelphia (2005)


41. Moreau, P., Lochegnies, D., Oudin, J.: An inverse method for prediction of the prescribedtemperature distribution in the creep forming process. Proc. Inst. Mech. Eng. J. Mech. Eng.Sci. 212, 7–11 (1998)

42. Moreau, P., Lochegnies, D., Oudin, J.: Optimum tool geometry for flat glass pressing. Int. J.Forming Process. 2, 81–94 (1999)

43. Nagtegaal, T.M.: Optical method for temperature profile measurements in glass melts. Ph.D.thesis, Eindhoven University of Technology (2002)

44. Noh, W.F., Woodward, P.: Simple line interface calculations. In: Vooren, A.I.V.D., Zandbergen,P.J. (eds.) Proceedings of the Fifth International Conference on Numerical Methods in FluidDynamics, Lecture Notes in Physics, vol. 59, pp. 330–340. Springer, New York (1976)

45. Oevermann, M., Klein, R., Berger, M., Goodman, J.: A projection method for two-phase in-compressible flow with surface tension and sharp interface resolution. Tech. rep. ZIB-report,no. 00-17, Konrad-Zuse-Zentrum fur Informationstechnik Berlin (2000)

46. Olsson, E., Kreis, G.: A conservative level set method for two phase flow. J. Comp. Phys. 210,225–246 (2005)

47. Peng, D., Kreis, G.: A pde-based fast local level set method. J. Comp. Phys. 155, 410–438(1999)

48. Pijl, S.P.V.D., Segal, A., Vuik, C., Wesseling, P.: A mass-conserving level-set method for mod-elling of multi-phase flows. Int. J. Numer. Meth. Fluids 47, 339–361 (2005)

49. Pye, L.D., Montenero, A., Joseph, I.: Properties of Glass-Forming Melts. CRC, Boca Raton(2005)

50. Rienstra, S.W., Chandra, T.D.: Analytical approximations to the viscous glass-flow problem inthe mould-plunger pressing process, including an investigation of boundary conditions. J. Eng.Math. 39, 241–259 (2001)

51. Rosseland, S.: Note on the absorption of radiation within a star. Mon. Not. R. Astron. Soc. 84,525–528 (1924)

52. Cesar de Sa, J.M.A.: Numerical modelling of glass forming processes. Eng. Comput. 3,266–275 (1986)

53. Cesar de Sa, J.M.A.: Numerical modelling of incompressible problems in glass forming andrubber technology. Ph.D. thesis, University College of Swansea (1986)

54. Cesar de Sa, J.M.A., Natal Jorge, R.M., Silva, C.M.C., Cardoso, R.P.R.: A computationalmodel for glass container forming processes. In: Europe Conference on ComputationalMechanics Solids, Structures and Coupled Problems in Engineering

55. Sadegh, N., Vachtsevanos, G.J., Barth, E.J., Pirovolou, D.K., Smith, M.H.: Modelling the glassforming process. Glass Technol. 38, 216–218 (1997)

56. Scardovelli, R., Zaleski, S.: Direct numerical simulation of free-surface and interfacial flow.Annu. Rev. Fluid Mech. 31, 567–603 (1999)

57. Sethian, J.A.: A fast marching level set method for montonically advancing fronts. Proc. Nat.Acad. Sci. 93(4), 1591–1595 (1996)

58. Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press,New York (1999)

59. Shakib, F.: Finite element analysis of the incompressible euler and navier-stokes equation.Ph.D. thesis, Stanford University (1989)

60. Sharp, D.E., Ginther, L.B.: Effect of composition and temperature on the specific heat of glass.J. Am. Ceram. Soc. 34, 260–271 (1951)

61. Shelby, J.E.: Introduction to Glass Science and Technology, 2nd edn. The Royal Society ofChemistry, Cambridge (2005)

62. Shepel, S.V., Smith, B.L.: New finite-element/finite-volume level set formulation for modellingtwo-phase incompressible flows. J. Comp. Phys. 218, 479–494 (2006)

63. Simmons, J.H., Simmons, C.J.: Nonlinear viscous flow in glass forming. Am. Ceram. Soc.Bull. 68(11), 1949–1955 (1989)

64. Storck, K., Loyd, D., Augustsson, B.: Heat transfer modelling of the parison forming in glassmanufacturing. Glass Technol. 39(6) (1998)

65. Sussman, M., Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H., Welcome, M.L.: An adaptivelevel set approach for incompressible two-phase flows. J. Comp. Phys. 148, 81–124 (1999)


66. Sussman, M., Fatemi, E.: An efficient, interface-preserving level set re-distancing algorithmand its application to interfacial incompressible fluid flow. SIAM J. Sci. Comput. 20(4),1165–1191 (1995)

67. Sussman, M., Pucket, E.G.: A coupled level set and volume-of-fluid method for computing 3dand axisymmetric incompressible two-phase flows. J. Comp. Phys. 162, 301–337 (2000)

68. Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incom-pressible two-phase flow. J. Comp. Phys. 114, 146–159 (1994)

69. Tasic, B.: Numerical methods for solving ode flow. Ph.D. thesis, Eindhoven University ofTechnology (2004)

70. Thompson, E.: Use of pseudo-concentration to follow creeping viscous flow during transientanalysis. Int. J. Numer. Methods Fluids 7, 749–761 (1986)

71. Tooley, F.V.: The Handbook of Glass Manufacture, Vol II. Aslee Publishing Co, New York(1984)

72. Tornberg, A.K., Engquist, B.: A finite element based level-set method for multiphase flowapplications. Comput. Visual. Sci. 3(1), 93–101 (2000)

73. Waal, H.D., Beerkens, R.G.C. (eds.): NCNG Handboek voor de Glasfabricage, 2edn. TNO-TPD-Glastechnologie (1997)

74. Williams, J.H., Owen, D.R.J., Cesar de Sa, J.M.A.: The numerical modelling of glass formingprocesses. In: Bharwaj, H.C. (ed.) Collected Papers of the XIV International Congress onGlass, pp. 138–145. Indian Ceramic Society, Calcutta (1986)

75. Yigitler, K.: Numerical simulation of blowing of axisymmetric wide-mouth glass containers.In: Proceedings of the Colloquium on Modelling of Glass Forming

76. Zhou, H., Yan, B., Li, D.: Three-dimensional numerical simulation of the pressing process intv panel production. Simulation 82(3), 193–203 (2006)

Radiative Heat Transfer and Applicationsfor Glass Production Processes

Martin Frank and Axel Klar

1 Introduction

In glass manufacturing, a hot melt of glass is cooled down to room temperature.The annealing has to be monitored carefully in order to avoid excessive temperaturedifferences which may affect the quality of the product or even lead to cracks in thematerial. In order to control this process it is, therefore, of interest to have a math-ematical model that accurately predicts the temperature evolution. The model willinvolve the direction-dependent thermal radiation field because a significant part ofthe energy is transported by photons. Unfortunately, this fact makes the numericalsolution of the radiative transfer equations much more complex, especially in higherdimensions, since, besides position and time variables, the directional variables alsohave to be accounted for. Therefore, approximations of the full model that are com-putationally less time consuming but yet sufficiently accurate have to be sought. Itis our purpose to present several recent approaches to this problem that have beenco-developed by the authors.

This manuscript is organized as follows. In Sect. 2, we derive the underlyingkinetic equation model for radiative transfer in glass. This model is supplementedby initial and boundary conditions. In addition, several versions of this model,that are later used, are introduced. For later reference and for the reader whowants to skip the derivation, the basic model is summarized in Sect. 2.6. Section 3deals with direct numerical methods for the solution of the radiative transfer equa-tions. These methods will later be used to compute benchmark results. Thus, wepresent convergence and robustness results. The rest of the discussion focuses ontwo approximation methods that have been co-developed by the authors, namely

M. Frank (�)University of Kaiserslautern, Erwin-Schrodinger-Strasse, 67663 Kaiserslautern, Germanye-mail: [email protected]

A. KlarUniversity of Kaiserslautern, Erwin-Schrodinger-Strasse, 67663 Kaiserslautern, Germany andFraunhofer ITWM, Fraunhofer Platz 1, 67663 Kaiserslautern, Germanye-mail: [email protected]


57

[email protected]

[email protected]

58 M. Frank and A. Klar

higher-order diffusion (Sect. 4) and moment methods (Sects. 5 and 6). These modelsare compared numerically in Sect. 7, where we also present results specificallyrelated to glass cooling. Parts of this work have been taken from the articles[18, 23–25, 27, 47, 48, 76, 85].

2 Radiative Heat Transfer Equations for Glass

Radiative transfer has to compete with the two other modes of energy transfer,namely heat conduction and convection. In everyday life, these three effects canbe seen at a cup of hot coffee. The cup itself gets warmer because of heat conduc-tion between the coffee and the cup material. The warmth felt near the outside wallsof the cup is due to radiation and the vapor emerging from the top of the cup carriesenergy by convection. The distinguishing features of the three modes are given inTable 1.

Radiation consists of electromagnetic waves, which have the same nature asvisible light. The elementary particle of the radiation field is the photon. Heat isconducted in solids and fluids by free electrons and phonon–phonon interactions,whereas convection is energy transport by material transport.

While radiation can also be transported through the vacuum, conduction andconvection need a medium. The conductive and convective heat flux is di-rectly proportional to temperature differences. On the other hand, the celebratedStefan–Boltzmann law states that the radiative heat flux is proportional to the dif-ference of the fourth powers of the temperature. Because of this, radiation becomesthe dominant effect at large temperatures.

Conduction and convection are local phenomena, which occur at the atomiclength scale of approximately 10−9 m. Radiation on the other hand is a non-localphenomenon. The average distance a photon travels between two collisions canvary between the atomic length scale of 10−9 m up to 1010 m (distance earth-sun)and even more. As a consequence, the commonly used mathematical descriptionsof radiative heat transfer and conduction/convection are different.

Table 1 Modes of energy transfer

Radiation Conduction Convection

Energy transport by photons free electrons, material transportPhonon interaction

Medium required no yes yesTemperature dependence q∼ T 4−T 4

∞ q∼ ∇T q∼ T −T∞Mean free path 10−9 ∼ 1010 m ∼ 10−9 m ∼ 10−9 mDepends on x, t,Ω ,ν x, t x, t

Radiative Heat Transfer and Applications for Glass Production Processes 59

Furthermore, the physical quantities describing the radiation field depend onspace, time, direction and frequency, while those used to describe conduction andconvection depend only on space and time.

2.1 Fundamental Quantities

In this section we want to define the fundamental physical quantities describing theradiation field.

2.1.1 Intensity

The basic variable is the spectral intensity

ψ(t,x,Ω ,ν), (1)

the radiative energy flow per time, per area normal to the rays, per solid angle andper frequency. This means that ψ dt dA dΩ dν has the dimension of energy flux andis proportional to the number of photons. The spectral intensity depends on positionx ∈R3, time t ∈R, direction Ω ∈ S2 = {x ∈R3 : ‖x‖= 1}, and frequency ν ∈R+.

The total intensity

ψ(t,x,Ω) =∫ ∞

0ψ(t,x,Ω ,ν)dν (2)

is the spectral intensity integrated over the whole spectrum.

2.1.2 Energy Flux

The total energy flux is defined as

E(t,x) = ϕ(t,x) =∫

S2ψ(t,x,Ω)dΩ =

∫

S2

∫ ∞

0ψ(t,x,Ω ,ν)dνdΩ . (3)

In the context of moment models, this quantity is denoted by E , in the context ofdiffusion models it is traditionally denoted by ϕ or φ . The radiative energy is thezeroth order moment of the total intensity with respect to the direction Ω . Severalother moments will play an important role in the following.

2.1.3 Heat Flux

Consider an infinitesimal surface element with outward normal n. The ingoing andoutgoing spectral heat fluxes are


|F | =∣∣∣−|F in|+ |Fb|

∣∣∣

=∫

n·Ω<0(n ·Ω)ψdΩ +

∫

n·Ω>0(n ·Ω)ψdΩ

=∫

S2(n ·Ω)ψdΩ . (4)

Thus the spectral heat flux is

F(t,x,ν) =∫

S2Ωψ(t,x,Ω ,ν)dΩ . (5)

To obtain the total heat flux, we integrate over the spectrum,

F(t,x) =∫

S2

∫ ∞

0Ωψ(t,x,Ω ,ν)dνdΩ . (6)

2.1.4 Radiation Pressure

The heat flux into a surface element dA is, as above,

(n ·Ω)ψdAdΩ . (7)

Thus the beam carries momentum at a rate

1c(n ·Ω)ψndAdΩ . (8)

The fraction of momentum falling onto dA is |n ·Ω |. Therefore, the flow of momen-tum into dA in the normal direction is

1c

ψ |n ·Ω |2dAdΩ . (9)

This must be counteracted by a pressure force pdA leading to the spectral radiationpressure

p =1c

∫

S2ψ |n ·Ω |2dΩ . (10)

The spectral radiative pressure tensor Pν is defined by

nT Pn = p, (11)

thus

P(t,x,ν) =1c

∫

S2(Ω ⊗Ω)ψ(t,x,Ω ,ν)dΩ . (12)


Here, Ω ⊗Ω is the outer product (tensor product). The total radiative pressuretensor is

P(t,x) =1c

∫ ∞

0

∫

S2(Ω ⊗Ω)ψ(t,x,Ω ,ν)dΩdν. (13)

Not quite correctly, we will also call the second order moment of ψ , without thefactor 1

c , radiative pressure.

2.2 Blackbody Radiation

In this section we want to define a perfect absorber, also called blackbody, andderive the Planck equilibrium distribution. The Planckian plays a crucial role in thefollowing.

Consider an electromagnetic wave that hits the surface of a medium. The wavecan either be reflected at the surface or penetrate the medium. If the wave passesthrough the medium without attenuation, the medium is called transparent. If noradiation reemerges it is called opaque. Otherwise, in the case of partial attenuation,it is called semitransparent.

A blackbody or perfect absorber is defined to have an opaque surface that doesnot reflect any radiation. A blackbody is thus a maximal absorber. A simple thermo-dynamical argument [66] shows that it is also a perfect emitter at every frequencyand into any direction.

The blackbody emissive power spectrum has first been derived by Max Planckin his famous work on Quantum Statistics [65]. In standard textbooks on QuantumMechanics nowadays it is usually derived in the context of second quantization ofthe electromagnetic field. Here, we want to give a different derivation by entropyminimization/maximization which fits into the context of this work.

If N(x, p) is the average number of photons with position x and momentum p ina phase space element of volume h3, where h is Planck’s constant, then

∫ ∫N(x, p)

dxd ph3 (14)

is the number of photons in the phase space volume under consideration. Photonsare integer-spin particles and obey Bose–Einstein statistics. According to a standardresult [32] from statistical physics, the entropy of an ensemble of bosons is

S =−2k∫ ∫

(N logN− (N + 1) log(N + 1))dxd p

h3 . (15)

Another standard result [32] relates the spectral intensity ψ and the numberdensity N,

N =c2

2hν3 ψ . (16)


The momentum can be written in terms of frequency and direction (“sphericalcoordinates”) as

p =hνc

Ω , (17)

thus

d p =(

hc

)3

ν2dνdΩ . (18)

Consequently, the entropy density of the radiation field is

H =−∫ ∫

2kν2

c3 (N logN− (N + 1) log(N + 1))dνdΩ . (19)

We want to define the mathematical entropy as

HR =−S. (20)

According to the Second Law of Thermodynamics, the entropy is a non-decreasingfunction of time. Thus, the equilibrium distribution for a given temperature has tomaximize S, or equivalently minimize HR. This principle yields the Planck equilib-rium distribution

B(ν,T ) =2hν3

c2

1

exp( hνkT )−1

, (21)

which describes the emissive spectrum of a blackbody. Blackbody emissive powerspectrum in nondimensional coordinates. In this derivation we made use of theStefan–Boltzmann law,

B(T ) =∫ ∞

0B(ν,T )dν = σSBT 4, (22)

with the Stefan–Boltzmann constant σSB = 5.670 ·10−8 Wm2K4 , which gives the cele-

brated dependence of the total emissive power of a blackbody on the fourth powerof its temperature.

2.3 The Transfer Equation

If the medium through which radiative energy travels is participating, then any inci-dent beam will be affected by absorption and scattering while it travels through themedium. In the following, we want to consider a medium at rest (compared to thespeed of light) and with constant refractive index. Furthermore, it is assumed thatthe medium is nonpolarizing and that it is in local thermodynamical equilibrium.For a very thorough discussion of these limitations see [88].


First we want to derive a discrete transfer equation and then, by passing to thelimit, obtain the integro-differential equation describing radiative transfer.

Let us assume that there is only a finite set of directions (Ω j) into which thephotons can travel. Consider a beam into direction Ωi which travels a distance Δsthrough the medium. Several effects can lead to the augmentation and reduction ofthe beam.

2.3.1 Absorption

When a photon hits an atom or molecule inside the medium with the right amountof energy it can be absorbed, thus leading to an excited state of the atom/molecule.The amount of absorbed photons is directly proportional to the distance traveledand to the number of photons itself. Thus the change in the spectral intensity due toabsorption is

(Δψ)abs =−κψΔs. (23)

2.3.2 Scattering

The photons can hit atoms or molecules in the medium and change their direction.We assume that the energy (or frequency) of the photons does not change (elasticscattering). We denote the fraction of photons that change their direction from Ω j

to Ωi by Si j. Note that the normalization condition

∑i

Si j = 1 (24)

has to hold. The win/loss balance reads

(Δψ(Ωi))scat = σ

(−ψ(Ωi)+∑

j

Si jψ(Ω j)

)Δs. (25)

2.3.3 Emission

If the medium has a finite temperature then it also emits thermal radiation which isdistributed as blackbody radiation. The emitted intensity along a path is again pro-portional to the length of the path. If the spectral intensity of the photons ψ were aPlanckian itself there should be no net absorption/emission. Hence the proportion-ality constant must be κ . Thus the intensity change caused by emission is

(Δψ)em = κBΔs. (26)


2.3.4 Overall Balance

Drawing a balance of the different effects, we obtain the discrete transfer equation

ψ(s+ Δs,Ωi) = ψ(s,Ωi)

+Δs

(κ(B(T )−ψ(Ωi))+ σ

(∑

j

Si jψ(Ω j)−ψ(Ωi)

)). (27)

The sum on the right hand side can be interpreted as a numerical quadrature rule.The matrix Si j can be interpreted as the evaluation of a function,

Si j = s(Ωi,Ω j). (28)

If we assume that the set of directions is continuous, then the summation over alldirections becomes an integration over the unit sphere. We obtain for all Ω ∈ S2,

ψ(s+ Δs,Ω) = ψ(s,Ω)

+Δs

(κ(B(T )−ψ(Ω))+σ

(∫S2

s(Ω ,Ω ′)ψ(Ω ′)dΩ ′−ψ(Ω)))

.

(29)

The normalization property (24) becomes

∫

S2s(Ω ,Ω ′)dΩ ′ = 1. (30)

A beam travels a distance Δx in a time Δxc , where c is the speed of light. Thus we

have

ψ(t + Δx/c,x + ΩΔx,Ω) = ψ(t,x,Ω)

+ Δx

(κ(B(T )−ψ(t,x,Ω))+ σ

(∫S2

s(Ω ,Ω ′)ψ(t,xΩ ′)dΩ ′ −ψ(t,x,Ω)))

.

(31)

Taking the limit Δx→ 0 we arrive at the radiative transfer equation. The frequencyν can be incorporated as an additional parameter. At a position x and a time t, forall directions Ω ∈ S2, for all frequencies ν ∈ [0,∞] it holds

1c

∂tψ(t,x,Ω ,ν)+ Ω∇ψ(t,x,Ω ,ν)

= κ(B(ν,T )−ψ(t,x,Ω ,ν))+ σ(∫

S2s(Ω ,Ω ′)ψ(t,x,Ω ′,ν)dΩ ′ −ψ(t,x,Ω ,ν)

).

(32)


We will also consider the special case of a one-dimensional slab geometry. Weconsider a plate which is finite in one dimension and infinite in the other dimensions.Thus, the intensity depends only on one space variable and is axially symmetric.Hence the equation simplifies to

1c

∂tψ(t,x,μ ,ν)+ μ∂xψ(t,x,μ ,ν)

= κ(2πB(ν,T)−ψ(t,x,μ ,ν))+ σ(

12

∫ 1

−1ψ(t,x,μ ′,ν)dμ ′ −ψ(t,x,μ ,ν)

).

(33)

Here, μ is the cosine of the angle between direction and x-axis.There are several other versions of the transfer equation, that we will consider in

the following. First of all, in most applications, the time scale is much larger than thetime the radiation needs to propagate into the medium. Thus we neglect the time-derivative and thus obtain the steady transfer equation. The absorption coefficientκ and the scattering coefficient σ can in general also depend on position and time.In the following, we want to assume isotropic scattering. This means that the scat-tering kernel is actually a constant, s = 1

4π . For the purpose of glass, it often sufficesto consider only absorption.

If frequency-dependence is not important, the so-called grey approximation canbe used, meaning that all quantities are frequency-dependent. For glass manufactur-ing, however, frequency-dependence is important. Table 2 shows typical absorptioncoefficients for glass, depending on frequency. For smaller frequencies, absorptionbecomes very large. For all practical purposes, glass is perfectly opaque for frequen-cies smaller than a limit ν1.

2.4 Overall Energy Conservation

The radiation field, by emission, strongly depends on the temperature of themedium. On the other hand, by absorption, it also affects the temperature of themedium. For an overall energy balance we have to take into account this connection.

Table 2 Eight frequency bands for glass

Band ι νι νι+1 κι

1 ∞ 5 0.402 5 0.3333 0.503 0.3333 0.2857 7.704 0.2857 0.2500 15.455 0.2500 0.2222 27.986 0.2222 0.1818 267.987 0.1818 0.1666 567.328 0.1666 0.1428 7136.06

0.1428 0 Opaque


Also, we have to consider the two other modes of energy transfer, heat conductionand convection.

The general energy conservation equation for a moving compressible fluid maybe stated as [66]

ρmDuDt

= ρ(∂tu + v∇u) =−∇q− p∇v + μΦ + Q′′′, (34)

where u is the internal energy, v is the velocity vector, q is the total heat flux vector,Φ is the dissipation function, and Q′′′ is the heat generated within the medium.

If the medium interacts with the radiation field through emission, absorptionand scattering, then the heat flux term q in (34) contains the radiative heat flux.The radiative contributions to the internal energy and the pressure tensor can beneglected [66].

If we assume that du = cmdT and furthermore that Fourier’s law of heatconductivity holds,

q = qcon + F =−k∇T + F, (35)

then (34) becomes

ρmcm(∂tT + v∇T ) = ∇k∇T − p∇v + μΦ + Q′′′ −∇F. (36)

In the following we want to restrict ourselves to a fluid at rest or a solid, i.e. v = 0.Furthermore we want to assume Φ = 0 and Q′′′ = 0. Thus we consider

ρmcm∂tT = ∇k∇T −∇F. (37)

By integrating the transfer equation with respect to Ω and ν , we see that this can bewritten as

ρmcm∂tT = ∇k∇T −∫ ∞

0

∫

S2κ(ψ−B)dΩdν. (38)

2.5 Boundary Conditions

Consider a beam of photons hitting a slab, as shown in Fig. 1. Some of the irradiationwill be reflected at the surface. A fraction of the radiation which penetrates themedium will be absorbed, the remaining will be transmitted. Thus we define thequantities

Reflectivity ρ = reflected part of radiationincoming radiation

Absorptivity α = absorbed part of radiationincoming radiation

Transmittivity τ = transmitted part of radiationincoming radiation .


Fig. 1 Reflection,absorption, and transmission

By definition, ρ +α +τ = 1. A medium is called opaque if τ = 0, i.e. no radiationis transmitted. For a black body, α = 1 and ρ = τ = 0.

Radiative energy can also be emitted inside a medium and can be releasedthrough the surface. Since a blackbody is a perfect emitter, we define

emissivity ε = energy emitted from surfaceenergy emitted from a blackbody at the same temperature .

Radiative transfer is a long-range phenomenon. In principle, if we want to knowthe amount of radiation at one point x, we have to take into account radiation ar-riving from any direction and any point in space. Thus, an energy balance mustbe performed either over the whole space or over an enclosure bounded by opaquewalls. When speaking of a wall or surface we actually mean a small layer (comparedto the size of the enclosure) where radiation is reflected, absorbed and emitted.

Consider a domain bounded by an opaque surface and let n denote the outwardnormal vector. For a point on the boundary we have the following energy balancefor all incoming directions, i.e. all Ω with n ·Ω < 0,

ψ(x,t,Ω) = ρ(Ω ′)ψ(x,t,Ω ′)+ (1−ρ(Ω))ψb. (39)

Here, Ω ′ = Ω −2(Ω ·n)n is the outgoing direction that is reflected into Ω . Further-more, Iν,b is the amount of radiation emitted from the surface, cf. Fig. 2.

Mostly, we will assume that the incoming radiation is a Planckian at sometemperature,

ψb = B(Tb). (40)

The reflectivity ρ generally depends on the direction Ω . It can be computed usingSnell’s law. On the interface between two media with refraction indexes n1 and n2,the refraction angle (with respect to the normal) θ2 of the transmitted ray and theincident angle θ1 are related by

n1 sinθ1 = n2 sinθ2. (41)


Fig. 2 Boundary condition

The reflectivity is then given by Fresnel’s equation

ρ =12

[tan2(θ1−θ2)tan2(θ1 + θ2)

+sin2(θ1−θ2)sin2(θ1 + θ2)

]. (42)

In the case of total refection, ρ = 1.The heat equation (38) can for example be supplemented with the following

boundary conditions. Assuming that the opaque body surrounding the medium un-der consideration is a gas in a large reservoir, we can consider the heat flux throughthe boundary due to advection

k∂T∂n

= h(Tb−T). (43)

Here, Tb is the outside temperature. We want to note that the value of the parameterh has to be determined by experiment. Also, we have to emphasize that the modelingof heat exchange by advection is actually quite sophisticated and still a subject of re-search. However, we will to assume in the following the simple boundary conditionsstated above.

2.6 Summary

In order to model glass cooling we consider the following radiative transfer problem.For a point x in the domain V ⊂ R3

cmρm∂T∂ t

= ∇ · k∇T −∫ ∞

ν1

∫

S2κ(B− I) dΩdν, (44a)

∀ν > ν1 : Ω ·∇ψ = κ(B−ψ)+ σ(φ−ψ), (44b)


where

φ(x,Ω ,ν) =1

4π

∫

S2s(Ω ,Ω ′)ψ(x,Ω ′,ν)dΩ ′

is the scattered intensity. Here, we have explicitly included the opaque band forfrequencies in the interval [0,ν1], where absorption is infinitely high and thus ψ ≡B.

On the boundary, for x ∈ ∂V , the ingoing radiation is prescribed by semi-transparent boundary conditions

I(Ω) = ρ(n ·Ω)I(Ω ′)+(1−ρ(n ·Ω)

)Ib(Ω), ∀n ·Ω < 0, (44c)

while the temperature is assumed to obey

k n ·∇T = h(Tb−T )+ απ(n2

n1

)2 ∫ ν1

0B(ν,Tb)−B(ν,T ) dν. (44d)

The additional term comes from the fact that in the opaque band 0 < ν < ν1 themedium behaves like a perfect black body.

At initial time t = 0, the temperature shall be given by

T (x,0) = T0(x). (44e)

In these equations, ψ(t,x,Ω ,ν) denotes the specific radiation intensity at pointx ∈V traveling in direction Ω ∈ S2 with frequency ν > 0 at time t ≥ 0. The out-side radiation Ib is assumed to be known for the ingoing directions (i.e. n ·Ω < 0)on the boundary. We denote the outward normal on ∂V by n. Furthermore, T (t,x)denotes the material temperature and Tb is the exterior temperature on the bound-ary. The equations contain as parameters the opacity κ , the scattering coefficient σ ,the heat conductivity k and the convective heat transfer coefficient h. Moreover, Bdenotes Planck’s function

B(ν,T ) = n21

2hPν3

c2

(e

hPνkBT −1

)−1

for black body radiation in glass which involves Planck’s constant hP, Boltzmann’sconstant kB and the speed of light in vacuum c. The integration in the second termof the temperature boundary condition (44d) is done on the opaque interval of thespectrum [0,ν1], where radiation is completely absorbed. At the interface betweenglass and surrounding air with refractive indices n1 > n2, respectively, light rays arereflected and refracted. This is modeled by the so-called semi-transparent boundaryconditions (44c). The reflectivity ρ ∈ [0,1] is the proportion of radiation that isreflected. It is equal to 1 if total reflection occurs i.e. if θ1 > θc where θc is the criticalangle given by sinθc = n2/n1. Otherwise ρ is calculated according to Fresnel’sequation

ρ(μ) =12

( tan2(θ1−θ2)tan2(θ1 + θ2)

+sin2(θ1−θ2)sin2(θ1 + θ2)

),


where the refraction angles θ1 and θ2 are given by cosθ1 = |n ·Ω |= μ and Snell’slaw of refraction

n1 sinθ1 = n2 sinθ2.

The solid angles of the reflected ray in (44c) is

Ω ′ = Ω −2(n ·Ω)n.

Finally, the hemispheric emissivity α of the boundary surface in (44d) is related tothe reflectivity ρ by

α = 2n1

∫ 1

01−ρ(μ) dμ .

For these equations and other applications in glass manufacturing problems we refer,for example, to [39, 40, 46, 87], and the monographs [31] and [60].

Analytical results concerning the existence and uniqueness of solutions to thetransfer equation itself and to the radiative heat transfer equations, where also energyconservation and additionally heat conduction are considered, have been obtainedby many authors. A rather recent review on methods for transport equations can befound in [6], cf. also [5]. The transfer equation together with energy conservationis considered in [28, 58]. The issue of heat conduction is addressed in [37, 43, 44].Convection, conduction and radiation is treated in [53, 69].

3 Direct Numerical Methods

The main difficulties in solving numerically the integro-differential equation (44)are the large set of unknowns and the coupling between the transport and the in-tegral operators. For instance, ψ is a function of time variable t, space variable x,frequency variable ν , and direction variable Ω . Solving the large linear system ofalgebraic equations induced by discretizing these variables is computationally verydemanding.

Here, we focus on the solution of steady-state, mono-energetic, frequency decou-pled, isotropic radiative transfer problems in three space dimensions. However, allthe methods presented in this paper can be straightforwardly extended to the moregeneral problem (44). Hence the radiative transfer equation we consider reads

Ω ·∇ψ +(σ + κ)ψ = σφ + κB (45)

with boundary values ψ = g. The (45) models the changes of an intensity ψ(x,Ω) asparticles are passing through the domain V at position point x = (x,y,z)T in the di-rection Ω = (μ ,η ,ξ )T and are subject to loses due to absorption κ and scattering σ ,while their number grows due to the source B inside the domain V . We assume thatσ and κ are nonnegative functions and we introduce the mean intensity φ as

φ(x) =1

4π

∫

S2ψ(x,Ω ′)dΩ ′. (46)


We also define the scattering ratio γ and the opacity coefficient ϑ as

γ = maxx∈V

(σ(x)

σ(x)+ κ(x)

)and ϑ = min

x∈V

(σ(x)+ κ(x)

)diam(V ). (47)

Here diam(V ) is the diameter of the space domain V . In applications, γ and ϑ areused to characterize the convergence rates of the iterative methods and the diffusionlimits in the optically thick medium.

Many numerical methods have been used to solve the (45). For a review on someof these methods see [52,75]. It is well known [1] that the standard Source Iteration(SI) becomes extremely costly when the scattering ratio γ ≈ 1. The standard Dif-fusion Synthetic Acceleration (DSA) has been widely used to accelerate the sourceiteration [4, 10]. The SI and DSA methods can be seen respectively, as Richardsoniteration and preconditioned Richardson iteration with the diffusion approach aspreconditioner.

We implement SI, DSA and a Krylov subspace method to solve the (45). Wealso propose a fast multilevel algorithm [36] which uses the approximate inverseoperator as a preconditioner and solves the linear system only in the coarse meshes.Numerical results show this algorithm to be faster than DSA in many regimes. Therobustness, efficiency and convergence rates of these methods are illustrated by sev-eral numerical test examples in both one and two space dimensions. Comparisonof the results obtained by different methods is also included in this section. Thematerial in this section is taken from [76].

3.1 Ordinates and Space Discretizations

We start with a discrete ordinates discretization in angle. This corresponds to ex-panding the integrals on the unit sphere S2 in terms of N weighted quadrature rules,

∫

S2ψ(x,Ω)dΩ

N

∑l=1

wlψ(x,Ωl), (48)

where Ωl = (μl,ηl ,ξl)T , for all l = 1, . . . ,N, with N = n(n+2), and n is the numberof direction cosines. Since Ωl ∈ S2, we have

μ2l + η2

l + ξ 2l = 1, for all l = 1,2, . . . ,N.

We assume n an even number of quadrature points so that the points (μl ,ηl,ξl) arenonzero, symmetric with respect to the x-, y- and z-axis and they are invariant under90◦ rotations. Furthermore they satisfy the relation

ξ 2i = ξ 2

1 + 2i−1n−2

(1−3ξ 21 ),

for i = 1,2, . . . ,n/2 and 0 < ξ1 < 1/3.


In (48) wl are the corresponding weights chosen to be positive and satisfy

N

∑l=1

Ωl = 4π ,N

∑l=1

Ωlμl = 0,N

∑l=1

Ωlηl = 0 andN

∑l=1

Ωlξl = 0.

In practice we choose wl = 4π/N and (μl,ηl ,ξl) are set in such a way the aboveconditions are guaranteed. Let SN be a given set of N discrete directions in S2, thena semi-discrete formulation of (45) is

μl∂ψl

∂x+ ηl

∂ψl

∂y+ ξl

∂ψl

∂ z+(σ + κ)ψl = σφ(x)+ κq(x), (49)

ψl(x) denotes approximation to ψ(x,μl ,ηl ,ξl) and φ is given by

φ(x) =1

4π

N

∑l=1

wlψl(x).

To discretize the (49) in space we suppose for simplicity, that the spatial domain isa box, V = [ax,bx]× [ay,by]× [az,bz]. Then we cover the domain V with a uniformnumerical mesh defined by

Vh ={

xi jk = (xi,y j,zk)T , xi = iΔx, y j = jΔy, zk = kΔz,

i = 0,1 . . . , I, j = 0,1 . . . ,J, k = 0,1 . . . ,K},

where x0 = ax, xI = bx; y0 = ay, yJ = by; z0 = az, zK = bz; and h denotes the maxi-mum cell size. We define the averaged grid points

Δx = xi− xi−1, Δy = y j− y j−1, Δz = zk− zk−1,

xi− 12

=xi−1 + xi

2, y j− 1

2=

y j−1 + y j

2, zk− 1

2=

zk−1 + zk

2,

for i = 1 . . . , I, j = 1 . . . ,J and k = 1 . . . ,K. By using the notation fi jk to denote theapproximation value of the function f at the grid point (xi,y j,zk), the fully discreteapproximation for the (45) can be written as

μlψl,i jk−ψl,i−1 jk

Δx + ηlψl,i jk−ψl,i j−1k

Δy + ξlψl,i jk−ψl,i jk−1

Δ z

+(σi− 1

2 j− 12 k− 1

2+ κi− 1

2 j− 12 k− 1

2

)ψl,i− 1

2 j− 12 k− 1

2

= σi− 12 j− 1

2 k− 12φi− 1

2 j− 12 k− 1

2κi− 1

2 j− 12 k− 1

2qi− 1

2 j− 12 k− 1

2, (50)

where the cell averages values of ψ are given by


ψl,i−1 jk =1

Δx

∫ y j

y j−1

∫ zk

zk−1

ψl(xi,y,z)dydz,

ψl,i j−1k =1

Δy

∫ xi

xi−1

∫ zk

zk−1

ψl(x,y j,z)dxdz,

ψl,i jk−1 =1

Δz

∫ xi

xi−1

∫ y j

y j−1

ψl(x,y,zk)dxdy,

ψl,i jk =1

ΔxΔyΔz

∫ xi

xi−1

∫ y j

y j−1

∫ zk

zk−1

ψl(x,y,z)dxdydz, (51)

In this paper we use the Diamond difference method to approximate the fluxes in(51). The method consists on centred differences and approximating the functionvalues at the cell centres fl,i− 1

2 j− 12 k− 1

2by the average of their values at the eight

neighbouring nodes as

fl,i− 12 j− 1

2 k− 12

=18

[fl,i−1 j−1k−1 + fl,i−1 jk−1 + fl,i−1 j−1k + fl,i−1 jk

+ fl,i j−1k−1 + fl,i jk−1 + fl,i j−1k + fl,i jk

]. (52)

Hence the discrete mean intensity φi− 12 j− 1

2 k− 12

in (50) is given by

φi− 12 j− 1

2 k− 12

=N

∑l=1

wlψl,i− 12 j− 1

2 k− 12.

Other discretizations using Legendre polynomial collocation in ordinates and finiteelement or Petrov–Galerkin methods in space can be used in the same manner, werefer to [10, 52, 81, 82] for details. For the discretization of the boundary conditionsin (49) we can proceed as follows:

when x = ax, the normal n(x0 jk) = (−1,0,0)T , then n(x0 jk) ·Ωl = −μl, and forμl > 0 we have ψl,0 jk = g0 jk

when y = ay, the normal n(xi0k) = (0,−1,0)T , then n(xi0k) ·Ωl = −ηl , and forηl > 0 we have ψl,i0k = gi0k

when z = az, the normal n(xi j0) = (0,0,−1)T , then n(xi j0) ·Ωl = −ξl , and forξl > 0 we have ψl,i j0 = gi j0

The other three cases can be discretized in a similar way. Needless to say that for agiven l = 1,2, . . . ,N no component of Ωl is ever zero and only three of the above sixcases can hold. Furthermore, in the discretization (50) there are (I +1)(J+1)(K+1)unknowns ψl,i jk and IK + JI + JK + I + J + K + 1 boundary equations.

3.2 Linear System Formulation

In order to simplify the notations and to get closer to a compact linear algebra for-mulation of (50), we first define the matrix entries


dl,i− 12 j− 1

2 k− 12

=|μl|Δx

+|ηl |Δy

+|ξl |Δz

+σi− 1

2 j− 12 k− 1

2+ κi− 1

2 j− 12 k− 1

2

8,

el,i− 12 j− 1

2 k− 12

=−|μl|

Δx+−|ηl|

Δy+−|ξl|

Δz+

σi− 12 j− 1

2 k− 12+ κi− 1

2 j− 12 k− 1

2

8.

and

ul,i− 12 j− 1

2 k− 12

=|μl|Δx

+|ηl|Δy

+−|ξl|Δz

+σi− 1

2 j− 12 k− 1

2+ κi− 1

2 j− 12 k− 1

2

8,

ul,i− 12 j− 1

2 k− 12

=|μl|Δx

+−|ηl|

Δy+|ξl|Δz

+σi− 1

2 j− 12 k− 1

2+ κi− 1

2 j− 12 k− 1

2

8,

ul,i− 12 j− 1

2 k− 12

=|μl|Δx

+−|ηl|

Δy+−|ξl|

Δz+

σi− 12 j− 1

2 k− 12+ κi− 1

2 j− 12 k− 1

2

8,

vl,i− 12 j− 1

2 k− 12

=−|μl|

Δx+|ηl |Δy

+|ξl|Δz

+σi− 1

2 j− 12 k− 1

2+ κi− 1

2 j− 12 k− 1

2

8,

vl,i− 12 j− 1

2 k− 12

=−|μl|

Δx+|ηl |Δy

+−|ξl|

Δz+

σi− 12 j− 1

2 k− 12+ κi− 1

2 j− 12 k− 1

2

8,

vl,i− 12 j− 1

2 k− 12

=−|μl|

Δx+−|ηl|

Δy+|ξl|Δz

+σi− 1

2 j− 12 k− 1

2+ κi− 1

2 j− 12 k− 1

2

8,

Next, we define the vectors

Ψl ≡

⎛⎜⎝

Ψl,0...

Ψl,K

⎞⎟⎠ ∈ R(I+1)(J+1)(K+1), with

Ψl,k ≡

⎛⎜⎝

Ψl,0k...

Ψl,Jk

⎞⎟⎠ ∈ R(I+1)(J+1), Ψl, jk ≡

⎛⎜⎝

ψl,0 jk...

ψl,I jk

⎞⎟⎠ ∈ R(I+1);

Φ ≡

⎛⎜⎜⎝

Φ1− 12

...ΦK− 1

2

⎞⎟⎟⎠ ∈ RIJK , with

Φk− 12≡

⎛⎜⎜⎝

Φ1− 12 k− 1

2...

ΦJ− 12 k− 1

2

⎞⎟⎟⎠ ∈ RIJ , Φ j− 1

2 k− 12≡

⎛⎜⎜⎝

φ1− 12 j− 1

2 k− 12

...φI− 1

2 j− 12 k− 1

2

⎞⎟⎟⎠ ∈ RI;


Q≡

⎛⎜⎜⎝

Q1− 12

...QK− 1

2

⎞⎟⎟⎠ ∈ RIJK , with

Qk− 12≡

⎛⎜⎜⎝

Q1− 12 k− 1

2...

QJ− 12 k− 1

2

⎞⎟⎟⎠ ∈ RIJ , Q j− 1

2 k− 12≡

⎛⎜⎜⎝

q1− 12 j− 1

2 k− 12

...qI− 1

2 j− 12 k− 1

2

⎞⎟⎟⎠ ∈ RI;

In what follows we define the matrix Hl (known as sweep matrix) for the first sweepcase μl < 0, ηl < 0, ξl < 0 and the other seven sweep cases can be derived similarly.In order to simplify the notation, we drop hereafter the space grids subscripts fromthe matrix entries unless otherwise stated. Thus,

Hl ≡

⎛⎜⎜⎜⎜⎜⎝

Dl El. . .

. . .

Dl El

Dl SS

⎞⎟⎟⎟⎟⎟⎠∈ R(I+1)(J+1)(K+1)×(I+1)(J+1)(K+1), with

Dl ≡

⎛⎜⎜⎜⎝

Dl Ul. . .

. . .

Dl Ul

S

⎞⎟⎟⎟⎠ ∈R(I+1)(J+1)×(I+1)(J+1), with

Dl ≡

⎛⎜⎜⎜⎝

d u. . .

. . .

d u1

⎞⎟⎟⎟⎠ ∈ R(I+1)×(I+1), Ul ≡

⎛⎜⎜⎜⎝

u u. . .

. . .

u u1

⎞⎟⎟⎟⎠ ∈ R(I+1)×(I+1);

El ≡

⎛⎜⎜⎜⎝

Vl Wl. . .

. . .

Vl Wl

S

⎞⎟⎟⎟⎠ ∈ R(I+1)(J+1)×(I+1)(J+1), with

Vl ≡

⎛⎜⎜⎜⎝

v v. . .

. . .v v

1

⎞⎟⎟⎟⎠ ∈R(I+1)×(I+1), Wl ≡

⎛⎜⎜⎜⎝

v e. . .

. . .v e

1

⎞⎟⎟⎟⎠ ∈ R(I+1)×(I+1).


S ≡

⎛⎜⎜⎜⎝

S S. . .

. . .

S SS

⎞⎟⎟⎟⎠ ∈ R(I+1)(J+1)×(I+1)(J+1), with

S≡

⎛⎜⎜⎜⎝

1 1. . .

. . .1 1

1

⎞⎟⎟⎟⎠ ∈ R(I+1)×(J+1).

Σl ≡

⎛⎜⎜⎜⎜⎝

Σl,1− 12

. . .

Σl,K− 12

0

⎞⎟⎟⎟⎟⎠∈ R(I+1)(J+1)(K+1)×IJK , with

Σl,k− 12≡

⎛⎜⎜⎜⎜⎝

Σl,1− 12 k− 1

2. . .

Σl,J− 12 k− 1

2

0

⎞⎟⎟⎟⎟⎠∈ R(I+1)(J+1)×IJ, and

Σl, j− 12 k− 1

2≡

⎛⎜⎜⎜⎜⎜⎝

σ1− 1

2 j− 12 k− 1

2+κ

i− 12 j− 1

2 k− 12

8. . .

σi− 1

2 j− 12 k− 1

2+κ

i− 12 j− 1

2 k− 12

80

⎞⎟⎟⎟⎟⎟⎠∈R(I+1)×I.

Using these definitions with Ψ and Φ being the unknowns, the fully discreteequation (50) can be written in matrix form as

⎛⎜⎜⎜⎝

H1 −Σ1. . .

...HN −ΣN

−w14π S . . . −wN

4π S I

⎞⎟⎟⎟⎠

⎛⎜⎜⎜⎝

Ψ1...

ΨN

Φ

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎝

Q1...

QN

0

⎞⎟⎟⎟⎠ , (53)

where I is the IJK× IJK identity matrix and 0 is the IJK null vector. The linearsystem (53) can be rewritten in common linear algebra notation as

Ax = b (54)


with

A≡

⎛⎜⎜⎜⎝

H1 −Σ1. . .

...HN −ΣN

−w14π S . . . −wN

4π S I

⎞⎟⎟⎟⎠ , x≡

⎛⎜⎜⎜⎝

Ψ1...

ΨN

Φ

⎞⎟⎟⎟⎠ , and b≡

⎛⎜⎜⎜⎝

Q1...

QN

0

⎞⎟⎟⎟⎠ .

3.3 Preconditioning Techniques

In computational radiative transfer the desired quantity is usually the mean intensityΦ which is a function only of position x. Therefore we use the Gaussian eliminationto eliminate the intensity Ψ1, . . . ,ΨN from (53) and the reduced equation

(I− 1

4π

N

∑l=1

wlSH−1l Σl

)Φ =

14π

N

∑l=1

wlSH−1l Ql , (55)

is solved for Φ . We rewrite (55) in compact form as

(I−A

)Φ = f , (56)

where the Schur matrix A and the right hand side f are given by

A =1

4π

N

∑l=1

wlSH−1l Σl and f =

14π

N

∑l=1

wlSH−1l Ql.

In this section we briefly discuss some numerical methods used in the literature tosolve the linear system (56).

3.3.1 Source Iteration

The most popular iterative method to solve (55) is the Richardson iteration knownin the radiative transfer community as Source Iteration (SI) method. Given an initialguess Φ(0), the (m+ 1)-iterate solution is obtained by

Φ(m+1) =1

4π

N

∑l=1

wlSH−1l

(Ql + ΣlΦ(m)

), m = 0,1, . . . . (57)

It is easy to see that iteration (57) is equivalent to preconditioned block Gauss–Seidel method applied to (54), where the preconditioner is the block lower triangleof the matrix A. Thus, if M is the block lower triangle of A,


M ≡

⎛⎜⎜⎜⎝

H1. . .

HN

−w14π S . . . −wN

4π S I

⎞⎟⎟⎟⎠ ,

x and b are as given in (54), then

Mx(m+1) = (M−A)x(m) + b,

and

x(m+1) = (I−M−1A)x(m) + M−1b. (58)

Therefore the (m+ 1)-iterate mean intensity satisfy

Φ(m+1) =1

4π

N

∑l=1

wlSH−1l Ψ (m+1)

l =1

4π

N

∑l=1

wlSH−1l

(Ql + ΣlΦ(m)

),

which is identical to (57).Formal results from linear algebra [29, 35] demonstrate that the preconditioned

Richardson iteration (58) converges rapidly as long as the norm of the matrix (I−M−1A) is small. This condition is ensured by taking the scattering ratio γ small,compare [1] for analysis. For γ � 1 the SI method converges rapidly, but for γ ≈ 1(large optical opacity) convergence becomes slow and may restrict the efficiency ofthe SI algorithm. The SI algorithm can be implemented as follows

Algorithm 1: SI algorithm

given the initial guess Φ(0)

do m = 0,1, . . . , itmaxdo l = 1,2, . . . ,N

a. set wl ←− Ql + ΣlΦ(m)

b. solve for yl: Hlyl = wl

c. set wl ←− Syl

end do

d. compute Φ(m+1) =1

4π

N

∑l=1

wlwl

e. compute r(m) = Φ(m+1)−Φ(m)

if

(‖r(m)‖L2

‖r(0)‖L2≤ τ

)stop

end do

Here itmax is the maximum number of the iterations m, τ is a given tolerance and‖.‖L2 is the discrete L2-norm. The step (b) can be solved directly using Gaussianelimination known in computational radiative transfer as sweeping procedure. Addi-tionally, for each direction Ωl in SN only one of the eight possible sweeps is needed.


3.3.2 Diffusion Synthetic Acceleration

Among the methods used to accelerate the SI algorithm are the syntheticacceleration procedures [1, 4]. The procedures consist on splitting the SI in to two-step iterations. Thus, we denote ψ(m+ 1

2 ) as first iteration for the SI in the continuousform of the problem (45),

Ω ·∇ψ(m+ 12 ) + (σ + κ)ψ(m+ 1

2 ) =σ4π

∫

S2ψ(m)(x,Ω ′)dΩ ′+ κq(x),

ψ(m+ 12 )(x,Ω) = g(x), (59)

and an equation for ψ(m+1) is required in such a way to be more accurate approxi-mation to ψ than ψ(m+ 1

2 ). To perform this step with synthetic acceleration method,we subtract (59) from (45),

Ω ·∇(ψ−ψ(m+ 12 ))+(σ + κ)

(ψ−ψ(m+ 1

2 )) =σ4π

∫

S2

(ψ−ψ(m))(x,Ω ′)dΩ ′,

(ψ−ψ(m+ 1

2 ))(x,Ω) = 0, (60)

then (60) are replaced by an approximate problem. The Diffusion Synthetic Accel-eration (DSA) method [4] approximates the (60) by the diffusion problem

−∇ ·( 1

3(σ + κ)∇ϕ

)+ κϕ =

σ4π

∫

S2

(ψ−ψ(m))(x,Ω ′)dΩ ′, x ∈V,

Here ϕ(x) is an approximation to the mean intensity

ϕ(x)≈ 14π

∫

S2

(ψ(m+1)−ψ(m+ 1

2 ))(x,Ω ′)dΩ ′.

Thus the (m+ 1)-iterate mean intensity is given by

φ (m+1) = φ (m+ 12 ) + ϕ .

Note that (61) does not depend on the angle variable Ω , is linear elliptic equation andsimple to solve numerically with less computational cost and memory requirement.

In order to build a discretization for the diffusion problem (61) which is consis-tent to the one used for the radiative transfer equation (45), we consider the samegrid structure and the same notations as those used in Sect. 2. Hence a space dis-cretization for the (61) reads as

−D2h

( 13(σ + κ)

ϕ)

i jk+ κi− 1

2 j− 12 k− 1

2ϕi− 1

2 j− 12 k− 1

2= pi− 1

2 j− 12 k− 1

2, (61)


where pi− 12 j− 1

2 k− 12

= σi− 12 j− 1

2 k− 12

(φ (m+ 1

2 )i− 1

2 j− 12 k− 1

2− φ (m)

i− 12 j− 1

2 k− 12

)and the difference

operator D2h is given by D2

h = D2x +D2

y +D2z , with

D2x (β ϕ)i jk =

βi jk + βi+1 jk

2ϕi+1 jk−ϕi jk

(Δx)2 − βi−1 jk + βi jk

2ϕi jk−ϕi−1 jk

(Δx)2 ,

D2y (β ϕ)i jk =

βi jk + βi j+1k

2

ϕi j+1k−ϕi jk

(Δy)2 − βi j−1k + βi jk

2

ϕi jk−ϕi j−1k

(Δy)2 ,

D2z (β ϕ)i jk =

βi jk + βi jk+1

2

ϕi jk+1−ϕi jk

(Δz)2 − βi jk−1 + βi jk

2

ϕi jk−ϕi jk−1

(Δz)2 .

The functions κi− 12 j− 1

2 k− 12, ϕi− 1

2 j− 12 k− 1

2and pi− 1

2 j− 12 k− 1

2appeared in (61) are given

as in formula (52). The gradient in the boundary conditions is approximated byupwinding without using ghost points. For example, on the boundary surface x = ax

of the domain V , the boundary discretization is

ϕ 12 j+ 1

2 k+ 12− 2

3(σ 12 j− 1

2 k− 12+ κ 1

2 j− 12 k− 1

2)

ϕ 32 j− 1

2 k− 12−ϕ 1

2 j− 12 k− 1

2

Δx= 0,

and similar work has to be done for the other boundaries. All together, the abovediscretization leads to a linear system of form

Dϕ = p, (62)

where D is IJK× IJK nonsymmetric positive definite matrix obtained from the dif-ference diffusion operator (61) with boundary conditions included, and p is IJKvector containing the right hand side term.

Once again the DSA method can be viewed as preconditioned Richardson itera-tion for the linear system (54) with the diffusion matrix D like preconditioner,

x(m+1) =(I−D−1A

)x(m) + D−1b,

and D−1 is obtained by solving the diffusion linear system (62). In terms of Φ thisis equivalent to

Φ(k+1) =(

I− (I−D−1)A)

Φ(k) + (I−D−1)b.

The implementation of DSA method to approximate the solution of the radiativetransfer equation (45) is carried out in the following algorithm

Algorithm 2: DSA algorithm

given the initial guess Φ(0)

do m = 0,1, . . . , itmax(a) – (d) are similar to Algorithm 1 for the intermediate solution Φ(m+ 1

2 )


(e) compute p = σ(Φ(m+ 1

2 )−Φ(m))

(f) solve for ϕ : Dϕ = p

(g) set Φ(m+1) = Φ(m+ 12 ) + ϕ

(h) compute r(m) = Φ(m+1)−Φ(m)

if

(‖r(m)‖L2

‖r(0)‖L2≤ τ

)stop

end do

Note that the first lines in Algorithm 2 are similar to the Algorithm 1. How-ever, the source iteration algorithm gives only the intermediate solution Φ(m+ 1

2 )

which has to be corrected by adding the solution ϕ obtained by the diffusion ap-proach. Furthermore, if iterative methods are used for the diffusion approach, thenan inner iteration loop has to be added to the iteration used by the SI algorithm andan outer SI iteration may require less accuracy from the inner iterations.

3.3.3 Krylov Subspace Methods

In general the matrices A and A in (54) and (56) respectively are nonsymmetricand not diagonally dominant. Furthermore, since σ and κ are nonnegative functionsand SN has nonzero directions, the matrix A has positive diagonal elements andnonpositive off-diagonal elements. In addition, if el,i− 1

2 j− 12 k− 1

2≤ 0, for all l, i, j,k,

then the matrix A is weakly diagonally dominant. This condition is equivalent to

h = max(Δx,Δy,Δz

) ≤maxi jk

(8|μl|

σi− 12 j− 1

2 k− 12+ κi− 1

2 j− 12 k− 1

2

,

8|ηl|σi− 1

2 j− 12 k− 1

2+ κi− 1

2 j− 12 k− 1

2

,8|ξl|

σi− 12 j− 1

2 k− 12+ κi− 1

2 j− 12 k− 1

2

), ∀ l, (63)

which means physically that the cell size is no more than eight mean free paths ofthe particles being simulated. Needless to say that the condition (63) gives the boundof the coarser mesh should be used in the computations.

In this paper we propose two Krylov subspace based methods, namely theBI-Conjugate Gradient Stabilized (Bicgstab) [86] and the Generalized MinimalResidual (Gmres(m)) [71], where m stands for the number of restarts for Krylovsubspace used in the orthogonalization. Bicgstab method has been applied early in[82] to solve (56). The main idea behind these approaches is that the Krylov sub-space methods can be interpreted as weighted Richardson iteration

x(m) =(

I−αP−1A)

x(m−1) + αP−1b, 0 < α < 2, m = 1, . . . , (64)

applied to the linear system (54), where the relaxation parameters α and the precon-ditioner P are variables within each iteration step. Note that when α = 1 and P = Mthe iteration (64) is reduced to the SI method.


The Bicgstab and Gmres(m) algorithms to solve the linear system (56) can beimplemented in the conventional way as in [36,71,86], with the only difference thatthe dense matrix A can not be explicitly stored. All what is needed, however, is asubroutine that performs a matrix-vector multiplication as shown in the followingalgorithm

Algorithm 3: Matrix-vector multiplication

given a vector u, to apply the matrix A to u we proceed asdo l = 1, . . . ,N

a. set vl ←− Σlub. solve for wl : Hlwl = vl

c. set vl ←− Swl

end do

d. set u←− u− 14π

N

∑l=1

wlvl

Note that only three vectors (u, vl and wl) are needed to perform the multiplicationof the matrix A to the vector u. Moreover, only three calls for the algorithm 3 arerequired from the Bicgstab or Gmres(m) subroutines.

Preconditioned Bicgstab or Gmres(m) methods can be also used. For instance,in the case when the matrix A is diagonally dominant, the Bicgstab or Gmres(m)methods can be accelerated by using the diagonal as a preconditioner. This approachwhich requires additional computational work can be easily implemented. It is worthto say that incomplete Cholesky or ILU type preconditioners can not be used to solve(56) because the matrix A is never formed explicitly.

3.4 A Fast Multilevel Preconditioner

We describe in this section multilevel solvers for the linear system (56) using anapproximate inverse operator as preconditioner on each level of the multigrid hier-archy. Multilevel methods were first applied to radiative transfer problems in [36].The author proposes two different type of smoothings to approximate solutions forthe one dimensional version of (45) in slab geometry.

To formulate multilevel solvers we first modify our notation slightly. Using thediscretizations introduced in Sect. 2 we assume for simplicity, a given sequence ofuniform, equidistant nested grids

V1 ⊂V2 ⊂ ·· · ⊂VL−1 ⊂VL = Vh,

on V with respective mesh sizes Δx = Δy = Δz = 2−l, l = 1, . . . ,L. We use thesubscripts l and L to refer to the coarse and fine level respectively. Therefore theproblem statement (56) becomes

(I−AL

)ΦL = fL. (65)


With M being the iteration matrix, multilevel can be written as follows

MΦ(m+1)L +

(I−AL−M

)Φ(m)

L = fL.

This formulation is equivalent to,

Φ(m+1)L = Φ(m)

L + M−1(

fL− (I−AL)Φ(m)L

)= Φ(m)

L + M−1r(m),

where r denotes the residual associated to (65) and is defined by

r = fL− (I−AL)ΦL.

The preconditioner we consider in this section is the Atkinson–Brakhage approxi-mate inverse [8] given as

M−1 = BLl = I +(I−Al)−1AL. (66)

Then the (m+ 1)-iterate solution for (65) is simply

Φ(m+1)L = Φ(m)

L + BLl r(m).

Analysis of convergence for this kind of multilevel methods has been done in [36].The central ideas in this analysis are the strong convergence and collective compact-ness of the operators generated by Al .

Let us first define the two-level (Grid2) algorithm. Applying the multilevel pre-conditioner (66) to the problem (65) we need the fine-to-coarse grid transfer operatorR l

L defined byR l

LΦL = Φl,

and the coarse-to-fine grid transfer operator PLl defined by

PLl Φl = ΦL.

A natural way to choose these operators is, bilinear interpolation for PLl and simple

injection for R lL as in the standard multigrid literature [30]. However, for radiative

transfer equation with discontinuous variables these operators have to be changedto those given in [3] which are specially designed for problems with jumping coef-ficients. Note that to use these operators we require that any discontinuity of κ , σor q in (45) is a spatial mesh point.

The Grid2 algorithm for solving (65) is detailed in the following steps

Algorithm 4: Grid2 algorithm

given the fine level {L,AL, fL}, the coarse level {l,Al, fl}, the initial guess

Φ(0)L and the tolerance τ

do m = 0,1, . . . , itmax


a. compute the residual r(m) = fL− (I−AL)Φ(m)L

b. set uL←−ALr(m)

c. restriction ul ←−R lLuL

d. solve on the coarse level for vl: (I−Al)vl = ul

Nystrom interpolatione. compute wl ←−Alvl

f. prolongation wL←−PLl wl

g. set zL←− uL + wL

h. compute the preconditioner p←− r + zL

i. update the solution Φ(m+1)L = Φ(m)

L + p

j. compute the residual r(m+1) = fL− (I−AL)Φ(m+1)L

if

(‖r(m+1)‖L2

‖r(0)‖L2≤ τ

)stop

end do

The step (d) usually solves the coarse problem exaclty using direct methods. How-ever, since Al is dense matrix which is never explicitly computed nor stored,iterative solvers are required to perfom the step (d). In our numerical exampleswe used Gmres(m) method which has been discussed in Sect. 4. Note that in ourcontext the notation Grid2 for two-level algorithm does not necessarily mean thatwe consider two levels of mesh refinements. Thus, Algorithm 4 is also appli-cable in cases where we have two different space discretizations on the same mesh(L �= l + 1).

The fully multilevel algorithm (Gridnest) or nested iteration as refered to in [30]can be implemented recursively as follows

Algorithm 5: Gridnest algorithm

given the finest level {lmax,Almax , flmax}, the coarsest level{lmin,Almin , flmin} and the tolerances {τl}, l = lmin, . . . , lmax

a. Solve on the coarsest level for Φlmin : (I−Almin)Φlmin = flmin

do k = lmin + 1, . . . , lmax

b. set l←− k−1c. set L←− kd. compute the right hand side fL

e. set Φ(0)L ←−PL

l Φl

f. set τ←− τl

g. call Grid2 to solve for ΦL: (I−AL)ΦL = fL

end do

Some comments are in order. The steps (b)–(f) are needed only to set the inputs,

fine level {L,AL, fL}, coarse level {l,Al, fl}, initial guess Φ(0)L and tolerance τ to

the algorithm Grid2. Recall that Gridnest uses coarse levels to obtain improved ini-tial guesses for fine level problems. The tolerance parameters {τl}, which determinehow many iterations of the multilevel algorithm to do on each level, can be consid-ered either fixed or adaptively chosen during the course of computation.


3.5 Numerical Results

The methods discussed in the above sections are tested on a PC with AMD-K6200 processors using Fortran compiler, see [76] for details. In all these methods theiterations are terminated when

‖r(m)‖L2

‖r(0)‖L2≤ 10−5. (67)

To solve the diffusion problem in DSA we used a preconditioned Bicgstab whereas,Gmres(10) is used in Grid2 and Gridnest to solve the coarse problems. These inneriterations are stopped as in (67) but with 10−2 instead of 10−5. In all our computa-tions for the two space dimension cases we used a discrete SN-direction set with 60directions from [22].

3.5.1 Radiative Transfer Equation in 1D Slab Geometry

Our first example is the (45) in 1D slab geometry

μ∂ψ∂x

+(σ + κ)ψ =σ2

∫ 1

−1ψ(x,μ ′)dμ ′+ κq(x) (68)

ψ(0,μ) = 1, μ > 0, and ψ(1,μ) = 0, μ < 0.

We set q = 0, we used 64 Gauss quadrature nodes in the discrete ordinates and afine mesh of 512 gridpoints in the space dicrestization. The convergence results fortwo different values of σ and κ are shown in Fig. 3. The fast convergence of Grid2is well demonstrated in both cases. Although the scattering ratio for the two casesis 0.99, Grid2 method shows strong reduction of number of iterations comparing tothe other methods. Same observation is shown when the regime is optically thick(σ = 99, κ = 1).

Next we want to compare the efficiency of these methods in terms of CPU timeand number of iterations when the scattering ratio γ runs in the range (0,1). To thisend we set σ = 10 and we vary κ keeping the fine gridpoints fixed to 512. In Fig. 4we plot the scattering ratio versus the number of iterations in the left and versus theCPU time in the right. Grid2 and Gmres(40) preserve roughly the same amount ofcomputational work (referring to number of iterations and CPU time) in the wholeinterval, while the SI and DSA become costly for values of γ near 1.

3.5.2 Radiative Transfer Equation with Thermal Source

The second example is the (45) in the unit square with a thermal source q = B(T ),with B is the frequency-integrated Planckian B = aRT 4 with aR is the radiationconstant (aR = 1.8× 10−8). We fix the temperature to have the linear profile


0 5 10 15 20 25

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

Number of iterations

L2−

norm

of r

esid

ual

σ = 0.99, κ = 0.01

SIDSAGmres(40)Grid2

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

L2−

norm

of r

esid

ual

σ = 99, κ = 1

0 10 20 30 40 50 60Number of iterations

SIDSAGmres(40)Grid2

Fig. 3 Convergence plots for the radiative transfer equation in 1D slab geometry

T (x,y) = 800x + 1,000 and the boundary function g(x, y) = B(T (x, y)). In Fig. 5we show the convergence plots for different values of σ and κ using a fine meshof 257× 257 gridpoints. In all cases Grid2 algorithm presents faster convergencebehaviour than DSA even in the diffusion regime (σ = 100 and κ = 1).

We summarize in Table 3 the CPU time consumed for each method to performthe computations with the different values of σ and κ . In Table 4 we report, thenumber of gridpoints I× J at each level, the iteration counter m for that level, thenumber of iterations in Gmres(10) iGmres, the L∞-norm of the residual, ‖r(m)‖L∞ , andthe factor ‖r(m)‖L∞/‖r(m−1)‖L∞ .

The results presented in Table 4 do not include those obtained for σ = 1, κ = 1or σ = 100, κ = 1 because the number of iterations m in Grid2 is very large.For instance, for σ = 100, κ = 1 this number surpasses 15 on the coarsest mesh.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

350

400a

Scattering ratio

CP

U ti

me

SIDSAGmres(40)Grid2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.705

1015202530

Clipping

bSIDSAGmres(40)Grid2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Scattering ratio

CP

U ti

me

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4Clipping

Fig. 4 Plots of scattering ratio γ versus the number of iterations in (a) and the CPU time (seconds)in (b). In both figures σ is fixed to 10

Nevertheless, in all these test cases we have observed that there is very little variationin the number iGmres and the reduction factor ‖r(m)‖L∞/‖r(m−1)‖L∞ as the meshes arerefined.

The results of these tables and Fig. 5 show various interesting features about thebehaviours of the preconditioner used by each method. First, where the scatteringratio γ = 0.99 (σ = 100 and κ = 1), it is clear that the SI method is unacceptablyslow to converge. The convergence rate is improved significantly by Gmres(40), andit is improved even more by DSA method but at the cost of extra work and storage.


0 1 2 3 4 5 6

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7


L2−

norm

of r

esid

ual

σ = 1, κ = 10

SIDSAGmres(40)Grid2

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

L2−

norm

of r

esid

ual

0 10 20 30 40 50 60


σ = 100, κ = 1

SIDSAGmres(40)Grid2

Fig. 5 Convergence plots for the radiative transfer equation with thermal source

Table 3 CPU time (in seconds)

SI DSA Gmres(40) Grid2

σ = 1, κ = 10 17.8 29.1 15.5 17.1σ = 1, κ = 1 33.7 77.4 20.8 23.5σ = 100, κ = 1 2389.8 82.3 174.4 121.7

The most effective method for solving this example, however, is the multilevelGrid2 and Gridnest methods. Second, the number of iterations iGmres in Gridnsetremain nearly the same in all levels and is bounded by the number in the coarsestlevel.


Table 4 Results obtained by Gridnest for σ = 1 and κ = 10 at different levels

I× J m iGmres ‖r(m)‖L∞ ‖r(m)‖L∞ /‖r(m−1)‖L∞

33×33 0 0.73E+001 2 0.32E-01 0.43E-012 2 0.12E-02 0.37E-013 3 0.56E-05 0.46E-04

65×65 0 0.11E-011 2 0.33E-03 0.30E-012 2 0.12E-05 0.36E-04

129×129 0 0.83E-021 2 0.67E-03 0.81E-012 2 0.53E-05 0.79E-04

257×257 0 0.22E-021 2 0.73E-04 0.33E-012 2 0.15E-06 0.20E-04

0 2

1

0.23

0.44 0.88 1.22

σ = 1

κ = 0

σ = 0

κ = 10 κ = 0

σ = 0 κ = 0.001 σ = 0 κ = 100

s = 1

s = 0 s = 0

s = 0s = 0

σ = 100

Fig. 6 Geometry and values of σ , κ , s = κq for the discontinuous equation

3.5.3 Radiative Transfer Equation with Discontinuous Variables

The aim of this example is to test the multilevel algorithm for radiative transfer prob-lem with jumping coefficients. Thus, the problem statement is the (45) augmentedby discontinuous scattering, absorption and source term [2]. The space domain ge-ometry and the values of σ , κ and s = κB for each subdomain are given in Fig. 6.We take in the first run of this example vacuum boundary conditions whereas, in thesecond run we use the nonhomogeneous boundary condition g,

g(x, y) =

⎧⎪⎪⎨⎪⎪⎩

1 when x = 0 and 0≤ y≤ 1,1 when y = 0 and 0 < y≤ 0.44,1 when y = 1 and 0 < y≤ 0.44,0 otherwise.


0 50 100 150 200

100

10010−1

10−2

10−4

10−6

10−2

10−3

10−4

10−5

10−6


L2−

norm

of r

esid

ual

0 5 10 15 20 25 30

Clipping

SIDSAGmres(40)Grid2

0 50 100 150 200 250 300

100

10010−1

10−2

10−4

10−6

10−2

10−3

10−4

10−5

10−6


L2−

norm

of r

esid

ual

0 5 10 15 20 25 30

Clipping

SIDSAGmres(40)Grid2

Fig. 7 Convergence rates for the discontinuous equation subject to vacuum boundary conditions(top) and nonhomogeneous boundary conditions (bottom)

Table 5 CPU time (in seconds)

SI DSA Gmres(40) Grid2

Vacuum boundary condition 794.04 304.42 192.25 205.22Nonhomogeneous boundary condition 1100.18 397.53 110.07 201.91

The space domain is discretized uniformly into 400×200 gridpoints at the finestlevel. We display in Fig. 7 the convergence rates for the two runs. Table 5 providesthe running time used by each method for these computations. A simple inspection


Table 6 Gridnest results for the discontinuous problem with vacuum boundary

I× J m iGmres ‖r(m)‖L∞ ‖r(m)‖L∞ /‖r(m−1)‖L∞

100×50 0 0.13E+011 9 0.50E+00 0.39E+002 9 0.13E+00 0.27E+003 8 0.24E-01 0.19E+004 7 0.14E-02 0.57E-015 7 0.45E-05 0.32E-02

200×100 0 0.51E-011 8 0.10E-02 0.21E-012 7 0.78E-05 0.73E-023 7 0.32E-07 0.41E-02

400×200 0 0.79E-021 7 0.40E-04 0.51E-022 7 0.13E-06 0.33E-02

800×400 0 0.38E-021 7 0.24E-04 0.64E-022 7 0.94E-07 0.39E-02

of Fig. 7 shows that Grid2 algorithm solves this problem more effectively than SI orGmres(40) methods and with less iterations than DSA method.

Furthermore, we note that the Gmres(40) method performs poorly after the sev-enth iteration in both runs. This may be partly due to the fact that the discontinuousσ and κ coefficients change the matrix structure very badly. While, it is not surpris-ing that the SI algorithm performs very poorly in this case. An examination of theCPU time in Table 5 reveals that Gmres(40) consumes less computational work thanthe other methods. We have observed that the main part of the CPU time needed inDSA or Grid2 is used by Bicgstab or Gmres(10) to solve the diffusion problem inDSA or the coarse linear system in Grid2, respectively. However, by limiting thenumber of iterations in Bicgstab and Gmres(10) to 1, or increasing the tolerancefrom 10−2 to 10−1, the results change favourably with significant advantage forGrid2.

Table 6 tabulates the results obtained by Gridnest using four levels. It can beclearly seen that there is very little variation in the number iGmres and the reductionfactor ‖r(m)‖L∞/‖r(m−1)‖L∞ as the meshes are refined.

3.5.4 Radiative Transfer Equation with Frequency Dependence

Our final example is the frequency-dependent problem

Ω ·∇ψ(x,Ω ,ν)+(σ + κ

)ψ(x,Ω ,ν) =

σ4π

∫

S2ψ(x,Ω ′,ν)dΩ ′+ κB(T,ν), (69)


with boundary condition ψ = B(T (x),ν). Here ψ = ψ(x,Ω ,ν), T = T (x),σ = σ(x,ν) and κ = κ(x,ν) denote respectively, the radiation intensity, the tem-perature, the scattering and the opacity within the frequency ν > 0.

In order to discretize the (69) with respect to the frequency variable ν , we assumeN frequency bands [νι ,νι+1], ι = 1, . . . ,N with piecewise constant absorption,

κ = κι , ∀ ν ∈ [νι ,νι+1] ι = 1, . . . ,N.

We define the frequency-averaged intensity in the band [νι ,νι+1] by

ψι =∫ νι+1

νιψ(x,Ω ,ν ′)dν ′.

Then, the (69) are transformed to a system of N radiative transfer equations of theform

Ω ·∇ψι(x,Ω)+(σι + κι

)ψι (x,Ω) =

σι4π

∫

S2ψι (x,Ω ′)dΩ ′+ κι

∫ νι+1

νιB(T,ν ′)dν ′.

(70)To approximate the frequency integrals we used trapezoidal formula. In our numer-ical simulations we use eight frequency bands [νι ,νι+1], ι = 1, . . . ,8 from glassmanufacturing [47]. These frequencies are given in Table 2 in the introduction.

We compute the solution of the (69) in a unit square, on the refined grid with100×100 gridpoints, and 64 discrete ordinates. Hence the number of unknowns foreach frequency band is 64×104, and these computations are done for the 8 frequen-cies such that the overall number of equations amounts 5.12× 106. The scatteringparameter σ is varying in the set {1,10,100}. For every frequency band we calcu-late it corresponding scattering ratio and, we store the number of iterations and therunning time obtained by each method. The results are given in Table 7.

All the algorithms iterate the solution in a large iteration numbers for the firstfrequency bands (with large scattering ratio), then these numbers go decreasing asthe frequency bands grow until their reach the minimum for the last frequency band.In this case when the size of scattering ratio is changing dramatically over the fre-quency bands, only the DSA and Grid2 algorithms lead to satisfying results for allfrequencies and also in diffusive limit (σ = 100). The superiority of Grid2 is clearlydemonstrated in Table 7.

4 Higher-Order Diffusion Approximations

Approximations that are widely used are the PN approximations, cf. Sect. 5.1. Amajor drawback in higher dimensions and for complicated problems is the largenumber of equations which have to be solved. We propose the SPN approximationsas alternatives to the full glass equations. This class of approximations uses diffusionequations instead of the radiative transfer equations. The number of equations is


Table 7 Number of iterations and CPU time for the eight frequency-bands problem

Band Scattering ratio SI Gmres(40) DSA Grid2

1 0.71428 13 6 6 32 0.66666 13 6 6 33 0.11494 7 5 4 34 0.06077 6 4 4 3

σ = 1 5 0.03450 5 4 3 36 0.00371 4 3 3 27 0.00175 3 3 3 28 0.00014 3 2 2 2

CPU(sec) ——— 30.45 26.36 39.38 29.29

1 0.96153 117 17 8 52 0.95238 114 17 8 53 0.56497 21 8 5 44 0.39292 14 7 5 3

σ = 10 5 0.26329 10 5 5 36 0.03597 5 3 3 37 0.01732 4 3 3 38 0.00173 3 2 2 2

CPU(sec) ——— 108.30 36.61 35.13 38.40

1 0.99601 1577 79 16 152 0.99502 1637 79 17 143 0.92850 151 22 15 104 0.86617 80 16 14 9

σ = 100 5 0.78137 48 12 15 86 0.27175 10 5 5 47 0.14985 8 4 4 48 0.01381 4 3 3 2

CPU(sec) ——— 1190.73 107.80 57.27 122.05

considerably reduced compared to the PN equations. The method originates fromneutron transport theory in nuclear physics where it was successfully introduced.Nevertheless, it suffered from a lack of theoretical foundation with the result that itwas not completely accepted in the field. This has been remedied, however, duringthe last decade, such that the method has now been substantiated.

We want to study the optically thick regime where the opacity κ is large and theradiation is conveyed in a diffusion-like manner. Therefore, we rewrite the aboveequations in non-dimensional form introducing reference values which correspondto typical values of the physical quantities. In order to obtain a diffusion scaling weimpose the relations

tre f = cmρmκre f x2re f

Tre f

Ire f, and kre f =

Ire f

κre f Tre f,


on these reference values and define the non-dimensional parameter

ε =1

κre f xre f(71)

which is small in the optically thick, diffusive regime. Then the rescaled equationsread (without marking the scaled quantities):

ε2 ∂T∂ t

= ε2∇ · k∇T −∫ ∞

ν1

∫

S2κ(B−ψ) dΩdν, (72a)

∀ν > 0,Ω ∈ S2 : εΩ ·∇ψ = κ(B−ψ). (72b)

Here, we have neglected scattering. This can be incorporated in a straightforwardway, however. The boundary condition for the temperature changes into

εk n ·∇T = h(Tb−T)+ απ(n2

n1

)2 ∫ ν1

0B(ν,Tb)−B(ν,T ) dν. (72c)

It is well known that an outer asymptotic expansion of (72a) and (72b) leads to equi-librium diffusion theory, which is, in the cases considered here, expected to be validin the interior of V , see e.g. [46,67,68]. The diffusion or Rosseland approximation is

∂T∂ t

= ∇ ·(

k + kr(T ))

∇T, with kr(T ) =4π3

∫ ∞

ν1

1κ

∂B∂T

dν.

However, this diffusion theory is not capable of describing boundary layers and thequestion arises whether more sophisticated approximations can suitably model theboundary layer effects. In the realm of neutron transport, such higher-order asymp-totic corrections to diffusion theory exist, and are reasonably well understood; theyare the so-called simplified PN (SPN) theories, see [9, 80]. These SPN theories are,in fact, diffusion in nature i.e. diffusion equations or coupled systems of diffusionequations are employed. They contain boundary layer effects and can be remark-ably accurate, much more accurate than the standard Rosseland approximation. Inpractice, these equations are viewed as an extended form of the classical diffusiontheory. No separate boundary layer treatment is necessary because the boundary lay-ers are included in the SPN equations. For other approximate theories for the aboveequations and applications, see for example [17, 49]. The material of this section istaken from [47].

4.1 Asymptotic Analysis and Derivation of the SPNApproximations

To solve (72a) in the domain V , we write this equation as

(1 +

εκ

Ω ·∇)

ψ(x,t,Ω ,ν) = B(ν,T ).


and apply Neumann’s series to formally invert the operator

ψ =(

1 +εκ

Ω ·∇)−1

B

∼=[1− ε

κΩ ·∇+

ε2

κ2 (Ω ·∇)2− ε3

κ3 (Ω ·∇)3 +ε4

κ4 (Ω ·∇)4 · · ·]B. (73)

Integrating with respect to Ω and using the result

∫

S2(Ω ·∇)n dΩ = [1 +(−1)n] :

2πn + 1

∇n,

where ∇2 = ∇ ·∇ = Δ is the Laplacian, we get

ϕ =∫

S2ψ dΩ = 4π

[1 +

ε2

3κ2 ∇2 +ε4

5κ4 ∇4 +ε6

7κ6 ∇6 · · ·]B +O(ε8).

Hence,

4πB =[1 +

ε2

3κ2 ∇2 +ε4

5κ4 ∇4 +ε6

7κ6 ∇6]−1

ϕ +O(ε8)

=

{1−

[ ε2

3κ2 ∇2 +ε4

5κ4 ∇4 +ε6

7κ6 ∇6]

+[ ε2

3κ2 ∇2 +ε4

5κ4 ∇4 +ε6

7κ6 ∇6]2

−[ ε2

3κ2 ∇2 +ε4

5κ4 ∇4 +ε6

7κ6 ∇6]3 · · ·

}ϕ +O(ε8),

so we have the asymptotic expansion

∀ν > 0 : 4πB =[1− ε2

3κ2 ∇2− 4ε4

45κ4 ∇4− 44ε6

945κ6 ∇6]ϕ +O(ε8). (74)

If we discard terms of O(ε4), O(ε6) or O(ε8) we obtain the SP1,SP2 and SP3 approx-imations, respectively. All these equations contain the frequency ν as a parameter.

4.1.1 SP1 and Diffusion Approximations

From (74), we obtain

4πB = ϕ− ε2

3κ2 ∇2ϕ +O(ε4)


and up to O(ε4) we may write the equation in the form

∀ν > 0 : −ε2∇ · 13κ

∇ϕ + κϕ = κ(4πB). (75a)

In this equation, ν is simply a parameter. Thus, in practice, these equations wouldbe solved independently for each frequency or frequency group and subsequentlycoupled via (72a). By (75a),

∫ ∞

ν1

∫

S2κ(B− I) dΩ dν =

∫ ∞

ν1

κ(4πB−ϕ) dν =−ε2∫ ∞

ν1

∇ · 13κ

∇ϕ dν +O(ε4).

Thus, the energy equation (72a) becomes up to O(ε2):

∂T∂ t

= ∇ · k∇T +∫ ∞

ν1

∇ · 13κ

∇ϕ dν. (75b)

Equations (75b) and (75a) are the SP1 approximation to (72a) and (72b). Since (75b)is only of order O(ε2) the approximation is O(ε2). Using ϕ = 4πB+O(ε2) in (75b)one obtains up to O(ε2)

∂T∂ t

= ∇ · k∇T +∫ ∞

ν1

∇ · 13κ

∇(4πB) dν

= ∇ · k∇T + ∇ ·(4π

3

∫ ∞

ν1

1κ

∂B∂T

dν)

∇T, (76)

i.e. we have obtained the conventional equilibrium diffusion or Rosselandapproximation (73). However, (75a) permits a boundary layer behaviour near theboundary ∂V that is not present in (76).

4.1.2 SP2 Approximation

From (74), we get for ε going to 0

4πB = ϕ− ε2

3κ2 ∇2ϕ− 4ε2

15κ2 ∇2( ε2

3κ2 ∇2ϕ)

+O(ε6).

This impliesε2

3κ2 ∇2B = ϕ−4πB +O(ε4).

Hence, with O(ε6) error, the expansion above gives

4πB = ϕ− ε2

3κ2 ∇2ϕ− 4ε2

15κ2 ∇2[ϕ−4πB] = ϕ− ε2

3κ2 ∇2[ϕ +

45(ϕ−4πB)

],


or equivalently,

−ε2∇ · 13κ

∇[ϕ +

45(ϕ−4πB)

]+ κϕ = κ(4πB). (77)

Equation (77) implies

∫ ∞

ν1

∫

S2κ(B− I) dΩ dν =−ε2

∫ ∞

ν1

∇ · 13κ

∇[ϕ +

45(ϕ−4πB)

]dν +O(ε4).

Thus, the energy equation (72b) becomes up to O(ε4)

∂T∂ t

= ∇ · k∇T +∫ ∞

ν1

∇ · 13κ

∇[ϕ +

45(ϕ−4πB)

]dν. (78)

These two equations can be written in a more advantageous way if we define

ξ = ϕ +45(ϕ−4πB). (79)

Introducing the new variable ξ into (77) and (78) we obtain the SP2 equations:

∂T∂ t

= ∇ · k∇T +∫ ∞

ν1

∇ · 13κ

∇ξ dν, and (80a)

−ε2∇ · 35κ

∇ξ + κξ = κ(4πB). (80b)

There is a remarkable similarity between the SP2 (80) and the SP1 (75). This isbecause the SP1 equations contain some, but not all, of the O(ε4) correction terms.In the realm of neutron transport, the SP2 approximation has not found favour be-cause, in the presence of material inhomogenities, it yields discontinuous solutions.However, it is obvious that (80b) and (80a) can not produce a discontinuous solution.

Also, in the realm of neutron transport, the SP1 and SP2 solutions are not capableof exhibiting boundary layer behaviour, while the more complicated SP3 solutiondescribed below does incorporate this in a remarkably accurate way. However, theradiative transfer SP1 and SP2 equations stated here can contain boundary layer be-haviour. The SP3 approximation derived in the following should capture significantradiative transfer boundary effects that are not captured by the SP1 and SP2 approxi-mations.

4.1.3 SP3 Approximation

Ignoring terms of O(ε8) in (74), we get


4πB = ϕ− ε2

3κ2 ∇2[ϕ +

4ε2

15κ2 ∇2ϕ +44ε4

315κ4 ∇4ϕ]+O(ε8)

= ϕ− ε2

3κ2 ∇2[ϕ +

(1 +

11ε2

21κ2 ∇2)( 4ε2

15κ2 ϕ)]

+O(ε8)

= ϕ− ε2

3κ2 ∇2[ϕ +

(1− 11ε2

21κ2 ∇2)−1( 4ε2

15κ2 ϕ)]

+O(ε8) (81)

Hence, if we define

ϕ2 ≡(

1− 1121

ε2

κ2 ∇2)−1( 2ε2

15κ2 ϕ), (82)

then (81) becomes up to O(ε8):

4πB = ϕ− ε2

3κ2 ∇2(ϕ + 2ϕ2)

or

∀ν > 0 : −ε2∇ · 13κ

∇(ϕ + 2ϕ2)+ κϕ = κ(4πB), (83a)

while (82) becomes

− 11ε2

21κ2 ∇2ϕ2 + ϕ2 =215

ε2

κ2 ∇2ϕ =25

( ε2

3κ2 ∇2ϕ)

=25

[−4πB + ϕ− 2ε2

3κ2 ∇2ϕ2

]

or ( 415− 11

21

) ε2

κ2 ∇2ϕ2 + ϕ2 =25(ϕ−4πB)

or eventually

∀ν > 0 : −ε2∇ · 935κ

∇ϕ2 + κϕ2− 25

κϕ =−25

κ(4πB). (83b)

By (83a) we get up to O(ε6)

∫ ∞

ν1

∫

S2κ(B− I) dΩ dν =−ε2

∫ ∞

ν1

∇ · 13κ

∇(ϕ + 2ϕ2) dν.

Thus, the energy equation (72b) becomes

∂T∂ t

= ∇ · k∇T +∫ ∞

ν1

∇ · 13κ

∇(ϕ + 2ϕ2) dν. (83c)


Equation (83c) together with the two approximate (83a, 83b) form the SP3

approximation to the system (72a) and (72b).These equations can be rewritten in a computationally more advantageous way.

Let us calculate θ{(83a)}+{(83b)} :

−ε2∇ · 1κ

∇{

θ3

(ϕ + 2ϕ2)+9

35ϕ2

}+ κ

{θϕ + ϕ2− 2

5ϕ}

= κ(

θ − 25

)(4πB).

We seek linear combinations of both equations such that the two functions in thebrackets on the left are scalar multiples. More explicitly, we look for θ that fulfillsthe condition

θ3

(ϕ + 2ϕ2)+9

35ϕ2 = μ2

(θϕ + ϕ2− 2

5ϕ), (84)

where μ2 > 0 is a constant to be determined later. Equation (84) holds for arbitraryϕ and ϕ2 iff

θ3

= μ2(

θ − 25

)and

2θ3

+9

35= μ2.

The second of these equations may be solved for θ and we get a quadratic equationin μ2:

12

μ2− 970

= μ2(2

3μ2− 11

14

).

Its discriminant is positive and thus we obtain two positive real solutions

μ21 =

37− 2

7

√65, and μ2

2 =37

+27

√65,

and the corresponding values of the scalar θ are

θ1 =9

35− 3

7

√65, and θ2 =

935

+37

√65.

Now relation (84) implies, for n = 1,2,

(−ε2∇ · 1

κ∇μ2

n + κ)[

θnϕ + ϕ2− 25

ϕ]

=(

θn− 25

)κ(4πB). (85)

This suggests that we define two new independent variables for n = 1,2

In =θnϕ + ϕ2−2/5ϕ

θn−2/5= ϕ +

1θn−2/5

ϕ2 = ϕ + γnϕ2 (86)

where

γn =1

θn−2/5=

57

[1 +(−1)n3

√65

].


The two equations in (85) are now

−ε2μ21 ∇ · 1

κ∇I1 + κI1 = κ(4πB), (87a)

−ε2μ22 ∇ · 1

κ∇I2 + κI2 = κ(4πB). (87b)

The advantage of this form of the SP3 equations is that the diffusion equations areuncoupled. It will be seen below that there remains, however, a weak coupling inthe boundary conditions.

The linear transformation of variables above is inverted according to the formulae

ϕ =γ2I1− γ1I2

γ2− γ1, and ϕ2 =

I2− I1

γ2− γ1. (88)

Defining three constants

w0 =1

γ2− γ1=

730

√56

=7

36

√65, and (89a)

w1 =γ2

γ2− γ1=

16

(3 +

√56

), w2 =

−γ1

γ2− γ1=

16

(3−

√56

)(89b)

we can write ϕ = w1I1 + w2I2 and ϕ2 = w0(I2− I1) and we have furthermore

13(ϕ + 2ϕ2) =

13(w1−2w0)I1 +

13(w2 + 2w0)I2 = a1I1 + a2I2.

Here again we introduced constants

a1 =w1−2w0

3=

130

(5−3

√56

), and a2 =

w2 + 2w0

3=

130

(5 + 3

√56

).

In this way, the SP3 energy equation (83c) above becomes:

∂T∂ t

= ∇ · k∇T +∫ ∞

ν1

∇ · 1κ

∇(a1I1 + a2I2) dν. (90)

4.2 Boundary Conditions for SPN Approximations

The boundary conditions for the SPN equations in neutron transport come from avariational principle, see [9, 80]. Here, we use the boundary conditions developedfor the transport case to state (and rewrite in a more suitable form) the boundaryconditions for SP1,SP2 and SP3 approximations to the transport problem (72b) withthe boundary condition (44c).


We consider the transport equation (72b)

∀ν > 0 : εΩ ·∇ψ(x,Ω)+ κI(x,Ω) = κB, x ∈V,

with semi-transparent boundary conditions on ∂V

ψ(x,Ω) = ρ(n ·Ω)ψ(x,Ω ′)+(1−ρ(n ·Ω)

)ψb(x,Ω), n ·Ω < 0.

Let us define the scalar flux as before

ϕ(x) =∫

S2ψ(x,Ω) dΩ ,

and define two integrals of the influx into the domain for m = 1,3

ψm(x) =∫

n·Ω<0

(1−ρ(n ·Ω)

)Pm(|Ω ·n|)Ib(x,Ω) dΩ , x ∈ ∂V. (91)

Here, the Legendre polynomials of order 1 and 3 are used:

P1(μ) = μ , and P3(μ) =52

μ3− 32

μ .

Furthermore, it is convenient in the sequel to have the following integrals at hand:

r1 = 2π∫ 1

0 μρ(μ) dμ ,

r2 = 2π∫ 1

0 μ2ρ(μ) dμ ,

r3 = 2π∫ 1

0 μ2ρ(μ) dμ ,

r4 = 2π∫ 1

0 μP3(μ)ρ(μ) dμ ,

r5 = 2π∫ 1

0 P3(μ)ρ(μ) dμ ,

r6 = 2π∫ 1

0 P2(μ)P3(μ)ρ(μ) dμ ,

r7 = 2π∫ 1

0 P3(μ)P3(μ)ρ(μ) dμ ,

The boundary conditions in [9, 80] were derived for the case ρ = 0 = const. Forsemi-transparent boundary conditions the same arguments apply and the calcula-tions can be analogously carried out, the only difference beeing modifications in thecoefficients. We therefore content ourselves with stating the resulting equations. Inthe SP1 approximation (75a), the boundary condition for ϕ is:

∀ν > 0 : (1−2r1)ϕ(x)+ (1 + 3r2)2ε3κ

n ·∇ϕ(x) = 4ψ1(x). (92)

The boundary condition for ϕ in the SP2 approximation (80b) is, see [80]:

(1 − 2r1)ϕ +(1 + 3r2)2ε3κ

n ·∇[ϕ +

45(ϕ−4πB)

]

+ (1−4(3r3− r1))12(ϕ−4πB) = 4ψ1. (93)


These equations reduce to (92) if one deletes the (ϕ − 4πB) terms in them. Theycan be written in a more advantageous way if we define ξ as in Sect. 2 which isequivalent to

ϕ =59

ξ +49(4πB).

One obtains

(1 − 2r1)(5

9ξ +

49(4πB)

)+(1 + 3r2)

2ε3κ

n ·∇ξ

+ (1−4(3r3− r1))12

(59

ξ +49(4πB)−4πB

)= 4ψ1,

or, using the abbreviations

α1 =56(1−4r3) α2 =

16(−1 + 12r1−20r3)

the SP2 boundary conditions for ξ in (80b) can be written:

∀ν > 0 : α1ξ (r)+ (1 + 3r2)2ε3κ

n ·∇ξ (r) = α24πB(T(x))+ 4ψ1(x). (94)

The boundary conditions (92) and (93) were derived variationally, not from a bound-ary layer analysis. They should be accurate if Ib(x,Ω) is a reasonably smoothfunction of Ω , but they could be inaccurate if Ib is not smooth.

Finally, the SP3 boundary conditions for ϕ and ϕ2 in (83a) and (83b) are, see [9]:for all frequencies ν > 0 and x ∈ ∂V there must hold

(1 − 2r1)14

ϕ(x)+ (1−8r3)5

16ϕ2(x)+ (1 + 3r2)

ε6κ

n ·∇ϕ(x)

+(1 + 3r2

3+

3r4

2

) 2ε3κ

n ·∇ϕ2(x) = ψ1(x), (95a)

− (1 + 8r5)1

16ϕ(x)+ (1−8r6)

516

ϕ2(x)+ 3r4ε

6κn ·∇ϕ(x)

+(

r4 +3

14(1 + 7r7)

) εκ

n ·∇ϕ2(x) = ψ3(x). (95b)

or formally

A1ϕ(x)+ A2ϕ2(x)+ A3εκ

n ·∇ϕ(x)+ A4εκ

n ·∇ϕ2(x) = ψ1(x)

B1ϕ(x)+ B2ϕ2(x)+ B3εκ

n ·∇ϕ(x)+ B4εκ

n ·∇ϕ2(x) = ψ3(x).


We still have to derive boundary conditions for I1 and I2. Using the formulae in (88),we can transform the boundary conditions for ϕ and ϕ2 into boundary conditionsfor I1 and I2. The equations above then become

(A1γ2w0−A2w0)I1 +(−A1γ1w0 + A2w0)I2

+(A3γ2w0−A4w0)εκ

n ·∇I1

+(−A3γ2w0 + A2w0)εκ

n ·∇I2 = ψ1

(B1γ2w0−B2w0)I1 +(−B1γ1w0 + B2w0)I2

+(B3γ2w0−B4w0)εκ

n ·∇I1

+(−B3γ2w0 + B2w0)εκ

n ·∇I2 = ψ3

or, again formally rewritten for writing convenience,

C1I1 +C2I2 +C3εκ

n ·∇I1 +C4εκ

n ·∇I2 = ψ1

D1I + D2I2 + D3εκ

n ·∇I1 + D4εκ

n ·∇I2 = ψ3.

We eliminate the gradient term n ·∇I2 in the first equation and n ·∇I1 in the secondin order to get boundary conditions for the I1 and I2 equations, respectively. We find

(C1D4−D1C4)I1 +(C3D4−D3C4)εκ

n ·∇I1

=−(C2D4−D2C4)I2 +(D4 ψ1−C4 ψ3)

−(C2D3−D2C3)I2 +(C3D4−D3C4)εκ

n ·∇I2

= (C1D3−D1C3)I1− (D3 ψ1−C3 ψ3)

so, if we set D = C3D4−D3C4 and define constants

α1 = (C1D4−D1C4)/D,

β1 = −(C1D3−D1C3)/D,

α2 = −(C2D3−D2C3)/D,

β2 = (C2D4−D2C4)/D,

then we end up with SP3 boundary conditions in the following form:

α1I1(x)+εκ

n ·∇I1(x) =−β2I2(x)+ (D4 ψ1(x)−C4 ψ3(x))/D, (96a)

α2I2(x)+εκ

n ·∇I2(x) =−β1I1(x)− (D3 ψ1(x)−C3 ψ3(x))/D. (96b)


Equations (96a) and (96a) are the boundary conditions to go with the diffusionequations (87a) and (87b), respectively. The coupling of I1 and I2 in the boundaryconditions is very weak.

Consider, for example, the simple case when there is no reflection ρ = 0 andψb(x,Ω) = ψb(x) is isotropic. Then by (91),

ψm(x) = 2πψb(x)1∫

0

Pm(μ) dμ = πψb(x){

1, for m = 1,14 , for m = 3.

The constants r1, . . . ,r7 are all zero such that we find after some calculations D =1

144

√6/5 and the constants in the boundary conditions are

α1 = 596

(34 + 11

√6/5

),

β1 = 596

(2−√6/5

),

α2 = 596

(34 + 11

√6/5

),

β2 = 596

(2 +

√6/5

).

Finally, the source terms in (96a) and (96b) become in this case

[6ψ1(x)−2

(3±5

√65

)ψ3(x)

]= πψb(x)

[6+2

(3±5

√65

)14

]

=52

πψb(x)

[3±

√65

].

The approximate SPN-theories stated above are simpler than transport theory be-cause they do not contain the angular variable Ω . However, they do contain thefrequency variable ν . It is formally possible to derive simpler theories in which thefrequency is eliminated. We refer to [48].

5 Moment Models

First we briefly review the basics of the moment approach. Consider again the trans-port equation (32) for the radiation. We will assume isotropic scattering here. Thisequation is in fact a system of infinitely many coupled integro-differential equationsthat describes the distribution ψ of all photons in time, space and velocity space. Onthe one hand this system is computationally very expensive and on the other handwe are not interested in the photon distribution itself but in macroscopic quantitieslike the mean energy or mean flux of the radiation field. For instance, only the gradi-ent of the radiative flux enters into the energy balance. The macroscopic quantitiesare moments of the distribution function. Let

〈 · 〉 :=∫

S2· dΩ (97)


denote the average over all directions. The energy, flux vector and pressure tensorof the radiation field are defined, respectively, as

E := 〈ψ〉, F := 〈Ωψ〉, P := 〈(Ω ⊗Ω)ψ〉. (98)

To derive equations for the macroscopic quantities we multiply the transport equa-tion by 1 and Ω and average over all directions. We obtain the conservation laws

∇F = κ(〈B〉−E) (99)

∇P =−(κ + σ)F. (100)

These are four equations (the first is a scalar equation, the second has threecomponents) for 10 unknowns (E scalar, F 3-component vector, P symmetric 3×3-matrix). Hence we have to pose an additional condition. Usually this condition isa constitutive equation for the highest moment P, expressed in terms of the lowermoments E and F . This is referred to as the closure problem. The simplest approxi-mation, the so-called P1 approximation, is obtained if we assume that the underlyingdistribution is isotropic. Thus, we obtain P = 1

3 E and therefore

∇F = κ(〈B〉−E) (101)

∇13

E =−(κ + σ)F. (102)

The general PN closure is usually derived in a different way.

5.1 Spherical Harmonics

The Spherical Harmonics approach is one of the oldest approximate methods forradiative transfer [20, 33]. For the sake simplicity, we restrict our explanation to thecase of slab geometry. The derivation for three-dimensional case can be found forexample in [12] and also in standard textbooks [14,41,62]. The idea of the sphericalharmonics approach is to express the angular dependence of the distribution functionin terms of a Fourier series,

ψ(μ) =∞

∑l=0

ψSHl

2l + 12

Pl(μ), (103)

where Pl are the Legendre polynomials. These form an orthogonal basis of the spaceof polynomials with respect to the standard scalar product on [−1,1],

∫ 1

−1Pl(μ)Pk(μ)dμ =

22l + 1

δlk. (104)


In more space dimensions, one uses spherical harmonics, which are an orthogonalsystem on the unit sphere.

If we truncate the Fourier series at l = N we have

ψSH(μ) =N

∑l=0

ψSHl

2l + 12

Pl(μ). (105)

One can obtain equations for the Fourier coefficients

ψSHl =

∫ 1

−1ψSH(μ)Pl(μ)dμ (106)

by testing (32) with Pl(μ) and then integrating. Thus we get

∇∫ 1

−1μPl(μ)ψSH(μ)dμ = κ(2〈B〉δl0−ψSH

l )+ σ(ψ0δl0−ψSHl ) (107)

for the moments ψSHl of the distribution function. Using the recursion relation

(l + 1)Pl+1(μ)+ lPl−1(μ) = (2l + 1)μPl(μ) (108)

we obtain

∇(

l + 12l + 1

ψSHl+1 +

l2l + 1

ψSHl−1

)= κ(2〈B〉δl0−ψSH

l )+ σ(ψ0δl0−ψSHl ). (109)

This is a linear system of first order partial differential equations. For a criterion onhow many moments are sufficient for a given problem see [78].

The two most widely used boundary conditions are Mark [54, 55] and Marshak[56] boundary conditions. The idea of the Mark boundary conditions is to assign thevalues of the distribution at certain directions μi which are the zeros of the Legendrepolynomial of order N +1. That this is in fact a natural boundary condition becomesclear in the next section.

Marshak’s boundary conditions, on the other hand, demand that the ingoing halfmoments of the distribution are prescribed, i.e. for the left boundary

∫ 1

0Pl(μ)ψ(μ)dμ . (110)

This, in some sense, reflects the boundary conditions for the full equations.


5.2 Minimum Entropy Closure

The approximations based on the expansion of the distribution function into apolynomial or the equivalent diffusion approximations suffer from serious draw-backs. First, anisotropic situations are not correctly described. This becomesapparent most drastically for a ray of light, where |P| = E . Also, the distributionfunction can become negative and thus the moments computed from the distribu-tion can become unphysical. Second, boundary conditions cannot be incorporatedexactly. At a boundary we usually prescribe the ingoing flux only. Here we have toprescribe values for the full moments. These moments contain the unknown outgo-ing radiation. Moreover, a polynomial expansion cannot capture discontinuities inthe angular photon distribution. Krook [42] remarks that at the boundary there isusually a discontinuity in the distribution between in- and outgoing particles.

In this section, we want to describe one idea which resolves the first problem. Theidea is to use an Entropy Minimization Principle to obtain the constitutive equationfor P. This principle has become the main concept of Rational Extended Thermo-dynamics [61].

We want to explain the Entropy Minimization Principle and its practical appli-cation by means of our simple moment system (99–100). To close the system wedetermine a distribution function ψME that minimizes the radiative entropy

H∗R(ψ) =∫

S 2

∫ ∞

0h∗R(ψ)dνdΩ (111)

with

h∗R(ψ) =2kν2

c3 (n logn− (n + 1) log(n + 1)) where n =c2

2hν3 ψ (112)

under the constraint that it reproduces the lower order moments,

〈ψME〉= E and 〈ΩψME〉= F. (113)

The entropy is the the well-known entropy for bosons adapted to radiation fields[63, 70]. At first sight, it is not clear why the distribution should minimize the en-tropy when all that is known for non-equilibrium processes is that there exists anentropy inequality. But it can be shown [15] that the minimization of the entropy forgiven moments and the entropy inequality are equivalent.

The above minimization problem can be solved explicitly and the pressure canbe written as [16]

P = D( f )E. (114)

Here, f = FE is the relative flux,

D( f ) =1− χ( f )

2I +

3χ( f )−12

f ⊗ f| f |2 (115)


is the Eddington tensor and

χ( f ) =5−2

√4−3| f |23

(116)

is the Eddington factor. The Eddington tensor can always be written in the form(115) under the assumption that the intensity is symmetric about a preferred direc-tion [50]. The minimum entropy Eddington factor satisfies the natural constraints

tr(D) = 1 (117)

D( f )− f ⊗ f ≥ 0 (118)

f 2 ≤ χ( f )≤ 1 (119)

In the literature, the Eddington factor (116) has been derived based on many, appar-ently not connected, ideas. Levermore [50] assumed that there existed a referenceframe in which the distribution was exactly isotropic and used the covariance of theradiation stress tensor. Anile et al. [7] derived it by collecting physical constraintson the Eddington factor and supposing the existence of an additional conservationlaw, where the conserved quantity behaves like the physical entropy near radiativeequilibrium. The minimum entropy system was thoroughly investigated in [16, 84].Further variable Eddington factors have been proposed, cf. [50, 59] and referencestherein.

The closed system has several desirable properties. The flux is limited in a naturalway, i.e. | f | < 1. Physically, this corresponds to the fact that information can-not travel faster than the speed of light. Furthermore, the underlying distributionfunction is always positive. Also, the system can be transformed to a symmetrichyperbolic system [7], which makes it accessible to a general mathematical the-ory [21]. Again, Marshak type boundary conditions can be derived.

5.3 Flux-Limited Diffusion and Entropy Minimization

The classical diffusion approximation is a linear parabolic partial differentialequation. In this equation, information is propagated at infinite speed. This can alsobe seen from the fact that the flux |F | is not bounded by the energy E (relative fluxf < 1). But this should hold, due to the definition of the moments. Thus the classicaldiffusion approximation contradicts fundamental physical concepts.

Therefore the concept of flux-limited diffusion has been introduced. A diffusionequation is called flux-limited if

|F | ≤ E. (120)

The following is a summary of [50]. We begin by writing the moment equations inthe form


∇F = κ(〈B〉−E) (121)

∇(DE) =−(σ + κ)F, (122)

with the Eddington tensor D. Two assumptions in the derivation of the classicaldiffusion equation will be modified. First, the Eddington tensor is only identicallyequal to 1

3 for isotropic radiation. For a ray of light (“free-streaming”), on the otherhand, we should have |DE| = E . Second, one should not neglect ∂tF . Instead, wenote that in the diffusive as well as in the free-streaming regime, the spatial andtemporal derivatives of the relative flux f = F

E and the Eddington tensor D can beneglected.

Rewriting the equations in terms of f and E we get

∇( f E) = κ(〈B〉−E) (123)

∇(DE) =−(σ + κ) f E. (124)

The second equation becomes

∇(DE) =−(σ + κ) f E. (125)

Inserting (123) into (125), we obtain

f ∇ f + ∇((D− f ⊗ f )E)+ σ f E = 0 (126)

with σ = κ〈B〉+σEE . If we drop the derivatives of f and D, we arrive at

(D− f ⊗ f )∇E + σ f E = 0, (127)

or

(D− f ⊗ f )R = f with R =− 1σ

∇EE

. (128)

The idea is now to

1. Choose D as a function of f2. Solve (D− f ⊗ f )R = f for f3. Insert f (R) into the first moment equation to obtain a diffusion approximation

The first step shows how the concept of flux-limited diffusion is related to a (non-linear) moment closure. If

D =1− χ

2I +

3χ−12

f ⊗ f| f | (129)

then f is an eigenvector of D and also of (D− f ⊗ f ) with

(D− f ⊗ f ) f = (χ−| f |2) f . (130)


Hence the equation (D− f ⊗ f )R = f has the solution

R =f

χ− f 2 . (131)

Solving this equation for f and writing the result as

f = λ (R)R (132)

we arrive at the closure

F =− 1σ

λ(

1σ

∇EE

)∇E. (133)

If one chooses for D the minimum entropy Eddington factor then [50]

λ =3(1−β 2)2

(3 + β 2)2 (134)

where β is implicitly given by

R =4β (3 + β 2)(1−β 2)2 . (135)

The same boundary conditions as for the diffusion approximation can be used.

5.4 Partial Moments

In spite of its advantages the minimum entropy system still suffers from a majordrawback. In Fig. 8 we show a numerical test case [11] with two colliding beams.The parameters are κ = 2.5, σ = 0. The temperature inside the medium is zero.

At both sides, beams with a radiative temperature TR :=(

EσSB

)1/4, where σSB is

Stefan–Boltzmann’s constant, of 1000 and relative fluxes of f = ±0.99, respec-tively, enter. Figure 8 shows the radiative energy. The full moment model has aqualitatively wrong solution with two shocks. This is not surprising since this Ed-dington factor, as stated above, is related to radiation which is isotropic in a certainframe [50]. This assumption is violated in the test case above. The unphysical be-havior can be remedied by combining Minimum Entropy with the partial momentidea described in the following.

The partial moment idea is somehow intermediate between the Discrete Ordi-nates approach and Moment Models. In Discrete Ordinates models the integral overall directions is discretized with a numerical quadrature rule. This yields a cou-pled system of finitely many transport equations, each describing transport into onedirection.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

1

1.2

1.4

1.6

1.8

2x 104

x

E =

1/ (

2 π)

∫ I

dΩ

TransportHSM1M1

Fig. 8 Radiative energy. Artificial radiative shock wave in the full moment entropy (M1) model

Let A be a partition of the unit sphere S2, where A ∈ A denotes the set of theangular integration. Instead of integrating over all directions we average over eachA ∈A separately. Thus we define the average

〈 · 〉A :=∫

A· dΩ . (136)

Again, we multiply the transport equation by 1 and Ω and average over each A∈Ato obtain

∇FA = 〈S〉A (137)

∇PA = 〈ΩS〉A. (138)

We define the corresponding partial moments by

EA = 〈I〉A (139)

FA = 〈Ω I〉A (140)

PA = 〈(Ω ⊗Ω)I〉A. (141)

To close this system we have to find an equation for the partial pressures PA asfunctions of the partial energies EA and partial fluxes FA.

Examples for the choice of A , which are used later, are

• For the full moment model we have A = S2, i.e. the integral is over the full sphere.


• For the half moment model we have A ∈ {S2+,S2−}. Here, S2

+ = {Ω ∈ S2 :Ωx > 0} is the positive half sphere, where the x-component of Ω is positive, andS2− = {Ω ∈ S2 : Ωx < 0} analogously is the negative half sphere.

• For the quarter moment model we have A ∈ {S2++,S2

+−,S2−−,S2−+}. Here,S2

++ = {Ω ∈ S2 : Ωx > 0,Ωy > 0} is the quarter sphere in the first quadrant.Analogously, S2

+− = {Ω ∈ S2 : Ωx > 0,Ωy < 0} etc.

One could also choose other sets for the angular integration.

5.5 Partial Moment PN Closure

The basic idea of the PN closure is to expand the photon distribution into a polyno-mial. Here we use the same idea, but separately for both half ranges. This approachhas been investigated in the literature in different forms and contexts and mostlyin connection with boundary conditions, for example recently in [11]. Schuster andSchwarzschild [73, 74] introduce two constant distributions for left- and rightgoingphotons (P0 approximation). Krook [42], based on ideas of Sykes [79], considershalf moment in one space dimension with a PN closure. Sherman [77] comparesfull-PN and half-PN numerically in 1D. Ozisik et al. [64] derive a half moment P1

closure in spherical geometry. Further references can be found in [57], where alsoan octuple P1 closure in cylindrical geometry is introduced. Similar ideas appear inrelated subjects, like gas dynamics, cf. [13] and references therein.

For the half moment P1 system in one space-dimension, for instance, we assumethat in each half range the distribution can be represented by a polynomial of degreeone. The coefficients of the polynomial are determined by the constraint that thelower order half moments should be reproduced. The half moment P1 system reads,

∂xF+ = κ(

12〈B〉−E+

)+ σ

(12(E+ + E+)−E+

)(142)

∂x(χ+( f+)E+) = κ(

14〈B〉−F+

)+ σ

(14(E+ + E+)−F+

)(143)

∂xF− = κ(

12〈B〉−E−

)+ σ

(12(E+ + E+)−E+

)(144)

∂x(χ−( f−)E−) = κ(−1

4〈B〉−F−

)+ σ

(−1

4(E+ + E+)−F−

). (145)

The partial Eddington factors are

χ±( f±) =−16± f± with f± =

F±E±

. (146)

We note that this is a hyperbolic system. The eigenvalues associated to the “+”moments are positive, while the eigenvalues associated to the “−” moments are


negative, in accordance with physical intuition. This structure makes the formulationof accurate boundary conditions easy. We simply prescribe the ingoing halfmoments, in accordance with the conditions for the full equations. For more discus-sions, including existence and uniqueness results, and the explicit quarter momentP1 closure in two space-dimensions we refer the reader to [72].

5.6 Partial Moment Entropy Closure

The partial moment entropy closure was introduced for radiative heat transfer in [19]and developed in [18, 25, 85]. For the sake of completeness we recall the procedureexplained earlier. We have to find a distribution function ψME that minimizes theradiative entropy H∗R given by (111–112), under the constraint that it reproduces thelower order partial moments,

〈ψME〉A = EA and 〈ΩψME 〉A = FA (147)

for all A ∈A . The minimizer is given by

ψME = ∑A∈A

1

α4A(1 + βA ·Ω)4

1A, (148)

where αA and βA are Lagrange multipliers corresponding to the constraints. Thisformula differs from the one given in [19] since we consider frequency-averagedquantities here. It can be obtained from the minimizer in [19] by integration over ν .

In the case of A = {S2+,S2−}, the half moments over this distribution can be

computed explicitly.Note that E± ≥ 0,F+≥ 0,F− ≤ 0. Multiplying the transfer equation with m(μ) =

1+,1−,μ+,μ−) and integrating with respect to ν and μ we get

ε∂xF+ = κ(

12〈B〉−E+

)+ σ

(12(E+ + E+)−E+

)(149)

ε∂x〈(μ+)2ψ〉= κ(

14〈B〉−F+

)+ σ

(14(E+ + E+)−F+

)(150)

ε∂xF− = κ(

12〈B〉−E−

)+ σ

(12(E+ + E+)−E+

)(151)

ε∂x〈(μ−)2ψ〉= κ(−1

4〈B〉−F−

)+ σ

(−1

4(E+ + E+)−F−

). (152)

This system is closed by an entropy minimization principle. Then, the minimizerψME is determined by

H∗R(ψME) = minψ{H∗R(ψ) : 〈m(μ)ψ〉= (E+,E−,F+,F−)} ,


i.e. ψME minimizes the entropy under the constraint of a reproduction of the halfspace moments with E+ ≥ 0,E− ≥ 0,F+ ≥ 0,F− ≤ 0 as above.Solutions of similiar minimization problems are discussed in [16]. A straightforwardcomputation shows that in the present case the unique solution ψME = ψME(T,μ) =ψME(T,μ ,ν) is given by

ψME(T,μ) =2hν3

c2

1

exp( hνkT (α−(1−+ β−μ−)+ α+(1+ + β+μ+)))−1

,

where α+ > 0,α− > 0 and β+ >−1,β− < 1 are determined by the constraints

〈1±ψME〉= σSB

πT 4 β 2±±3β±+ 3

3α4±(1±β±)3= E±

〈μ±ψME〉= σSB

πT 4 β±±3

6α4±(1±β±)3= F±.

The temprature T is introduced here as a normalization parameter to measure thedeviation from the usual Planckian. We mention in passing that in general maxi-mization of entropy is a touchy business, see for example [34].

Having solved the minimization problem, one obtains

1c〈(μ±)2ψME〉= σ

cπT 4 1

3α4±(1±β±)3= χ±( f±)E±,

if we define the relative fluxes f± = F±cE± and the Eddington factors

χ±( f±) =8 f 2±

1±6 f±+√

1±12 f±−12 f 2±.

Approximating 〈(μ±)2ψ〉 ∼ 〈(μ±)2ψME〉 in (149)–(152) and using the abovecomputation we arrive at

∂xF+ = κ(

12〈B〉−E+

)+ σ

(12(E+ + E+)−E+

)(153)

∂x(χ+( f+)E+) = κ(

14〈B〉−F+

)+ σ

(14(E+ + E+)−F+

)(154)

∂xF− = κ(

12〈B〉−E−

)+ σ

(12(E+ + E+)−E+

)(155)

∂x(χ−( f−)E−) = κ(−1

4〈B〉−F−

)+ σ

(−1

4(E+ + E+)−F−

). (156)


The system (153–156) is a stationary hyperbolic equation with relaxation terms. Ananalysis of the equations has been performed in [26].

The eigenvalues associated to E+ and F+ are always positive while theeigenvalues associated to E− and F− are always negative. This result agrees withintuition since E+ and F+ describe transport to the right, while E− and F− describetransport to the left. Therefore we have to prescribe two boundary conditions on theleft and right hand side, respectively. We use at x = 0:

F+ = 〈μ+ [ρ(μ)ψME(T,−μ)+ (1−ρ(μ))B(Tout)]〉 (157)

χ+E+ = 〈(μ+)2 [ρ(μ)ψME(T,−μ)+ (1−ρ(μ))B(Tout)]〉 (158)

and the analogous conditions at x = 1. Equations (153)–(158) are solved togetherwith the temperature equation in the form

ε2∂tT = ε2k∂xxT −2πκ(

2σSB

πT 4− (E+ + E−)

)(159)

and corresponding boundary conditions.The Partial Moment Entropy approximation has a lot of desirable physical and

mathematical properties. The underlying distribution function is always positive.Hence the relative flux and the speed of propagation are limited. The system issymmetriziable hyperbolic. This makes it accessible to a powerful mathematicaltheory guaranteeing well-posedness locally in time. Like the full moment entropyapproximation [16], the system correctly approaches the diffusive limit and the free-streaming limit. The eigenvalues of the half moment and quarter moment entropyaproximation have a special structure. For the half moment case, the eigenvaluesof the “+” direction are always positive, the eigenvalues of the “−” direction arealways negative. Both are bounded in modulus by the speed of light c. This propertymakes very simple and accurate numerical schemes possible, for example kineticschemes or upwind schemes. The formulation of accurate boundary conditions isagain straight-forward.

6 Frequency-Averaged Moment Equations

Moment models are obtained by testing (32) with functions depending on direction,in our case (1,μ)T , then integrating the result over all the directions and frequencies.Then, the system does only depend on time and space variables, and is hence farcheaper to solve. However, this has a cost since we are not always able to reproduceneither frequency dependent problems nor very stiff directional configurations suchas the collision of two opposite beams [11, 19].

In order to solve this difficulty, we do not average over all directions and allfrequencies but distinguish photons going to the left and to the right and differentfrequency bands.


Let

〈g〉+m =

νm+ 1

2∫

νm− 1

2

∫ 1

0gdμdν and 〈g〉−m =

νm+ 1

2∫

νm− 1

2

∫ 0

−1gdμdν (160)

denote the average over all right/left-going photons in the mth frequency band[νm− 1

2,νm+ 1

2[. We denote the bands as half-open intervals to have mathemati-

cally disjoint sets. However, since only integrals over the bands matter, one couldalso use closed intervals. The moments E+

R,m = 〈ψ〉+m , F+R,m = 〈Ωψ〉+m and P+

R,m =〈(Ω ⊗Ω)ψ〉+m are respectively the radiative energy, the radiative flux and the radia-tive pressure inside the mth group and the positive half-space. The quantities for thenegative half space are defined in analogy.

Testing (32) with (1,μ)T and averaging with the above defined averages we get

∂xF+R,m = κ+

m aθ 4m,+− κ+

m E+R,m + σ+

m

(E+

R,m+E−R,m2 −E+

R,m

)(161)

∂xP+R,m = κ+

ma2 θ 4

m,+− κ+m F+

R,m− σ+m

(E+

R,m+E−R,m4 −F+

R,m

)(162)

and

∂xF−R,m = κ−m aθ 4m,−− κ−m E−R,m + σ−m

(E+

R,m+E−R,m2 −E−R,m

)(163)

∂xP−R,m =−κ−m a2 θ 4

m,−− κ−m F−R,m− σ−m

(−E+

R,m+E−R,m4 −F−R,m

), (164)

where we have used the following frequency averages of the frequency dependentquantites κ and σ :

κ+m =

〈κB〉+m〈B〉+m , κ+

m =〈κψ〉+m〈ψ〉+m , κ+

m =〈κμψ〉+m〈μψ〉+m and σ+

m =〈σψ〉+m〈ψ〉+m . (165)

6.1 Entropy Minimization

For each m (161)–(164) is a system of 4 equations for 6 unknown moments. Toobtain a well-posed system one usually expresses the highest moment, here P±R,m, asa function of the lower order moments, here E±R,m and F±R,m. This is referred to as“closure” of the system.

To close the system here, we use entropy minimization, see [7, 16, 51, 61]. Com-pare [19] for the grey half space model and [83] for the multigroup full space model.

Let us first recall the definition of the radiative entropy,

hR(I) =2kν2

c3

[nI lnnI− (nI + 1) ln(nI + 1)

](166)


where

nI =c2

2hν3 ψ . (167)

According to the entropy minimization principle, we determine a distributionfunction that minimizes the radiative entropy under the constraint that it reproducesthe lower order moments,

HR(ψME) = minψ

{H(ψ) = ∑

m(〈hR(ψ)〉+m + 〈hR(ψ)〉−m) : : |

∀m : 〈ψ〉±m = E±m and c〈μψ〉±m = F±m}. (168)

This gives the closure function,

ψME(Ω ,ν)=∑m

1[νm−1

2;ν

m+12[2hν3

c2

[exp(

hνk

(α+m (1+β +

m μ+)+α−m (1+β−m μ−))−1]−1

(169)where α±m ,β±m are Lagrange multipliers, that are defined to reproduce the moments.

6.2 Inversion of the System

The next step is to express the Lagrange multipliers α±m ,β±m as functions of E±R,m,F±R,m and to substitute

P±R,m ≈ 〈ψME(α±m ,β±m )〉±m = 〈ψME(E±R,m,F±R,m)〉±m . (170)

Hence we obtain a system for E±R,m and F±R,m.For the grey half space model [19], the Lagrange multipliers as functions of the

moments can be computed explicitly. However, with the introduction of multigroupvariables this is not the case anymore. Integrations require the knowledge of thefollowing function,

Ξ(η) =η∫

0

ξ 3[exp(ξ )−1]−1dξ . (171)

For example,

E+m =

1c

∫ 1

0

∫ νm+ 1

2

νm− 1

2

ψME dνdμ

=∫ 1

0

2k4

h3c3 (α+m (1 + β +

m μ+))−1(Ξ(ν ′m+ 1

2)−Ξ(ν ′

m− 12))dμ (172)


with ν ′ = hνk α+(1 + β +μ+). Unfortunately, except for η = 0 and η = +∞ there is

no analytic expression of Ξ . A numerical calculation would be too expensive sincewe have to be very accurate. Therefore, in [83] an approximation was introduced,which we can also be used here,

Ξ(η) C∞ + exp(−C∗η)imax

∑i=0

Ciη i (173)

The constants Ci are chosen so that the approximation has a very good behaviour inthe vicinity of η = 0. For our applications, taking imax = 5 is sufficient.

Once this approximation is made, it is possible to integrate and hence to computethe Lagrange multipliers of the minimization problem as functions of the moments.Then, we are able to compute the radiative pressures as functions of the radiativeenergies and fluxes. Moreover, we can show that we can write the pressures inEddington form, P±R = D±R E±R , where

D±m =(1− χ±m )

2: Id +

(3χ±m −1)2

:F±m ⊗F±m∥∥F±m

∥∥2 . (174)

The scalars χ±m are called Eddington factors.

6.3 Properties

The multigroup half space model keeps the interesting properties of the other mo-ment models closed by entropy minimization, that is to say

• The main physical properties remain: conservation of the total energy and dissi-pation of the total entropy. Moreover, the addition of multigroup allows to havea better balanced-energy in the case of strongly frequency-dependant problems.

• The model naturally limits the flux. This property can be expressed as follows:

∀m, :F±mE±m

< 1 (175)

This means that the photons cannot travel faster than the speed of the light. Wenote that this important property is often not satisfied by macroscopic models.

• For 1D problems, it is very easy to make a simple numerical scheme that canefficiently solve every possible angular configuration. This is done only by usingupwind schemes (see [19]). We chose to develop only a four-moments model toobtain a simple and very competitive model. However, in some situations onemight need more moments to capture the physical solution [78].

• The cost of the method is low and can be lowered to be less than the number ofgroups times the cost of the half space model by doing a pressure precalculation.


These properties are the most important ones but it is to note that the multigrouphalf space model keeps all the properties (and limitations) of both the half space [19]and multigroup full space [83] models.

7 Numerical Comparisons

7.1 Numerical Results

The approximations presented above have different mathematical structures. TheDiscrete Ordinates, Spherical Harmonics and partial PN equations are linear firstorder partial differential equations. The minimum entropy and the partial momententropy system are nonlinear hyperbolic first order partial differential equations. Onthe other hand the diffusion and flux-limited diffusion equations are parabolic equa-tions, whereas the SPN equations are elliptic/parabolic. We remark that, althoughthey are closely related, the minimum entropy moment model and flux-limited dif-fusion with the same Eddington factor are not completely equivalent, but can in facthave very different solutions. For example, the solutions for the minimum entropysystem can have shocks whereas this is impossible for flux-limited diffusion.

In the following Figures we show some numerical comparisons of the differentmodels. The abbreviations in the legends mean

• S40/Transport: Discrete Ordinates Solution with 40 directions• P1: P1 approximation with Marshak boundary conditions• SP1: SP1/Diffusion approximation with Marshak boundary condition• FLD: flux-limited diffusion with minimum entropy Eddington factor and

Marshak boundary conditions• HSP1: half P1 approximation• HSM1: half moment entropy approximation• Quarter Space: quarter moment entropy approximation

The transport solution has been obtained with a direct discretization as describedabove. The parabolic equations SP1 and FLD have been discretized with a stan-dard finite difference scheme. For the balance laws P1, HSP1, HSM1 and QuarterSpace we used kinetic schemes based on the distribution function from the momentclosure. All of the latter systems have eigenvalues in modulus less than the speedof light. Thus, similar CFL conditions hold. To be valid in the diffusive limit, thekinetic schemes can be modified to become asymptotic preserving, cf. [19] for asimple analysis in 1D.

7.2 Grey Transport

First, we investigate the transport equation (with fixed temperature) withoutfrequency dependence. In Fig. 9 we consider a given temperature profile in the unit


0 0.05 0.1 0.15 0.2 0.25 0.30.8

0.9

1

1.1

1.2

1.3

1.4

1.5x 105

x

S40HSP1HSM1P1SP1FLD

E =

1/ (

2 π)

∫ I

dΩ

Fig. 9 Steady radiative energy for a fixed temperature profile T (x) = 1000+800x in the interval[0,1], κ = 1, σ = 0.1

interval [0,1], T (x) = 1000 + 800x. This temperature enters into the Planck sourceterm 〈B〉 via Stefan–Boltzmann’s law

〈B〉(T ) = σSBT 4. (176)

At the boundary we prescribe black body radiation at the corresponding temperatureas ingoing radiation. In Fig. 9 we see that the high order Discrete Ordinates solution(considered as benchmark result) and the half moment approximations agree verywell, whereas P1, SP1 and flux-limited diffusion differ significantly.

This becomes more striking in the 2D example in Fig. 10. The P1 and SP1 ap-proximations are unable to capture the simple anisotropy in this test case, whereasthe quarter moment model and the solution of the full equations agree very well.

7.3 Grey Cooling

Here we apply the above methods to a cooling problem. We use an initial tempera-ture of 1,000 K and an outside temperature of 300 K. The parameters a, ε , κ are setequal to 1. α and the reflectivity are chosen equal to 0. k is chosen equal to 1 and 0.1,i.e. we consider two situations where heat conduction and radiation are dominating,respectively. u is chosen equal to zero. The gridsize is Δx = 0.01, Δ t = 10−4. Forthe radiative transfer solution a Gaussian quadrature with 64 points is used for theangular discretization. We use the above first order finite difference discretization


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

4

6

8

10

12

14

16x 104

x

1/4

π ∫ I

dΩ

κ=0.01, σ=0.1

TransportQuarter SpaceSP3P1

Fig. 10 Steady radiative energy for a fixed temperature profile T (x) = 1000+400(x+y) in [0,1]2,κ = 0.01, σ = 0.1. Cut along the diagonal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

590

600

610

620

630

640

650

660

670

T

Half SpaceTransportSP3P1

Fig. 11 Temperatures at time t = 0.01 with k = 1

in space and a Newton iteration to obtain an approximate solution of the nonlin-ear equations (153–156). The calculated temperatures are shown in Figs. 11 and 13.The mean intensities, i.e. E+ + E− in the half moment case and < μI > for the ra-diative transfer solutions, are shown in Figs. 12 and 14. The results obtained withthe half space moment method, the P1 approximation, the SP3 approximation [47]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4x 104

x

E


Fig. 12 Mean intensity at time t = 0.01 with k = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

540

560

580

600

620

640

660

680

T


Fig. 13 Temperatures at time t = 0.01 with k = 0.1

and the solution of the full transport equation are compared. We note that the usualRosseland or diffusion approximation [45] gives in all cases results which are farless accurate than the solutions considered here. As can be seen in the figures thehalf moment method outperforms the other methods in both cases.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4x 104

x

E


Fig. 14 Mean intensity at time t = 0.01 with k = 0.1

7.4 Multigroup Transport

First we consider only the equation for the radiative intensity with a fixed mattertemperature profile. We divide the spectrum into four bands [λi− 1

2,λi+ 1

2[ (]νi+ 1

2,

νi− 12] respectively) with piecewise constant κi on [λi− 1

2,λi+ 1

2[. We used λ 1

2= 0 μm,

λ 32

= 1.035 μm, λ 52

= 2.07 μm, λ 72

= 7 μm and λ 92

= ∞ and σ = 0.

In Figs. 15–17 we compare the results obtained with the half space momentmodel to the solution of the full RHT equations using a source iteration as wellas diffusive P1 and SP3 approximations. For details on these equations we refer thereader to [47]. The classical Rosseland approximation gives in all cases consideredhere far less accurate results.

For the radiative energy

ER =∫ ∞

0

∫ 1

−1ψdμdν = ∑

m(E+

R,m + E−R,m) (177)

we define, in analogy to Stefan’s law, the radiative temperature

TR :=(

2πER

a

)1/4

. (178)

The parameters corresponding to Fig. 15 are κ1 = 100 m−1, κ2 = 1 m−1, κ3 = 10m−1, κ4 = ∞ and represent a rather diffusive, optically thick physical regime.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11000

1100

1200

1300

1400

1500

1600

1700

1800

x

TR

TransportHalf SpaceSP3P1

0 0.05 0.1

1100

1150

1200

Clipping

Fig. 15 Steady radiative temperature for a fixed matter temperature profile, T (x) = 1000+800x,Tb(0) = 1000, Tb(1) = 1800. Diffusive regime. Four frequency bands

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11450

1500

1550

1600

1650

1700

1750

1800

x

TR


0 0.05 0.1 0.15 0.21450

1500

1550Clipping

Fig. 16 Steady radiative temperature for a fixed matter temperature profile, T (x) = 500+1500x,Tb(0) = 500, Tb(1) = 2000. Transport regime. Four frequency bands

The half space model performs better than the diffusive approximations which aredesigned for this physical situation. The differences become more striking in Fig. 16where we chose a rather opposite physical regime with large photon mean free path,κ1 = 0.1 m−1, κ2 = 0.01 m−1, κ3 = 1 m−1, κ4 = ∞. We chose the same absorp-tion coefficients in Fig. 17. However, while in the first two cases the boundary


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

1540

1560

1580

1600

1620

1640

1660

1680

1700

1720

TR


Fig. 17 Steady radiative temperature for a fixed matter temperature profile, T (x) = 500, Tb(0) =1500, Tb(1) = 2000. Transport regime. Four frequency bands

temperature agreed with the interior matter temperature we chose here a muchhigher boundary temperature which corresponds to heat flux entering the medium.The half space model is far more accurate than the diffusive approximations.

7.5 Multigroup Cooling

In our next test case we consider the transport equation coupled to the heat equation.We use k = h = 1, α = 0 and ρ = 0. The outside temperature is Tb = 1,000 at theleft and Tb = 1,800 at the right boundary. The scattering and absorption coefficientsare chosen as in our second and third uncoupled test cases. In Fig. 18 we show thesteady radiative temperature. Again, the new half space model agrees best with thefull transport solution.

7.6 Adaptive methods for the Simulation of 2-d and 3-d CoolingProcesses

The application that we study here is the cooling of a glass cube representing a typ-ical fabrication step in glass manufacturing. We consider clean glass, which meansthat the treatment of scattering can be omitted. The frequencies are approximatedby an eight-band model. The values used are given in Table 2. Furthermore, we set

k = 1 , h = 0.001 , Tb = 300


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11440

1460

1480

1500

1520

1540

1560

1580

1600

1620

1640

x

TR


Fig. 18 Steady radiative temperature for the coupled equations, Tb(0) = 1,000, Tb(1) = 1,800.Transport regime. Four frequency bands

and start with a uniform temperature distribution T0(x)=1,000. The time integrationis stopped at t =0.001.

We use a space-time adaptive method described in detail in [38]. We validatethe SPN-solutions with numerical solutions to the full RHTE. The full RHTE issolved by a diamond differencing discretization coupled with a discrete ordinatemethod which uses 60 directions [10, 76]. This is for the present situation sufficientto obtain an accurate solution for the transport problem provided the spatial grid ischosen fine enough.

7.6.1 Two-Dimensional Glass Cooling

We consider an infinitely long square glass block which allows us to use a two-dimensional approximation on the scaled square domain Ω =[0,1]2.

In Fig. 19, we show temperature distributions at the final time te =0.001 obtainedfor the SP3-approximation. As expected, the strongest cooling takes place in the cor-ners of the computational domain. The meshes automatically chosen by our adaptiveapproach are highly refined at the boundary caused by the steep temperature gradi-ents there. In this case, a stable uniform discretization of the two-dimensional RHTErequires the solution of a linear system with more than 4.8 million unknowns in eachtime step, whereas the dimension of the linear algebraic systems for the adaptiveSP3-approximation is not greater than 272,000.

In Fig. 20 the SPN-solutions are compared to the full RHTE- and Rosselandapproximation. In particular, they reconstruct the temperature near the boundarymuch more accurately than the Rosseland approximation which is often used inengineering practice.


Fig. 19 Two-dimensional temperature distributions and spatial meshes on Ω = [0,1]2 resultingfrom SP3-approximations at te = 0.001. The temperature axis ranges from 300 to 1,000. Strongrefinement takes place in the boundary layer due to the large temperature gradients there


300

400

500

600

700

800

900

1000

0 0.2 0.4 0.6 0.8 1

TE

MP

ER

AT

UR

E

X

EPS=1.0

SP1

SP3

RHTE

ROSSELAND

700

750

800

850

900

950

1000

0 0.02 0.04 0.06 0.08 0.1 0.12

TE

MP

ER

AT

UR

E

X

EPS=1.0

SP1

SP3

RHTE

Fig. 20 Comparison of two-dimensional temperature distributions at te = 0.001 along the liney = 0.5 obtained from different radiation models. The SP3-solution matches very well with theRHTE solution inside the glass cube. Some differences are visible in the boundary region. BothSPN -approximations give much more accurate results than the Rosseland approximation

The time steps in the adaptive procedure increase rapidly by two orders of mag-nitude reflecting the ongoing diffusive smoothing in the boundary layer. Altogether9 and 24 time steps are needed. In contrast, a uniform time discretization yieldingthe same accuracy, requires 100 steps.


Concerning the computation times the parameters discussed above lead to thefollowing results: Using the method described above without adaptivity in space andtime for Rosseland, SP1 and SP3 the computational effort is approximately doubled

484 704.5 925

Fig. 21 Three-dimensional temperature distribution and adaptive spatial mesh on Ω = [0,1]3

resulting from the SP3-approximation at te =0.001. We removed one small cube to present detailsfrom inside the glass cube. Refinement takes place in the boundary layer due to the large tempera-ture gradients there. The adaptive mesh consists of 82,705 grid points


300

400

500

600

700

800

900

1000

0 0.2 0.4 0.6 0.8 1

TE

MP

ER

AT

UR

E

X

EPS=1.0

SP1

SP3

RHTE

ROSSELAND

700

750

800

850

900

950

1000

0 0.02 0.04 0.06 0.08 0.1

TE

MP

ER

AT

UR

E

X

EPS=1.0

SP1

SP3

RHTE

Fig. 22 Comparison of three-dimensional temperature distributions at te = 0.001 along the liney= z=0.5 obtained from different radiation models. The SP3-solution matches very well with theRHTE solution, whereas the Rosseland approximation gives quite poor results

from Rosseland to SP1 and from SP1 to SP3. The solution of the RHT problem usingthe multigrid method described in [76] takes again approximately twice as muchtime as the SP3 solution for the same accuracy. Adaptivity in space yields a factorof 3–5 in computation time for the present situation and adaptivity in time yields afactor of 10–50.


7.6.2 Three-Dimensional Glass Cooling

We consider a glass block which is represented by a scaled cube Ω =[0,1]3.Figure 21 displays the SP3-solution and the adaptive three-dimensional grid

chosen by our method for TOLx=0.01. As already observed in the two-dimensionalcase, the SPN-solutions approximate the temperature computed from the full RHTEvery well. In contrast to the Rosseland approximation, they exhibit physically cor-rect boundary layers as can be seen from Fig. 22. To accurately capture theseboundary layers, the use of local refinement is essential.

The mesh shown in Fig. 21 consists of 82,705 nodes, leading to a linear systemof order 1,405,985. A uniform method requires approximately 250,000 grid pointsto reach a comparable solution quality. The solution of the full RHTE is done on a100× 100× 100-grid, yielding a linear system with 480 million unknowns whichhas to be solved in each time step. The comparison of the computation times yieldssimiliar results as in 2-D.

To conclude, these investigations show that the SPN-equations and moment meth-ods described above are a relatively inexpensive way to improve the accuracy ofclassical diffusion models. Compared to the solution of full radiative heat transferequations, the complexity and computer time are considerably reduced. Further re-duction can be achieved by fully adaptive discretization methods steered by robusta posteriori error estimators.

Acknowledgements We wish to thank all our collaborators and co-authors, in particular B.Dubroca, T. Gotz, J. Lang, E.W. Larsen, M. Seaıd, G. Thommes, R. Turpault and R. Pinnau. Partsof this work have been taken from the articles [18, 23–25, 27, 38, 47, 48, 76, 85]. This work wassupported by German Research Foundation DFG under grants KL 1105/7 and 1105/14.

References

1. Adams, M.L., Larsen, E.W.: Fast iterative methods for deterministic particle transportcomputations. Prog. Nucl. Energy 40, 3–159 (2002)

2. Adams, M.L.: Subcell balance formulations for radiative transfer on arbitrary grids. Transp.Theory and Stat. Phys. 26, 385–431 (1997)

3. Alcouffe, R., Brandt, A., Dendy, J., Painter, J.: The multigrid method for diffusion equationswith strongly discontinuous coefficients. SIAM J. Sci. Stat. Comp. 2, 430–454 (1981)

4. Alcouffe, R.: Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations. Nucl. Sci. Eng. 64, 344–355 (1977)

5. Agoshkov, V.: On the existence of traces of functions in spaces used in transport theory prob-lems. Sov. Math. Dokl. 33, 628–632 (1986)

6. Agoshkov, V.: Boundary value problems for transport equations. Birkhauser, Boston (1998)7. Anile, A.M., Pennisi, S., Sammartino, M.: A thermodynamical approach to Eddington factors.

J. Math. Phys. 32, 544–550 (1991)8. Atkinson, K.E.: Iterative variants of the Nystrom method for the numerical solution of integral

equations. Numer. Math. 22, 17–31 (1973)9. Brantley, P.S., Larsen, E.W.: The simplified P3 approximation. Nucl. Sci. Eng. 134, 1–21

(2000)


10. Brown, P.N.: A linear algebraic development of diffusion synthetic acceleration for three-dimensional transport equations. SIAM. J. Numer. Anal. 32, 179–214 (1995)

11. Brunner, T.A., Holloway, J.P.: One-dimensional Riemann solvers and the maximum entropyclosure. J. Quant. Spectrosc. Radiat. Transfer 69, 543–566 (2001)

12. Cheng, P.: Dynamics of a radiating gas with applications to flow over a wavy wall. AIAA J.4, 238–245 (1966)

13. Clause, P.-J., Mareschal, M.: Heat transfer in a gas between parallel plates: Moment methodand molecular dynamics. Phys. Rev. A 38, 4241–4252 (1988)

14. Davison, B.: Neutron transport theory. Oxford University Press, Oxford (1958)15. Dreyer, W.: Maximisation of the entropy in non-equilibrium. J. Phys. A 20, 6505–6517 (1987)16. Dubroca, B., Feugeas, J.L.: Entropic moment closure hierarchy for the radiative transfer equa-

tion. C. R. Acad. Sci. Paris Ser. I 329, 915–920 (1999)17. Dubroca, B.: These d’Etat, Dept. of Mathematics, University of Bordeaux (2000)18. Dubroca, B., Frank, M., Klar, A., Thommes, G.: Half space moment approximation to the

radiative heat transfer equations. Z. Angew. Math. Mech. 83, 853–858 (2003)19. Dubroca, B., Klar, A.: Half moment closure for radiative transfer equations. J. Comput. Phys.

180, 584–596 (2002)20. Eddington, A.: The Internal Constitution of the Stars. Dover, New York (1926)21. Fischer, A.E., Marsden, J.E.: The Einstein evolution equations as a first-order quasi-linear

hyperbolic system I. Commun. Math. Phys. 26, 1–38 (1972)22. Fiveland, W.A.: The selection of discrete ordinate quadrature sets for anisotropic scattering.

ASME HTD. Fundam. Radiat. Heat Transf. 160, 89–96 (1991)23. Frank, M.: Partial Moment Models for Radiative Transfer. PhD thesis, TU Kaiserslautern

(2005)24. Frank, M.: Approximate models for radiative transfer. Bull. Inst. Math. Acad. Sinica (New

Series) 2, 409–432 (2007)25. Frank, M., Dubroca, B., Klar, A.: Partial moment entropy approximation to radiative transfer.

J. Comput. Phys. 218, 1–18 (2006)26. Frank, M., Pinnau, R.: Analysis of a half moment model for radiative heat transfer equations.

Appl. Math. Lett. 20, 189–193 (2007)27. Frank, M., Seaıd, M., Janicka, J., Klar, A., Pinnau, R.: A comparison of approximate models

for radiation in gas turbines. Prog. Comput. Fluid Dyn. 4, 191–197 (2004)28. Golse, F., Perthame, B.: Generalized solution of the radiative transfer equations in a singular

case. Commun. Math. Phys. 106, 211–239 (1986)29. Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)30. Hackbusch, W.: Multi-Grid Methods and Applications. Springer Series in Computational

Mathematics, vol. 4. Springer, New York (1985)31. Howell, R., Siegel, J.R.: Thermal Radiation Heat Transfer, 3rd edn. Taylor & Francis,

New York (1992)32. Huang, K.: Introduction to Statistical Physics. Taylor and Francis, New York (2001)33. Jeans, J.H.: The equations of radiative transfer of energy. Mon. Not. R. Astron. Soc. 78, 28–36

(1917)34. Junk, M.: Domain of definition of levermore’s five-moment system. J. Stat. Phys. 93,

1143–1167 (1998)35. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia

(1995)36. Kelley, C.T.: Multilevel Source Iteration Accelerators for the Linear Transport Equation in

Slab Geometry. Transp. Theory Stat. Phys. 24, 679–707 (1995)37. Kelley, C.T.: Existence and uniqueness of solutions of nonlinear systems of conductive

radiative heat transfer equations. Transp. Theory Stat. Phys. 25, 249–260 (1996)38. Klar, A., Lang, J., Seaid, M.: Adaptive solutions of SPN-Approximations to radiative heat

transfer in glass. Int. J. Therm. Sci. 44, 1013–1023 (2005)39. Klar, A., Schmeiser, C.: Numerical passage from radiative heat transfer to nonlinear diffusion

models. Math. Mod. Meth. Appl. Sci. 11, 749–767 (2001)


40. Klar, A., Siedow, N.: Boundary layers and domain decomposition for radiative heat transferand diffusion equations: Applications to glass manufacturing processes. Eur. J. Appl. Math.9–4, 351–372 (1998)

41. Korganoff, V.: Basic Methods in Transfer Problems. Dover, New York (1963)42. Krook, M.: On the solution of equations of transfer. Astrophys. J. 122, 488 (1955)43. Laitinen, M.T., Tiihonen, T.: Integro-differential equation modelling heat transfer in

conducting, radiating and semitransparent materials. Math. Meth. Appl. Sci. 21, 375–392(1998)

44. Laitinen, M.T., Tiihonen, T.: Conductive-radiative heat transfer in grey materials. Quart. Appl.Math. 59 737–768 (2001)

45. Larsen, E.W., Keller, J.B.: Asymptotic solution of neutron transport problems for small meanfree path. J. Math. Phys. 15, 75 (1974)

46. Larsen, E.W., Pomraning, G., Badham, V.C.: Asymptotic analysis of radiative transfer prob-lems. J. Quant. Spectr. Radiati. Transf. 29, 285–310 (1983)

47. Larsen, E.W., Thommes, G., Klar, A., Seaıd, M., Gotz, T.: Simplified PN approximations tothe equations of radiative heat transfer in glass. J. Comput. Phys. 183, 652–675 (2002)

48. Larsen, E.W., Thoemmes, G., Klar, A. : New frequency-averaged approximations to the equa-tions of radiative heat transfer. SIAM Appl. Math. 64 565–582 (2003)

49. Lentes, F.T., Siedow, N.: Three-dimensional radiative heat transfer in glass cooling processes.Glastech. Ber. Glass Sci. Technol. 72, 188–196 (1999)

50. Levermore, C.D.: Relating Eddington factors to flux limiters. J. Quant. Spectroscop. Radiat.Transf. 31, 149–160 (1984)

51. Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83 (1996)52. Lewis, E.E., Miller, W.F. Jr., Computational Methods of Neutron Transport. Wiley, New York

(1984)53. Lopez-Pouso, O.: Trace theorem and existence in radiation. Adv. Math. Sci. Appl. 10, 757–

773 (2000)54. Mark, J.C.: The spherical harmonics method, part I. Tech. Report MT 92, National Research

Council of Canada (1944)55. Mark, J.C.: The spherical harmonics method, part II. Tech. Report MT 97, National Research

Council of Canada (1945)56. Marshak, R.E.: Note on the spherical harmonic method as applied to the milne problem for a

sphere. Phys. Rev. 71, 443–446 (1947)57. Menguc, M.P., Iyer, R.K.: Modeling of radiative transfer using multiple spherical harmonics

approximations. J. Quant. Spectrosc. Radiat. Transf. 39 (1988), 445–461.58. Mercier, B.: Application of accretive operators theory to the radiative transfer equations.

SIAM J. Math. Anal. 18, 393–408 (1987)59. Minerbo, G.N.: Maximum entropy Eddington factors. J. Quant. Spectrosc. Radiat. Transf. 20,

541–545 (1978)60. Modest, M.F.: Radiative Heat Transfer, 2nd edn. Academic, San Diego (1993)61. Muller, I., Ruggeri, T.: Rational extended thermodynamics. Springer, New York (1998)62. Murray, R.L.: Nuclear reactor physics. Prentice Hall, New Jersey (1957)63. Ore, A.: Entropy of radiation. Phys. Rev. 98, 887 (1955)64. Ozisik, M.N., Menning, J., Halg, W.: Half-range moment method for solution of the

transport equation in a spherical symmetric geometry. J. Quant. Spectrosc. Radiat. Transf.15, 1101–1106 (1975)

65. Planck, M.: Distribution of energy in the spectrum. Ann. Phys. 4, 553–563 (1901)66. Pomraning, G.C.: The equations of radiation hydrodynamics. Pergamon, New York (1973)67. Pomraning, G.C.: Initial and boundary conditions for equilibrium diffusion theory. J. Quant.

Spectrosc. Radiat. Transf. 36, 69 (1986)68. Pomraning, G.C.: Asymptotic and variational derivations of the simplified PN equations. Ann.

Nucl. Energy 20, 623 (1993)69. Porzio, M.M., Lopez-Pouso, O.: Application of accretive operators theory to evolutive com-

bined conduction, convection and radiation. Rev. Mat. Iberoam. 20, 257–275 (2004)


70. Rosen, P.: Entropy of radiation. Phys. Rev. 96, 555 (1954)71. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving non-

symmetric linear systems. SIAM. J. Sci. Statist. Comput. 7, 856–869 (1986)72. Schafer, M., Frank, M., Pinnau, R.: A hierarchy of approximations to the radiative heat trans-

fer equations: Modelling, analysis and simulation. Math. Meth. Mod. Appl. Sci. 15, 643–665(2005)

73. Schuster, A.: Radiation through a foggy atmosphere. Astrophys. J. 21, 1–22 (1905)74. Schwarzschild, K.: Uber das Gleichgewicht von Sonnenatmospharen, Akad. Wiss. Gottingen.

Math. Phys. Kl. Nachr. 195, 41–53 (1906)75. Seaıd, M.: Notes on Numerical Methods for Two-Dimensional Neutron Transport Equation,

Technical Report Nr. 2232, TU Darmstadt (2002)76. Seaid, M., Klar, A.: Efficient Preconditioning of Linear Systems Arising from the Discretiza-

tion of Radiative Transfer Equation, Challenges in Scientific Computing. Springer, Berlin(2003)

77. Sherman, M.P.: Moment methods in radiative transfer problems. J. Quant. Spectrosc. Radiat.Transf. 7, 89–109 (1967)

78. Struchtrup, H.: On the number of moments in radiative transfer problems. Ann. Phys. 266,1–26 (1998)

79. Sykes, J.B.: Approximate integration of the equation of transfer. Mon. Not. R. Astron. Soc.111, 377 (1951)

80. Tomasevic, D.I., Larsen, E.W.: The Simplified P2 Approximation. Nucl. Sci. Eng. 122,309–325 (1996)

81. Turek, S.: An efficient solution technique for the radiative transfer equation. IMPACT, Com-put. Sci. Eng. 5, 201–214 (1993)

82. Turek, S.: A generalized mean intensity approach for the numerical solution of the radiativetransfer equation. Computing 54, 27–38 (1995)

83. Turpault, R.: Construction d’une modele M1-multigroupe pour les equations du transfert ra-diatif. C. R. Acad. Sci. Paris Ser. I 334, 1–6 (2002)

84. Turpault, R.: A consistent multigroup model for radiative transfer and its underlying meanopacities. J. Quant. Spectrosc. Radiat. Transf. 94, 357–371 (2005)

85. Turpault, R., Frank, M., Dubroca, B., Klar, A.: Multigroup half space moment approximationsto the radiative heat transfer equations. J. Comput. Phys. 198, 363–371 (2004)

86. Van der Vorst, H.A.: BI-CGSTAB: A fast and smoothly converging variant of BI-CG for thesolution of nonsymmetric linear systems. SIAM. J. Sci. Statist. Comput. 13, 631–644 (1992)

87. Viskanta, R., Anderson, E.E.: Heat transfer in semitransparent solids. Adv. Heat Transf. 11,318 (1975)

88. Viskanta, R., Menguc, M.P.: Radiation heat transfer in combustion systems. Prog. EnergyCombust. Sci. 13, 97–160 (1987)

Radiative Heat Transfer and Applicationsfor Glass Production Processes II

Norbert Siedow

1 IntroductionGlass is a man-made material which is used for many thousands of years. It playsan important role in our everydays life, in modern architecture, in science and manyother fields.

N. Siedow (�)Fraunhofer-Institut fur Techno- und Wirtschaftsmathematik Kaiserslautern, Germanye-mail: [email protected]


135

[email protected]

136 N. Siedow

Glass is one of the oldest materials in the world. The oldest finds date back tothe stone age in around 700 B.C.. In Egypt, the organized production of glass intojewelry items and small vessels commenced in around 3000 B.C.. From 1500 B.C.onwards, hollow glass was manufactured into ointment jars and oil containers inEgypt. Window glass dates back to the Gothic era of the twelfth century and the firstglass rolling process to 1688. More details about the history of glass and glass man-ufacturing can be found at the homepage of the Federal Association of the GermanGlass Industry (www.bvglas.de).

Today the glass industry is discovering new applications for glass based on stateof the art technology and recent scientific findings. One of the most recent applica-tions for glass is its use as a building material. Glass is also used as an insulatingmaterial in the form of glass fiber, it is used to make optical fibres for telephonecalls or TV in communications technology. Glass has become a key component indisplays and semi-conductors.

Glass products and their manufacturing require continuous adjustments to cus-tomer needs and constant process optimization is necessary to ensure and improvethe quality of the glass products. Beside expensive and time consuming laboratoryexperiments numerical simulation has become a key technology. “Strict numeri-cal treatment, i.e., the mathematical simulation of product behavior and all aspectsof production processes, is a must for every material-producing company” [5]. At aworkshop on “Modeling Needs of the Glass Industry” [8] Choudhary and Huff fromOwens–Corning, OH (USA), wrote: “In a complex process such as glassmaking,several variables are monitored and controlled ... . From the perspective of model-ing, the greatest interest is in the control of various temperatures.”

The knowledge of the temperature is important in almost all stages of the glassproduction and glass processing. During the melting process in the glass tank thetemperature influences the homogeneity of the glass melt. Undesired cracks couldbe the result of a wrong cooling of the glass. Thus, the knowledge of the righttemperature is important to guarantee the quality of the final products. Glass is asemitransparent material. Besides heat conduction and heat convection radiationplays an important role. Especially for high temperatures heat transfer by radia-tion is the dominant process. Conduction and convection are local phenomena. Themean free path for molecular collisions is generally very small. Thermal radiation,on the other side, is generally a global phenomenon. The average distance a pho-ton travels before interacting with a molecule may be very short (e.g. absorption inmetal), but can also be very long (e.g. sun rays).

The simulation of radiative heat transfer is a challenging problem for engineers,physicists and mathematicians since many years. The coupling of heat conductionand thermal radiation results in a high-dimensional non-linear system of partialdifferential equations. For the numerical solution of the partial differential equa-tions different methods can be used. Besides classical approaches like ray tracingtechniques or full discretization methods like the method of discrete ordinatesasymptotic approaches offer a good opportunity to derive alternative numericalmethods (Rosseland approximation, PN- and SPN-approximation, improved diffu-sion methods). From the practical point of view it is necessary that these methods

Radiative Heat Transfer and Applications for Glass Production Processes II 137

can be used together with commercial software packages. The methods should givesufficiently accurate results in acceptable time. In the first part of the lecture wewill discuss numerical models for fast radiative heat transfer simulation. The mostused method in industrial application is the Rosseland approximation, which treatsradiation as a correction of heat conduction. This fast method gives sufficiently ac-curate results only for optically thick materials. Therefore, we present two diffusionapproximation methods, which are nearly as fast as the Rosseland approximationbut valid also for semitransparent materials like glass. These methods conserve allgeometrical information about the glass domain under consideration.

Fast numerical methods are the fundamental basis for optimization and inverseproblems like the control of glass cooling to avoid high permanent thermal stressinside the glass or indirect measurements which are very often the only way to mea-sure the temperature inside hot glasses or glass melts. Spectral remote sensing is anexample for an indirect measurement method. Using a pyrometer or thermocamerathe spectral radiative intensity is measured for each wavelength. For the temperaturereconstruction a Fredholm’s integral equation of the first kind has to be solved. Fromthe mathematical point of view one has to solve an ill-posed inverse problem. In thesecond part of the lecture we will discuss ill-posed inverse problems – examples, thereason of ill-posedness and methods to overcome this difficulty. As an first examplecoming from glass industry we discuss spectral remote sensing. The second indus-trial application deals with the reconstruction of the initial temperature condition.Measuring temperature and heat flux at the whole boundary of the glass domain orparts of it we present an other possibility to reconstruct the temperature inside theglass body.

2 Models for Fast Radiative Heat Transfer Simulation

2.1 Introduction

For many industrial applications the modeling of heat transfer processes is of utmostimportance. This applies especially to radiative heat transfer in semitransparent ma-terials like glasses. The temperature is the most important parameter in almost allstages of glass making and glass processing. During the glass melting the tempera-ture influences the homogeneity of the glass melt, the drop temperature influencesthe following forming process and finally the cooling of the hot glass decides aboutthe frozen thermal stresses inside the glass product. Undesired effects like breakagecould be the result of a wrong cooling. Hence, the knowledge of the exact tem-perature distribution is necessary to control the production processes and the finalquality of the products. To determine the temperature one can use measurements orsimulate the heat transfer process using computers. In contrast to opaque materi-als, heat can be transported in semi-transparent glass by phononic vibrational heatconduction as well as via thermal radiation which is important particularly for high

138 N. Siedow

Fig. 1 Internal opticaltransmittance of a1-mm-thick glass pane

temperatures. Whereas heat conduction acts on a microscale in a nm−range thermalradiation acts on a macroscale from the nm− to the cm−range. Figure 1 taken from[5] shows the internal optical transmittance of a typical optical glass.

The radiative field is divided into an opaque, semitransparent, and a transparentwavelength range. In the wavelength range λ > 5μm the glass is opaque, that is noheat can be transported by radiation within the glass. The mean free path length ofphotons is very small. Only the surface of the glass can exchange heat via radiationwith the surroundings. In the semi-transparent wavelength range, (in Fig. 1: 2.5 <λ ≤ 5μm), heat can be transported by radiation within the glass. In the transparentwavelength range, (in Fig. 1: 0.5 < λ ≤ 2.5μm), the radiative field does not interactwith the glass.

Very good surveys about radiative heat transfer one can find in [1] and [7]. Fur-thermore we refer to [5], where the application to glass industry is in the focusof consideration, and to [13] and [14] for the numerical treatment of the radiativetransfer equation.

From the mathematical point of view for the simulation of the temperature dis-tribution of hot glasses one has to deal with a coupled system of partial differentialequations. Let D be a three-dimensional domain of absorbing and emitting glass.The heat transfer in D is described by the energy equation

cmρm∂T∂ t

(r,t) = ∇ · (kh∇T (r,t))−∇ ·qr(T ), (r, t) ∈ D× (0, t�], (1)

where cm is the specific heat, ρm the glass density, kh the heat conductivity, T thetemperature depending on the space position r and time t. qr(T ) denotes the radia-tive flux vector, whereas ∇ ·qr �= 0 only for the semitransparent wavelength region.The glass has a given initial temperature distribution

T (r,0) = T0(r). (2)


The net heat flux through the surface at the boundary ∂D is defined by heatconvection and diffuse surface radiation for those wavelength for which the glassis opaque.

kh∂T∂n

(rb, t) = γ[Ta−T(rb,t)]+ επ∫

opaque

[Ba(Ta,λ )−Ba(T (rb, t),λ )]dλ ,

rb ∈ ∂D, 0 < t ≤ t�, (3)

where n is the outer normal vector to the boundary point rb, γ denotes the heattransfer coefficient, Ba(Ta,λ ) Planck’s function (mentioned below by formula (7))depending on Ta the temperature of the surroundings and the wavelength λ , and εdenotes the emissivity of the glass. For a detailed derivation and discussion of theboundary condition we refer to [6].

The radiative flux vector qr is defined as the first moment with respect to thedirection Ω of the radiative intensity I(r,Ω,λ ), depending on space position r, di-rection Ω, and wavelength λ ,

qr(r) =∞∫

0

∫

S2

I(r,Ω,λ )ΩdΩdλ . (4)

S2 denotes the unit sphere. The radiative intensity is described by the radiative trans-fer equation

Ω ·∇I(r,Ω,λ )+ κ(λ )I(r,Ω,λ ) = κ(λ )Bg(T (r, t),λ ), (5)

with specular reflecting boundary conditions

I(rb,Ω,λ ) = ρ(Ω)I(rb,Ω′,λ )+ (1−ρ(Ω))n2gB(Ta,λ ), (6)

Ω′ = Ω−2(n ·Ω)n.

Here κ(λ ) denotes the absorption coefficient, ng the refractive index of glass andBg(T,λ ) Planck’s function for glass

Bg(T,λ ) =2hc2

0

λ 5n2g

(e

hc0ngλkBT −1

) (7)

(c0 = 2.998× 108 m/s speed of light, h = 6.626× 10−34Js Planck constant, kB =1.381×10−23J/K Boltzmann constant). Ba(T,λ ) denotes Planck’s function for airwith refractive index na = 1.

The system (1)–(6) is non-linear and highdimensional. To solve the systemnumerical methods are required which offer a sufficient accurate solution in ap-propriate computational time.

140 N. Siedow

A detailed survey about different numerical methods for solving the radiativetransfer equation can be found in [1]. The majority of methods can be divided intofour classes:

(a) Diffusion approximation methodsTypical examples are the Rosseland approximation or the below described meth-ods developed at ITWM.

(b) Methods, where the directional dependence is expressed by a series of specialbasis functionsRepresentants are the PN− and SPN−approximation.

(c) Methods using a discretization of the angular dependenceHere we mention the discrete ordinate method, which is the most popular onein literature.

(d) Monte Carlo methods

For more details we refer also to the lectures from A. Klar and M. Frank.

2.1.1 Fast Numerical Methods for Radiative Heat Transfer

The Rosseland approximation is widely used in industrial practice and describesthe radiation as a correction of the heat conductivity. There exist different methodsto derive the Rosseland approximation, for instance using asymptotic analysis. Werefer also to the lecture of A. Klar or the PhD thesis [18]. Originally the method wasderived in 1924 by S. Rosseland to investigate the radiation of stars. (see [9])

Here we will use the formal solution of the radiative transfer equation (5)

I(r,Ω,λ ) = I(rb,Ω,λ )e−κd(r,Ω) + κd(r,Ω)∫

0

B(T (r− sΩ),λ )e−κsds. (8)

d(r,Ω) is the distance from the point r to the boundary ∂D in direction −Ω. Usinga linear approximation of the Planck function in the point r

B(T (r− sΩ),λ )≈ B(T (r),λ )− s∂B∂T

(T (r),λ )Ω ·∇T (r),

and assuming that the desired glass is optically thick

κd(r,Ω)−→ ∞, (9)

the integral on the right hand side of (8) can be calculated analytically

I(r,Ω,λ ) = B(T (r,t),λ )− 1κ(λ )

∂B∂T

(T (r, t),λ )Ω ·∇T (r, t). (10)


Using now the definition of the radiative flux vector (4) we obtain

qr = kr∇T (r,t),

with kr =4π3

∞∫

0

1κ(λ )

∂B∂T

(T,λ )dλ . (11)

Instead of the combination of heat transfer equation (1) and radiative transferequation (5) one has to solve only the heat transfer equation with corrected heatconductivity

cmρm∂T∂ t

(r,t) = ∇ · ((kh + kr)∇T (r,t)) (r, t) ∈ D× (0, t�]. (12)

It is important to mention that the boundary condition (3) has to be changed too.Taking the energy balance over the whole wavelength region into account for theboundary condition yields (see [6])

(kh +

12

kr

)∂T∂n

(rb,t) = γ[Ta−T (rb,t)]+ σε[T4a −T 4(rb, t)],

rb ∈ ∂D, 0 < t ≤ t�. (13)

The Rosseland approximation treats thermal radiation as a correction of heat con-ductivity. The method is very fast and easy to implement into commercial softwarepackages. On the other side it is easy to realize from the derivation of the methodthat it is valid only for optically thick glasses. Especially near the boundary ∂D theRosseland approximation will give wrong simulation results.

The reason for this behavior is given by the requirement (9). Due to that require-ment the approximation for the spectral intensity (10) has lost almost all geometricalinformation included in the radiative transfer equation (5)–(6).

We want to present now two numerical methods developed at Fraunhofer ITWMand published first in [13] and [14].

These methods are similar to the Rosseland approximation but they will givemuch more accurate results because they conserve all geometrical information aswe will see later on.

The starting point is once more the formal solution of the radiative transfer equa-tion (8). As before we take the Taylor expansion to linearize the Planck function butwe skip the requirement (9). We calculate the integral in (8) analytically

I(r,Ω,λ ) = I(rb,Ω,λ )e−κd(r,Ω) + B(T(r, t),λ )(

1− e−κd(r,Ω))

− 1κ(λ )

[1− (1 + κd(r,Ω))e−κd(r,Ω)

] ∂B∂T

(T (r, t),λ )Ω ·∇T (r, t). (14)

142 N. Siedow

In contrast to the classical Rosseland approximation the formal solution was takenwithout any assumption for the optical thickness. This means that all geometricalinformation of the spectral radiative intensity is preserved. Applying the definitionof the radiative flux (4) to formula (14) for the divergence of the radiative flux vectoryields

∇ ·qr(r) = −∇ ·⎡⎣⎛⎝

∞∫

0

1κ(λ )

A (r,λ )∂B∂T

(T,λ )dλ

⎞⎠∇T (r, t)

⎤⎦

+∞∫

0

κ(λ )[B(T,λ )−B(Ta,λ )]∫

S2

(1−ρ(Ω))e−κd(r,Ω)dΩdλ

−∞∫

0

∂B∂T

(T,λ )∫

S2

(1−ρ(Ω))Ω ·∇Te−κd(r,Ω)dΩdλ , (15)

with a symmetric anisotropic diffusion tensor A (r,λ ). We approximate A (r,λ ) byan orthotropic diffusion tensor in form of a diagonal matrix

A (r,λ ) =

⎛⎝

a1(r,λ ) 0 00 a2(r,λ ) 00 0 a3(r,λ )

⎞⎠ , (16)

a j(r,λ ) =∫

S2

Ω 2j [1− (1 + κ(λ )d(r,Ω))e−κ(λ )d(r,Ω)]dΩ , j = 1,2,3

In contrast to the anisotropic diffusion tensor, such an orthotropic diffusion tensorcan easily be used in commercial software packages.

The first term on the right hand side of (15) represents a correction to theheat conduction due to radiation. The second term contains the boundary condi-tion and the third a thermal convection because it depends on the gradient of thetemperature T .

According to the reference [13] we call the method (15)–(16) Improved DiffusionApproximation.

Another method similar to the Improved Diffusion Approximation was presentedin [14] and used for the simulation of flat glass tempering. Starting point is as wellas before the formal solution (8) of the radiative transfer equation, and as well asbefore we skip the requirement (9). Integrating the radiative transfer equation (5)over all directions the divergence of the radiative flux vector can be calculated as

∇ ·qr(r) =∞∫

0

κ(λ )

⎛⎝4πB(T,λ )−

∫

S2

I(r,Ω,λ )dΩ

⎞⎠dλ (17)


Together with the formal solution (8) we obtain the so-called Formal SolutionApproximation

∇ ·qr(r) =∞∫

0

κ(λ )∫

S2

[B(T,λ )− I(r,Ω,λ )]e−κd(r,Ω)dΩdλ +

+∞∫

0

∂B∂T

(T,λ )∫

S2

[1− (1 + κ(λ )d(r,Ω))e−κd(r,Ω)]Ω ·∇T dΩdλ (18)

We want to demonstrate the introduced two new methods for three examplestypical for glass industry – the heating of a glass plate, the radiative heat transfer ina cavity with natural convection, and an application to flat glass tempering.

2.1.2 The Heating of a Glass Plate

A thin glass plate with thickness of 0.005 m and uniform initial temperature ofT0(x) = 200◦C is heated up from above and below by electrical heaters withTabove = 800◦C and Tbelow = 600◦C. The glass is described by physical parameterslisted in Table 1.

The considered glass is semitransparent in the region between 0.01 and 7.0μm,what is realistic for optical glasses. The used absorption coefficients are collected inTable 2.

Figure 2 shows the steady state temperature profile along the thickness of theglass plate simulated using the discrete ordinate method (DOM), the improveddiffusion approximation (ida) (15), and the formal solution method (fsa) (18). All

Table 1 Material parameters density 2,500kg/m3

specific heat 1,250J/kg/Kconductivity 1W/m/Krefractive index 1.5

Table 2 Absorptioncoefficients

λk−1 in μm λk in μm κk in 1/m0.01 0.2 0.400.2 3.0 0.503.0 3.5 7.703.5 4.0 15.454.0 4.5 27.984.5 5.5 267.985.5 6.0 567.326.0 7.0 7,136.067.0 ∞ opaque

144 N. Siedow

Fig. 2 Temperature distribution in a glass plate

three methods deliver comparable results. The maximum difference between thediscrete ordinate method and (15) is about 0.7◦C, whereas the difference betweenDOM and (18) is less than 0.03◦C. From the computational point of view the two“ITWM-approaches” are much faster than the discrete ordinate method.

2.1.3 Radiative Heat Transfer with Natural Convection

As a second example we simulate the radiative heat transfer with natural convectionin a two-dimensional cavity. The cavity has a hot right wall of 1,800◦C, a colderleft wall with 1,300◦C, and adiabatic walls at top and bottom. Gravity pointsdownwards. A buoyant flow develops because of thermally-induced density gra-dients. We compare our radiation model with DOM available in the commercialsoftware package FLUENT. We have implemented our improved diffusion methodinto FLUENT. Therefore, the FLUENT radiation is turned off and the volume radi-ation is added by a user defined function developed at Fraunhofer ITWM into thesource term. The main parameters are given in Table 3.

The walls are taken as diffusely reflecting. The glassmelt is considered as a greymaterial.

Figure 3 shows the temperature distribution inside the two-dimensional cavityusing the commercial software package FLUENT. The left picture was calculatedwith the discrete ordinate method available in FLUENT and the right with the im-proved diffusion approximation developed by ITWM. The temperature distributionis very similar but there are big differences in the computational time. FLUENT-DOM needs more than 5,000 iterations to converge whereas the ITWM-UDF needsonly 86 iterations.


Table 3 Material parametersof the cavity problem

density 2,500kg/m3

specific heat 1,200 J/kg/Kconductivity 1W/m/Kheat transfer coefficient 0W/m2/Kabsorption coefficient 40 1/mrefractive index 1.5viscosity 30 Pasthermal expansion coefficient 7 ·10−51/K

1.80e+031.78e+031.76e+031.74e+031.72e+031.70e+031.68e+031.66e+031.64e+031.62e+031.60e+031.58e+031.56e+03 1.54e+031.52e+031.50e+031.48e+031.46e+031.44e+031.42e+031.40e+031.38e+031.36e+031.34e+031.32e+031.30e+03

1.80e+031.78e+031.76e+031.74e+031.72e+031.70e+031.68e+031.66e+031.64e+031.62e+031.60e+031.58e+031.56e+03 1.54e+031.52e+031.50e+031.48e+031.46e+031.44e+031.42e+031.40e+031.38e+031.36e+031.34e+031.32e+031.30e+03

Fig. 3 Temperature profile in a two-dimensional cavity calculated with FLUENT. Left: usingFLUENT-DOM. Right: using ITWM-Approaches

2.1.4 Application to Flat Glass Tempering

Thermal tempering of flat glass plates consists of two stages: First the glass is heatedup at a temperature higher than the transition temperature (so-called “Tg”). In thesecond step it is cooled down rapidly by an air jet. This thermal treatment givesbetter mechanical and thermal strengthening to the glass by way of residual stressesgenerated along the thickness of the glass plate. The present example concerns thecalculation of stresses inside a sodalime silicate flat glass of thickness 6 mm. For adetailed description especially for the used parameters we refer to [14].

The cooling of the glass melt depends on the temperature distribution in time andspace. There exists a temperature range, where the glass changes from fluid to solidstate. The essential property is the viscosity of the glass. The viscosity is high forlow temperatures and the glass behaves like a linear-elastic material. High temper-atures cause high viscosity of glass, which behaves in that case like an Newtonianflow. The viscosity changes the density depending on the temperature. A change indensity influences the stress inside the glass. A numerical method for the calculation

146 N. Siedow

of transient and residual stresses in glass, including both structural relaxation andviscous stress relaxation, has been developed by Narayanaswamy. (See [14, 15])

In [14] we have investigated how the numerical calculation of the temperatureinfluences the calculation of the thermal stress inside the flat glass plate. Calculatingthe formal solution of the one-dimensional radiative heat transfer with 200 differ-ent directions and 30 wavelength bands we have got a reference which was calledto be exact. We have compared it with the Rosseland model, a model where wehave skipped the radiation totally, and the formal solution approximation (18). TheFigs. 4 and 5 show the evolution over time of the mid-plane and surface temperature

20

15

10

Tsu

rf,E

xact

- T

surf,C

ase

(K)

5

0

−5

25No Rad

50 100

Time (s)

150

Ross App ITWM

−10

Fig. 4 Evolution over time of surface temperature difference for the Exact solution compared withCase (where Case is on Of No Rad, Ross App, or ITWM models)

Tm

id,E

xact

- T

mid

,Cas

e (K

) 25

20

15

10

5

0

30

0 50 100 150

No Rad

Time (s)

Ross App ITWM

−5

Fig. 5 Evolution over time of mid-plate temperature difference for the Exact solution comparedwith Case (where Case is on Of No Rad, Ross App, or ITWM models)


differences of the exact solution and the other mentioned methods. It turns out thatthe ITWM approach gives the closest result to the exact temperature calculation.

Comparing the computational time the ITWM model is almost as fast asthe Rosseland approximation and about 100 times faster than the exact solutionprocedure (Table 4).

Using Narayanaswamy’s model [14] transient and residual stresses were com-puted along the glass thickness. The results are shown in Figs. 6 and 7. As forthe temperature itself the ITWM method gives the closest results to the exact

5

0

0 50

Time (s)

100 150

σsu

rf,E

xact

- σ

surf,C

ase

(Mpa

)

−5

10No Rad Ross App ITWM

−10

Fig. 6 Evolution over time of the stress differences on the glass surface between the Exact solutionand Case (where Case is on Of No Rad, Ross App, or ITWM models)

4

2

00 50 100

Time (s)

150

σmid

,Exa

ct -

σm

id,C

ase

(Mpa

)

−2

−4

6

No Rad Ross App ITWM

−6

Fig. 7 Evolution over time of the stress differences in the mid-plane between the Exact solutionand Case (where Case is on Of No Rad, Ross App, or ITWM models)

148 N. Siedow

Table 4 CPU time in s Exact solution model 806.50 sNo radiation 0.65 sRosseland approximation 1.65 sITWM model 7.50 s

solution for transient and residual stress calculation. It is interesting to mention thatfor the residual stress it is better to exclude radiation than to use the Rosselandapproximation.

2.1.5 Conclusions

Temperature is one of the most important parameters to make “good” glasses. Tosimulate the thermal behavior of glass radiation must be taken into account becauseof the fact that glass is a semitransparent material. From the mathematical point ofview one has to solve a non-linear and high-dimensional system of partial differ-ential equations, what can be done only numerically. Even the numerical solutionof the radiative transfer equation is very complex. For a domain discretization with20,000 space grid points, 60 different directions, and 10 wavelength bands one hasto solve using the Discrete Ordinate Method, which is favored in literature, a linearsystem of equations with 12 million unknowns. That is the reason why one has tolook for fast numerical methods. The above described Improved Diffusion Approx-imation and Formal Solution Approach are good alternatives, because they make abalanced compromise between exactness and quickness of the numerical solution.The methods are nearly as fast as the classical Rosseland approximation, whichtreats the thermal radiation as a correction of the thermal conductivity, and nearly asaccurate as the Discrete Ordinate Method, which is asymptotically exact. Comparedto the Rosseland approximation all geometrical information is conserved.

3 Indirect Temperature Measurement of Hot Glasses

3.1 Introduction

As discussed in the previous chapters temperature is the most important param-eter in almost all stages of glass production and glass processing. To determinethe temperature of hot glasses one can use numerical simulation or – what is themost conventional method – one measures the temperature. Measurement tech-niques are required that work in harsh environment of glass industry. The usage ofthermocouples which continously are in touch with glass or penetrate it disturbs themanufacturing process. Thermal pyrometers are used to measure the surface temper-ature of hot glasses, their usage to measure the inside temperature leads to wrongmeasuremet results. On the other side glass is semitransparent, i.e. it radiates not


T(z)

1000 1

Emissivity

0.8

0.6

0.4

0.2

00 1 2 3 4

950

9001 2

Depth [mm] λ [μm]30 4

[°C]

Fig. 8 The principle of remote spectral temperature sensing

only from the surface of a glass body but also from the inner parts. Hence one canuse this glass property to measure the temperature in the following way: The opticsof an instrument (spectrometer, spectral thermocamera) is directed towards a glassgob and records the wavelength dependent spectral intensity of the glass. From thesesignals the temperature distribution inside the glass gob has to be computed usinga mathematical algorithm. Figure 8 illustrates the principle of the remote spectraltemperature sensing. In the following we will develop and discuss a mathematicalalgorithm of the spectral remote sensing (see [5]).

3.2 The Basic Equation of Spectral Remote Temperature Sensing

The one-dimensional radiative transfer equation is the starting point for the indirecttemperature determination

μ∂ I∂ z

(z,μ ,λ )+ κ(λ )I(z,μ ,λ ) = κ(λ )B(T (z),λ ), (19)

where I(z,μ ,λ ) denotes the spectral radiative intensity depending on position z, ondirection μ = cos(θ ), and on wavelength λ . κ denotes the absorption coefficient,and B(T,λ ) Planck’s function defined by (7).

At the boundaries of the one-dimensional domain we consider specular reflection

I(zb,μ ,λ ) = ρ(μ)I(zb,−μ ,λ )+ (1−ρ(μ))n2gB(Tab,λ ). (20)

For μ > 0 we have zb = 0 and Tab = Ta1 and for μ < 0 zb = D and Tab = Ta2. D isthe length of the considered glass domain and ng the refractive index of glass.

150 N. Siedow

The formal solution I(z,μ ,λ ) of the equations (19), (20) can be calculated and isrelated to the measured spectral intensities Im(λ ) by

Im(λ ) = (1−ρ)I(0,−1,λ ). (21)

Therefore, for the measured intensities hold

Im(λ ) =1−ρ

1−ρ2e−2κD

⎧⎨⎩(1−ρ)n2

ge−κD[B(Ta2,λ )+ ρe−κDB(Ta1,λ )]

+D∫

0

κB(T (s),λ )[e−κs + ρe−2κD+κs]ds

⎫⎬⎭ . (22)

For measured spectral intensities Im(λ ) we have to determine the temperature profileT (z) inside the glass body 0≤ z≤D. Thus, (22) represents a non-linear Fredholm’sintegral equation of first kind. We have to solve an ill-posed inverse problem.

3.3 Some Basics of Inverse Problems

The field of inverse problems has become one of the most important and one of thefastest growing areas in applied mathematics during the last years driven by theneeds of industry as well as sciences. There is a vast literature on inverse and ill-posed problems. We refer here to [3] and [4].

Inverse problems are concerned with finding causes for observed or desired ef-fects. If one looks for the cause of an observed effect, we call it identification orreconstruction. If one looks for the cause of an desired effect, we call it control ordesign. At the beginning of this section we will show some examples of inverseproblems arising in various fields of industrial application.

3.3.1 Example 1: Numerical Differentiation

Assume that an input signal I is transformed by a “black box” function “ f ” to anoutput signal g which can be measured. The transformation is described by theconvolution

x∫

0

I(x− t) f (t)dt = g(x). (23)

If we take I ≡ 1 we obtain Volterra’s integral equation of first kind

x∫

0

f (t)dt = g(x),


0exact datanoisy data

f(x)

−0.1

−0.2

−0.3

0 0.2 0.4x

0.6 0.8 1

−0.4

−0.5

Fig. 9 Numerical differentiation with h = 0.1

and if, furthermore, we assume that g(x) is continuously differentiable and g(0) = 0we get the solution of the integral equation as the first derivative of g(x):

f (x) = g′(x).

g(x) represent data coming from measurement. In praxis the measured data are finiteand noisy:

gδi = g(xi)+ δ , h = xi− xi−1, i = 1,2, ...n,

where we assume that the points xi are equally distributed. The solution f (x) iscalculated for discrete points xi by numerical differentiation using

f δi = DhGδ

i =gδ

i+1−gδi−1

2h. (24)

It is known (see [2]) that the discretization error of central differences is of secondorder. As a test example we take g(x) = 2

3 x3− x2 + 1 and random noise of 1%. Fora uniform step size h = 0.1 we obtain the solution shown in Fig. 9.

Normally a smaller step size h = 0.01 leads to a smaller discretization error, butfor (24) that seams to be not true: Here a finer discretization leads to a bigger errorfor the reconstruction of f (x) as can be seen from Fig. 10.

3.3.2 Example 2: Identification of Heat Conductivity

As a second example we consider the one-dimensional heat transfer equation

∂∂x

(a(x)

∂u∂x

)=− f (x), 0 < x < l, a(0)

∂u∂x

(0) = g0, u(l) = g1. (25)

152 N. Siedow

0.4

0.2

0.2x

0.4 0.6 0.8 10

0

f(x)

−0.2

−0.4

−0.6

−0.8

−1

0.6exact datanoisy data

−1.2

Fig. 10 Numerical differentiation with h = 0.01

Knowing the temperature u(x) we want to determine the heat conductivity a(x). Ifwe assume that ∂u

∂x ≥ u0 > 0 we can transform (25) into

a(x) =g0−

x∫0

f (y)dy

∂u∂x

. (26)

For f (x) = 4x− 2, g0 = 4, g1 = 4, l = 1, and u(x) = −x2 + 4x + 1 we obtain theexact value for a(x)

a(x) = x + 1.

Instead of exact measurement u(x) we know uδ (x) with 0.01% random noise.Figure 11 shows the exact and noisy values u(x) and uδ (x).

As can be seen from Fig. 12 in that case it is not possible to reconstruct the rightvalue for the heat conductivity.

3.3.3 Example 3: An Other Heat Transfer Problem

As a third and last example we consider once more the heat transfer equation

∂∂x

(a(x)

∂u∂x

)=− f (x), 0 < x < l, a(0)

∂u∂x

(0) = 0, −a(1)∂u∂x

(1) = u(1) (27)

with Robbin-type boundary conditions and strongly varying heat conductivity 2 >a(x)≥ 10−8.


4

3.5

3

u(x)

2.5

2

1.5

4.5exact datanoisy data

10 0.2 0.4

x0.6 0.8 1

Fig. 11 Measurement with δ = 0.01%

30exact datanoisy data25

20

15

10

5a(x)

0

0 0.2 0.4x

0.6 0.8 1

−5

−10

−15

−20

Fig. 12 Reconstructed heat conductivity with δ = 0.01%

Instead of (27) we consider the finite volume approximation for three grid pointsx0 = 0, x1 = 0.5, and x2 = 1. The discretization of right hand side is given byf = (8,−4,0)T and the conductivity should have the values a = (1.99999801,1.99 ·

154 N. Siedow

10−6,1.0 · 10−8)T . Then the discrete solution of (27) is given by the solutionu = (1,0,0)T of the linear system

⎛⎝

1 −1 0−1 1.000001 −0.0000010 −0.000001 1.000001

⎞⎠⎛⎝

u1

u2

u3

⎞⎠=

⎛⎝

1−10

⎞⎠ . (28)

If we add a small disturbance δ = (0.01,0.01,0)T to the right hand side vector f ,so that f δ = f + δ we get as a new solution of the system (28) the vector uδ =(20001.03,20000.02,0.000002)T . Once more we notice that, a small error in themeasurement (right hand side) causes a big error for the reconstruction (solutionvector).

What is the reason for that behavior?A common property of all these three examples and of a vast majority of inverse

problems is their ill-posedness. Hadamard (1865–1963) has given a definition ofwell-posedness.

A mathematical problem is well-posed, if

(i) For all data, there exists a solution of the problem(ii) For all data, the solution is unique

(iii) The solution depends continuously on the data

A problem is ill-posed, if at least one of these three conditions is violated.

What is the reason for the ill-posedness of the considered examples?

3.3.4 Example 1: Numerical Differentiation

The function g was given with a small error

||g−gδ ||∞ ≤ δ .

For the numerical differentiation we used the central difference quotient with stepsize h. If g ∈C3[0,1] we get from the Taylor expansion

Dhg =g(xi+1)−g(xi−1)

2h= g′(x)+

h2

6g′′′(ξ ), ξ ∈ [0,1].

Thus the total error behaves like

|| f −Dhgδ ||∞ ≈ h2

6||g′′′||∞ +

δh

. (29)

For a fixed error level δ the first term on the right hand side is decreasing for de-creasing h while the second term is increasing. If the step size h becomes too small


0.02

0.018

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

00.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Fig. 13 Numerical error as function of step size. Green: Discretization error, Red: Measurementerror, Blue: Resulting error

the total error is increasing. The behavior of the errors is plotted in Fig. 13. Thereexists an optimal step size hopt , which can not be computed explicitly because itdepends on the smoothness of the unknown exact data.

3.3.5 Example 2: Identification of Heat Conductivity

The numerical differentiation in (26) of noisy data u(xi) is ill-posed like shown forthe first example. That is the reason why the second example is ill-posed too.

3.3.6 Example 3: An Other Heat Transfer Problem

Let us consider the third example, the linear system (28). It is easy to calculate(using for instance MATLAB) that the (numerical) eigenvalues of the system are

λ1 = 0.5 ·10−6, λ2 = 1.000001, λ3 = 2.0000005. (30)

Therefore, the condition number is

κ =λ3

λ1≈ 4 ·106.

156 N. Siedow

Let v1, v2, and v3 be the eigenvectors with respect to the eigenvalues. The solutionof (28) can be written as

u =3

∑i=1

λ−1i ( f ,vi)vi, (31)

where ( f ,vi) denotes the scalar product of the vectors f and vi. A small error in theright hand side f δ = f + δ leads to

uδ =3

∑i=1

λ−1i ( f ,vi)vi +

3

∑i=1

λ−1i (δ ,vi)vi. (32)

The interesting part is the second term in (32). The error is divided by theeigenvalues. For the smallest eigenvalue λ1 we obtain

δλ1≈ 10−2

0.5 ·10−6 = 20000. (33)

A small error in the right hand side leads to large error in the solution.What can be done to overcome the ill-posedness?

The answer to this question is regularization. In [3] is given a kind of definition:

Regularization is the approximation of an ill-posed problem by a family of neigh-boring well-posed problems.

There exist a lot of methods to construct such a family of neighboring well-posedproblems. We want to describe here only some of the most important methods. Fora more detailed overview we refer to [3] or [4].

3.3.7 Truncated Singular Value Decomposition

The simplest method to overcome the ill-posedness of a problem is to make a singu-lar value decomposition and skip the smallest singular values, i.e. for a fixed (largeenough) λ � we skip all

λ1 < λ2 < ...λ j < λ �.

Using that method in our third example we obtain for λ � = 0.5

uδ =3

∑i=2

λ−1i ( f ,vi)vi =

⎛⎝

0.5−0.5

−0.5 ·10−6

⎞⎠ (34)

It can be shown that the truncated singular value decomposition is identical to theminimization problem

J(u) = ||Au− f ||L2 →min (35)


and to take from all existing solutions of (35) that with minimal norm

||u||L2 →min .

3.3.8 Tichonov (Lavrentiev) Regularization

We look for a problem which is near by the original one but well-posed. Thereto weincrease the eigenvalues. Instead of (31) we obtain

uα =3

∑i=1

1λi + α

( f ,vi)vi, α > 0 (36)

For α we take a sequence αn→ 0. To decide which αn gives the best results one canuse different methods. Here we use the so-called L-curve method: We solve (36) fora sequence αn and look at the solution norm ||u||L2 versus residual norm ||Au− f ||L2

for different α . The result for the considered example is shown in Fig. 14. The graph

104

103

102

||u||

||Au-f||

101

100

10−1

10−3 10−2

X: 0.01979Y: 1.001

10−1 100

Fig. 14 The L-curve method

158 N. Siedow

is “L-shaped”. As the optimal regularization parameter α� we take the value in thecorner. In the example α� = 0.01977 for which we obtain

uα�=

⎛⎝

1.00040.01201 ·10−8

⎞⎠ ,

as the solution of (36), which is in good agreement to the exact solution of theproblem.

3.3.9 Landweber Iteration

Instead of the system (28) we consider the normal equation

A�Auδ = A� f δ

which we solve by a fixed point iteration

uδk+1 = uδ

k − τA�(Auδk − f δ ), uδ

0 given. (37)

Here the iteration number k plays the role of the regularization parameter α = 1k . As

the stopping rule we use the so-called discrepancy principle:

k�(δ , f δ ) = inf{

k ∈ N : ||Auδk − f δ ||L2 < ηδ

}, (38)

with some real value η ≥ η0 > 1.For the third example we find after four iterations (k� = 4) the minimal norm

solution uδk� = (0.5 −0.5 −1 ·10−7)T .

3.3.10 The classical Tichonov Regularization

The Russian mathematician A.N. Tichonov (1906–1993) proposed the regulariza-tion of the normalized system:

A�Auδα + αIuδ

α = A� f δ . (39)

It can be shown, that this procedure is equivalent to solve the minimization problem(see [3])

J(uδα ,α) = ||Auδ

α − f δ ||2L2+ α||u||2L2

→min (40)

Dealing with an ill-posed problem means to find the right balance between accuracyfor which stands the first term on the right hand side and stability for which standsthe second term.


Using the discrepancy or the L-curve principle one obtains for our example oncemore the minimal norm solution.

From all discussion above it follows that for a good regularization one has toinclude as much information as available. If we know, for instance, that the solutionof (28) has the norm ||u||2L2

= 1 the functional (40) can be modified

J(uδα ,α) = ||Auδ

α − f δ ||2L2+ α

(||u||2L2−1)→min,

which gives approximately the exact solution

uδα =

⎛⎝

1.001040.00104−9 ·10−9

⎞⎠ .

3.4 Spectral Remote Sensing

Now, we return to the integral equation (22). We write the non-linear equation as

G(T ) = Im (41)

After linearization

(G′(Tk))�G′(Tk)(Tk+1−Tk) = (G′(Tk))�(Im−G(Tk)) (42)

we have to apply regularization. Assume that G′(Tk) : X → Y , where X is thespace of temperature functions and Y the image space. As we discussed earlier theclassical Tichonov regularization can be written as a minimization problem:

infTk+1∈U

||(G′(Tk))�G′(Tk)(Tk+1−Tk)− (G′(Tk))�(Im−G(Tk))|| (43)

with some constraint condition U .If we assume that the temperature in the glass results from combined heat trans-

port by conduction and radiation, T (z) fulfills the stationary equation

∂∂ z

(kh

∂T∂ z−qr(T (z))

)= 0. (44)

qr(T (z)) represents the temperature-dependent radiative flux vector. The influenceof radiation can be calculated by the Rosseland approximation. Therefore, as a con-straint we choose

160 N. Siedow

U ={

Tk+1 ∈X :∂∂ z

((kh + kr)

∂Tk+1

∂ z

)= 0,

kr =4π3

λN∫

λ0

1κ(λ )

∂B∂T

(Tk+1(z),λ )dλ

⎫⎬⎭ . (45)

[λ0,λN ] denotes the semitransparent wavelength region and κ the absorption coeffi-cient depending on wavelength. The problem (43), (45) can be written as

||(G′(Tk))�G′(Tk)(Tk+1−Tk)− (G′(Tk))�(Im−G(Tk))||2

+α|| ∂∂ z

((kh + kr)

∂Tk+1

∂ z

)||2→min . (46)

We assume that the intensities are measured with some noise

||Im− Iδm||=

⎛⎝

λN∫

λ0

(Im(λ )− Iδ

m(λ ))2

dλ

⎞⎠

12

< δ .

The regularization parameter α is chosen according to the discrepancy rule. Wechoose a sequence {αn}→ 0 and take this α� as the optimal one for which

||Iδm−G(T α

k+1)||< ηδ , η > 1, (47)

is satisfied.We have tested the method for different applications from glass industry like the

heating of a quartz glass rod in a electrically heated tube furnace or the measurementof the temperature distribution in a glass drop during hot forming.

Figure 15 shows the experimental set-up of a laboratory experiment (see [5]). A280-mm-long quartz glass rod with diameter of 25 mm is placed in a tube furnace

Thermocouples

Furnace

Glass slab

Fig. 15 Experimental set-up for the laboratory experiment


450

350

Tem

pera

ture

°C

550

250

0.1Length in m

0.20

Inverse Reconstruction

Thermocouples

Eddington-Barbier

0.3

Fig. 16 Temperature inside the rod. Experimental measurements and reconstruction by spectralremote sensing

and heated up. The temperature profile of the furnace is transferred to the quartz rodduring the heating process. The corresponding temperature profile T (z) inside theglass was determined by means of spectral pyrometry. In six longitudinal groovesin the rod, eleven thermocouples were placed, whose sensor tips were equidistantlydistributed along the furnace axis. Figure 16 represents the comparison between themeasured temperature and the temperature calculated by spectral remote sensing.The thermocouple profile corresponds very well with the reconstructed profiles us-ing the above described algorithm. Using spectral remote sensing the temperatureinside the glass can be “measured” from the boundary to the hotest point inside thequartz rod.

For more details about the test examples we refer to [5].

3.5 Reconstruction of Initial Temperature

In the previous section we have discussed some examples of inverse problems. Ifinitial temperature distribution, boundary conditions and all physical parameters areknown, the forward problem can be solved by standard techniques. But in many in-dustrial applications like glassmaking and glass processing some of the informationis missing. In this section we will discuss the case where the initial temperature dis-tribution T0 is unknown. Instead of that one may be able to measure the temperatureand the heat flux at the whole boundary or parts of it for some time [t1, t2] ⊆ [0, t�],e.g. by infrared camera or pyrometer. The main question we want to investigate canbe stated: How to reconstruct the initial temperature distribution from additionalboundary measurements? Reconstructing the initial temperature from measuredboundary data is another possibility of indirect temperature determination for theinner part of a hot glass body (Fig. 17).

162 N. Siedow

Fig. 17 The boundarytemperature of a body ismeasured over a certain timeto “look into the body”

The material of this lecture was taken from [17] and [16]. Furthermor we refer to[11] and [12].

Like for the previous discussed problems we will show that the determinationof the initial temperature distribution from boundary data is ill-posed, i.e. smallmeasurement errors cause huge deviations in the result. Regularization techniqueshave to be used. We consider the three-dimensional (forward) heat transfer equation

∂T∂ t−∇ · (k∇T ) = f , in (0, t�)×D,

T |t=0 = T0, on D, (48)

k∂T∂n

+ γT = g, on St := (0, t�)× ∂D. (49)

As said before, it may be possible to measure the boundary temperature on the wholeboundary or parts of it. Let Γ be a subset of ∂D and SΓ := (0, t�)×Γ . We definethe extended forward operator A by

A(T0, f ,g) := T |SΓ .

The inverse problem states:

Given f, g and additionally (measured) boundary temperatures y on SΓ . Find T0

satisfyingA(T0, f ,g) = y. (50)


The inverse problem (50) is non-linear, but due to the linearity of the forwardproblem (48) one can eliminate the inhomogeneities f and g and one obtains

A(T0, f ,g) = A(T0,0,0)+ A(0, f ,g).

Evaluating A(0, f ,g) by solving a well-posed (forward) problem, instead of (50) wehave to solve the following inverse problem

(AT0) := A(T0,0,0) = y− A(0, f ,g) = y− v. (51)

Without loss of generality we want to discuss a simplified one-dimensional heattransfer equation to analyze the (51) in more detail. Consider

∂T∂ t

(x,t)− ∂ 2T∂x2 (x,t) = 0 in Dt = (0, t�]× (0,π), (52)

with boundary conditions

∂T∂x

(0,t) = 0,∂T∂x

(π ,t) = 0, t ∈ (0, t�], (53)

and additional measurements at the left boundary

T (0,t) = y(t), t ∈ (0, t�]. (54)

From these data we want to reconstruct the initial temperature distribution T (x,0) =T0(x), i.e. we want to solve

(AT0)(t) = y(t). (55)

Figure 18 illustrates the problem.In literature one can find two heat conduction problems related to the here pre-

sented one, namely the backward heat conduction problem and the sideways heatequation.

Fig. 18 Initial temperaturereconstruction

t*

00 π

164 N. Siedow

Fig. 19 Backward heatequation

t*

t

00 π

Fig. 20 Sideways heatequation

t*

00 π

Consider the heat transfer equation (52) with boundary conditions (53). If itis required to determine the temperature at some time t ∈ (0, t�) from measure-ments at the end time t�, i.e. T (x,t�), x ∈ [0,π ], then one has to solve the heatequation backward in time – the so-called backward heat equation. This problemis illustrated in Fig. 19. Figure 20 shows the second known inverse heat transferproblem.

Consider the heat transfer equation (52) with initial condition T (x,0) = T0(x)and boundary conditions at one side, for instance

T (0,t) = y(t),∂T∂x

(0,t) = 0, t ∈ (0, t�].

We assume t� = ∞. Now, it is required to find the temperature at x = π . This problemis called sideways heat equation.

Both inverse problems – backward heat and sideways heat – are ill-posed. Adetailed discussion and several numerical schemes of solving these problems can befound in [3] and [10].

Now, we return to the initial temperature reconstruction problem (55), which isdifferent to both – the backward and sideways heat equation. The solution of the


forward problem, i.e. with given initial condition T (x,0) = T0(x), can be calculatedanalytically by Fourier’s series

T (x,t) =∞

∑n=0

e−n2t (T0(x),wn(x))L2(0,π) wn(x), (56)

with

wn(x) =

⎧⎨⎩

√1π , n = 0√2π cos(nx), n > 0

and

(u(x),v(x))L2(0,π) =π∫

0

u(x)v(x)dx.

Using (56) the temperature at the boundary x = 0 can be written as

T (0,t) =1π

π∫

0

T0(x)dx +2π

∞

∑n=1

e−n2t

π∫

0

T0(x)cos(nx)dx

or in a more compact way as

T (0, t) =π∫

0

k(x,t)T0(x)dx, k(x,t) =∞

∑n=0

e−n2twn(0)wn(x), (57)

with kernel k(x, t) defined in (57). Therefore the inverse problem for calculatingT0(x) using boundary measurements (54) states

(AT0)(t) = y(t) =π∫

0

k(x,t)T0(x)dx. (58)

We have to solve an integral equation of the first kind.It can be shown, that there exists a unique solution T0(x) for (58) (see [17]). On

the contrary the continuous dependence on the right hand side is violated as can beseen from the following analytical consideration.

Let T0(x) be the unique solution of (AT0)(t) = y(t) and assume that instead ofy(t) we have measured

yk(t) = y(t)+

√2kπ

e−k2t , k = 1,2, ....

yk(t) deviates only slightly from y(t)

‖yk(t)− y(t)‖2L2(0,t�) =

2kπ

∫ t�

0e−2k2t dt =

1πk

(1− e−2k2t�

)k→∞−→ 0.

166 N. Siedow

The solution of (AT k0 )(t) = yk(t) is given by

T k0 (x) = T0(x)+

√kwk(x), (59)

what can be easily seen from

(AT k0 )(t) = (AT0)(t)+

√k(Awk)(t)

= y(t)+√

k

π∫

0

k(x, t)wk(x)dx

= y(t)+

√2kπ

e−k2t

= yk(t).

From (59) one obtains

‖T k0 (x)−T0(x)‖2

L2(0,π) = kk→∞−→ ∞

A small error in the measurements leads to a huge deviation in the reconstructedinitial condition. That is the reason why the problem is ill-posed.

We now turn back to the more general three-dimensional problem (51). For thesolution of ill-posed problems one can apply one of the regularization methodsdiscussed before. For given noisy measurement data

‖y− yδ‖L2(SΓ ) < δ

we use the Tichonov regularization to solve the discretized normalized system ofequations

(A�hAh + αI)T δ

0,α = A�hyδ ,

where T δ0,α is the regularized discrete solution. For the choose of α we apply the

discrepancy rule.The performance of the algorithm can be enhanced if an initial guess T δ

0 isknown, which is already close to T0. Let yδ (0) be the measurement at t = 0 on Γ .As the initial guess for T δ

0 we use the solution of the well-posed forward problem

−∇ · (k∇Tδ0 ) = f (0), in D,

T δ0 = yδ (0), on Γ ,

k∂T δ

0

∂n+ γTδ

0 = g(0), on ∂D\Γ (60)


Fig. 21 Initial temperature reconstruction for different measured times

As a numerical example we show a figure from [17], where the inverse problem(55) for the simplified one-dimensional forward problem (52)–(54) was considered.The normalized equation A�AT0 = y was solved using a conjugate gradient algorithmfor four different measurement times: t� = 0.05, t� = 0.2, t� = 0.5, and t� = 2.0.The number of iterations acts as regularization parameter. As a stopping rule thediscrepancy criterion was used. The exact initial temperature was chosen as T0(x) =cos(3x).

Near the boundary x ∈ [0,0.4] all reconstructions with different t� give satisfac-tory results. From Fig. 21 it can also be stated that the accuracy of the reconstructiondepends on the number of measured data. Whereas for t� = 0.05 only the left partof the reconstructed curve fits to the exact solution, the reconstruction with t� = 2.0fits for the whole interval x ∈ [0,π ].

In [16] and [11] the initial temperature reconstruction for a nonlinear heat equa-tion including thermal radiation was investigated. As a test example the cooling ofa hot glass plate was discussed:

cmρm∂T∂ t

(z, t) = kh∂ 2T∂ z2 (z,t)−κ

⎛⎝4σT 4(z,t)−2π

1∫

−1

I(z,μ)dμ

⎞⎠, in(0, t�]×(0,D),

kh∂T∂n

(z, t) = 0, z ∈ 0,D, t ∈ (0,t�], (61)

with measurements at the left boundary

T (0,t) = y(t), t ∈ (0, t�]. (62)

As before cm denotes the specific heat, ρm the density of the glass, and kh the glassconductivity. σ = 5.67051 ·10−8 W

m2K4 is the Stefan–Boltzmann constant. The radia-tive intensity I(z,μ) depending on position z and direction μ ∈ [−1,1] is defined as

168 N. Siedow

the solution of the radiative transfer equation

μ∂ I∂ z

(z,μ) = κ(σ

πT 4(z,t)− I(z,μ)

), z ∈ (0,D),

I(0,μ) =σπ

T 4a , μ > 0, I(1,μ) =

σπ

T 4a , μ < 0. (63)

From (61) to (63) we have to identify the initial temperature distribution T (z,0) =T0(z). The inverse problem is denoted as

A(T0, f ,g) = y, (64)

where in this case g = 0 and f = κ(

4σT 4(z,t)−2π1∫−1

I(z,μ)dμ)

is a non-linearity

depending via radiative intensity I on temperature T and hence on the initial tem-perature distribution T0.

Similar to (51) we make a decomposition of the non-linear equation

A(T0, f ,0) = A(T0,0,0)+ A(0, f (T0),0) = y

and use now a fixed-point iteration to determine the initial condition

(AT0,k+1) := A(T0,k+1,0,0) = y− A(0, f (T0,k),0) := y− v(T0,k). (65)

Instead of exact measurements y we consider yδ : ||y− yδ ||L2(SΓ ) < δ with somenoise level δ .

In [11, 16] the problem was solved using the Tichonov regularization

(A�A + αiI)Tδ ,αi

0,k+1 = A�(yδ − v(T δ ,αi0,k )) (66)

taking a series of decreasing regularization parameters αi = α0qi, q < 1, i = 1, ...,m.

For each αi we solve (66) and obtain T δ ,αi0,k+1. Among all

{T δ ,αi

0,k+1

}m

i=0we choose

Tδ ,α j

0,k+1 such that

‖T δ ,α j0,k+1−T

δ ,α j−10,k+1 ‖= min

{‖T δ ,αi

0,k+1−T δ ,αi−10,k+1 ‖L2(0,D), i = 1,2, ...,m

}. (67)

(67) is called quasi-optimality criterion, which does not depend on the noise level δ .Figures 22 and 23 show the reconstruction of the initial temperature profile,

where we have taken a typical situation of uniform cooling from both boundaries(see [11]) with random noise of 0.1% and 1.0%.

Both figures show satisfactory results of solving the inverse problem.


1300

1200

1100

exactreconstruction

1000

800

600

0 0.1 0.2 0.3 0.4 0.5z

0.6 0.7 0.8 0.9 1

Fig. 22 Initial temperature reconstruction with 0.1% noisy data

1300

1200

1100

exactreconstruction

1000

800

600

0 0.1 0.2 0.3 0.4 0.5z

0.6 0.7 0.8 0.9 1

Fig. 23 Initial temperature reconstruction with 1.0% noisy data

170 N. Siedow

3.6 Conclusions

Inverse problems are concerned with finding causes for an observed or desiredeffect. A common property of a vast majority of inverse problems is their ill-posedness. Very often the solution does not depend continuously on the data. Tosolve an ill-posed problem one has to use regularization techniques, that is, re-place the ill-posed problem by a family of neighboring well-posed problems. Theregularization has to be taken in accordance with the problem one wants to solve.The spectral remote sensing leads to an ill-posed integral equation. We use themethod to reconstruct the inner temperature of a hot glass body. That is why the us-age of the heat transfer operator with Rosseland approximation is an excellent wayto regularize the problem. The reconstruction of the initial temperature distributionfor a glass cooling process offers another possibility to reconstruct the temperatureprofile inside the hot glass. This ill-posed heat transfer problem is different to thosefrom literature – backward head and sideways heat equation. We have discussed asolution procedure even in that case when heat radiation has to be taken into ac-count. We solved the ill-posed problems with Tichonov regularization.

References

1. Modest, M.F.: Radiative Heat Transfer. Academic, San Diego (2003)2. LeVeque, R.J.: Finite difference methods for ordinary and partial differential equations: Steady-

state and time-dependent problems. SIAM (2007)3. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic,

Dordrecht (2000)4. Louis, A.K.: Inverse und schlecht gestellte Probleme, Teubner, Stuttgart (1989)5. Brinkmann, M., Siedow, N., Korb, T: Remote Spectral Temperature Profile Sensing. In:

Loch, H., Krause, D. (ed): Mathematical Simulation in Glass Technology. Springer, Berlin(2002)

6. Neunzert, H., Siedow, N., Zingsheim, F.: Simulation of the Temperature Behaviour of HotGlass during Cooling. In: Cumberbatch, E., Fitt, A. (ed): Mathematical Modelling. Case Stud-ies from Industry, pp. 181–198. Cambridge Univerity Press, Cambridge (2001)

7. Viskanta, R., Anderson, E.E.: Heat transfer in semitransparent solids. In: Irvine, T.F. Jr.,Harnett, J.P. (eds.) Advances in Heat Transfer, vol. 11, pp. 317–441. Academic, New York(1975)

8. Choudhary, M.K., Huff, N.T.: Mathematical modeling in glass industry: An overview of statusand needs. Glastech. Ber. Glass Sci. Technol 70, 363–370 (1997)

9. Rosseland, S.: Note on the absorption of radiation within a star. M.N.R.A.S. 84, 525 (1924)10. Berntsson, F.: Numerical Methods for Solving a Non-Characteristic Cauchy Problem for a

Parabolic Equation. Technical report LiTH-MAT-R-2001-17, Department of Mathematics,Linkoping University (2001)

11. Pereverzyev, S.S. Jr., Pinnau, R., Siedow, N.: Regularized fixed-point iterations fotr nonlinearinverse problems. Inverse Probl. 22,1–22 (2006)

12. Pereverzyev, S.S. Jr., Pinnau, R., Siedow, N.: Initial temperature reconstruction for a nonlinearheat equation: application to radiative and convective heat transfer. Inverse Problems in Scienceand Engineering. 16, 55–67 (2008)

13. Lentes, F.T., Siedow, N.: Three-dimensional radiative heat transfer in glass cooling processes.Glastech. Ber. Glass Sci. Technol. 72(6), 188–196 (1999)


14. Siedow, N., Grosan, T., Lochegnies, D., Romero, E.: Application of a New Method forRadiative Heat Transfer to Flat Glass Tempering. J. Am. Ceram. Soc. 88(8), 2181–2187 (2005)

15. Narayanaswamy, O. S.: A model of structural relaxation in glass. J. Am. Ceram. Soc. 54(10),491–498 (1971)

16. Pereverzyev, S.S. Jr.: Method of Regularized Fixed-Point and its Application. PhD Thesis,Technical University Kaiserslautern (2006)

17. Justen, L.: An Inverse Heat Conduction Problem with Unknown Initial Condition. DiplomaThesis, Technical University Kaiserslautern (2002)

18. Zingsheim, F.: Numerical Methods for Radiative Transfer in Semitransparent Media. PhDThesis, Technical University Kaiserslautern (1999)

Non-Isothermal Flow of Molten Glass:Mathematical Challenges and IndustrialQuestions

Angiolo Farina, Antonio Fasano, and Andro Mikelic

Abstract With specific reference to the process of glass fibers drawing we reviewthe models proposed to describe the various stages of the flow of molten glass fromthe furnace to the winding spool: the slow flow in the die, the jet formation un-der rapid cooling, the terminal fiber profile. In the course of our exposition we willpresent a general model for non-isothermal flows of mechanically incompressiblebut thermally expansible fluids (the basic model here assumed for glass), and theOberbeck–Boussinesq limit is discussed. Both the modelling and the mathemati-cal aspects will be illustrated in detail. An appendix is devoted to the question ofstability analysis.

Keywords Glass fiber drawing · Isochoric viscous fluids · Heat conducting ·Navier-Stokes equations · Justification of the Boussinesq approximation · Singularperturbation · Non-isothermal elongational free boundary flow

1 Introduction

In this chapter we present a mathematical theory for molten glass flow, with spe-cific reference to the industrial process of glass fibers drawing, roughly depictedin Fig. 1.

Molten glass is kept at a high temperature (1,200–1,400◦C) and goes by grav-ity through an array of hundreds of dies. From each drop a filament is pulled downwhich become thinner and thinner during its motion, reaching a final diameter which

A. Farina and A. Fasano (�)Universita degli Studi di Firenze, Dipartimento di Matematica “Ulisse Dini”, Viale Morgagni 67/A,I-50134 Firenze, Italye-mail: [email protected]; [email protected]

A. MikelicUniversite de Lyon, Lyon, F-69003, FRANCE; Universite Lyon 1, Institut Camille Jordan,UMR 5208 CNRS, Bat. Braconnier, 43, Bd du onze novembre 1918 69622 VilleurbanneCedex, FRANCEe-mail: [email protected]


173

[email protected]

[email protected]

[email protected]

174 A. Farina et al.

Fig. 1 A single glass fiberdrawing

reservoir

die

free liquid jel

rotating drum

can be three orders of magnitude less than the diameter of the die (various shapeshave been adopted for the die, but here we refer to a simple cylindrical geometry).As an obvious consequence of mass conservation the terminal speed of the glasscan be up to six orders of magnitude the one at which glass enters the die (typ-ical values can be: a fraction of 1 mm/s for the entrance velocity, and 30 m/s forthe terminal phase). The final product is as flexible as a textile fiber and is woundaround a rotating drum. The latter device provides the traction driving the entireprocess. From the die to the spool the flow conditions change in a dramatic way.While only moderate temperature changes take place within the die, at the exit ofthe die heat is removed at a high rate, both by radiation, and by an intense transver-sal air flow. Consequently, density will increase and, above all, viscosity will govery rapidly to quite large values (a phenomenon which makes the fiber tractionpossible). The shape of the boundary will be determined by surface tension (alsosensitive to temperature changes) and the flow conditions (strongly influenced bythe rapidly varying viscosity). Roughly speaking, we can distinguish four stages(see Fig. 1):

(a) The flow of molten glass at high temperature in the reservoir, feeding the fiberproduction system.

(b) The non-isothermal flow through the die, with rigid lateral boundaries.(c) The viscous jet flow with rapidly changing physical parameters, owing to the

fast cooling, up to the formation of a “fiber” (high viscosity, small variation ofthe axial velocity and very small radial velocity).

(d) The motion of the glass fiber, drawn down by a device called spinner (or spool).

Non-Isothermal Molten Glass Drawing 175

In our analysis we will not deal with stage (a), which is treated elsewhere in thisbook. Stage (b) terminates at the (unknown) point at which the jet is formed. Tobe more precise, depending on the glass temperature and on the pulling speed (ulti-mately determining the discharge), the glass may form the jet while it is still insidethe die, or it may fill completely the die region and even flow over its external rim(an undesired effect) before forming the jet. In order to describe the flow in the diewe have adopted the ideal scheme of a mechanically incompressible, but thermallyexpansible fluid, which is illustrated in detail in Sect. 2. Such an approach has beenheavily criticized in the past (see [2, 3]) on the basis of a linear stability analysis,showing that under such conditions the rest state is linearly unstable. Putting asidethe philosophical question of how significant linear instability can be, we will justemphasize that in the context of the approximation Ec ≈ 0 (Ec stands for Eckertnumber, which basically compares a typical kinetic energy of the unit mass withits typical heat content) such unpleasant property disappears (see Appendix 1). Inour case it will be clear that Ec is really negligible. Existence and uniqueness of asolution to the stationary fluid dynamical problem formulated in Sect. 2 is provedin Sect. 3, supposing that inflow and outflow temperature and velocity are known.In this same section we discuss the question of the Oberbeck–Boussinesq limit, arecurrent subject in non-isothermal flows that has been treated in a number of pa-pers in different contexts on the basis of the choice of a small parameter. In our case(flow in the die, where temperature excursion is small) the most sensible choice is aparameter combining the thermal expansion coefficient and the maximum tempera-ture variation. We will be able to obtain not only the corresponding O–B limit, butalso to estimate the higher order correction.

The third stage, i.e. stage (c), is the one of the jet flow in rapidly changing con-ditions. It starts at the point where the jet is formed (which is unknown) and itsendpoint is usually established according to some thumb rule. Of course one of themain difficulty of the problem as a whole is to make the solutions at the three stagesagree with each other. This requires a sequence of iterations, but the complete freeboundary problem is still open. The shape of the jet results from the action of surfacetension (via the surface curvature), creating the confining force to be balanced bythe stress tensor. The temperature dependence of both quantities is clearly a sourceof great difficulty.

The fourth stage is the terminal phase of the fiber motion, characterized by thefact that viscosity has already reached a very large value, the fiber is already consid-erably thin and further thinning is slow. The particular geometry allows to look forthe expansion of all relevant quantities in power series of the ratio between two typ-ical lengths in the transverse and in the longitudinal directions. We will derive the socalled Matovich–Pearson approximation on a rigorous basis and prove an existenceand uniqueness theorem for the limit problem.

One of the most delicate aspects of the whole theory is the selection of the con-fine between stage (c) and stage (d). Here we present an approach based on theMatovich–Pearson model (Sect. 4.1 and Appendix 2).


2 Mathematical Modelling

2.1 Definitions and Basic Equations

We start by recalling the mass balance, the momentum and energy equations, aswell as the Clausius–Duhem inequality in the Eulerian formalism. Following an ap-proach similar to the one presented in [25], pages 51–85, we denote by {ρ ,v,e,T,s}the density1, velocity, specific internal energy, absolute temperature and specific en-tropy, satisfying the following system of equations

DρDt

=−ρ div v, (1)

ρDvDt

=−ρge3 + div T, (2)

ρDeDt

=− div q+ D(v) : T, (3)

ρDsDt

+ div( q

T

)≥ 0, (4)

where:

• DDt

=∂∂ t

+ v ·∇ denotes the material derivative.

• D(v) =12

(∇v+(∇v)T

)is the rate of strain tensor.

• T is the Cauchy stress tensor. Further, D(v) : T= tr(D(v) TT

)= ∑i, j Di jTi j .

• q is the heat flux vector.• −ge3 is the gravity acceleration. Indeed e3 is the unit vector relative to the x3

axis directed upward.• In (3) the internal heat sources are disregarded.

Introducing now the specific Helmholtz free energy

ψ = e−Ts, (5)

inequality (4) becomes

ρ(

DψDt

+ sDTDt

)−T : D(v)+ q · ∇T

T≤ 0. (6)

1 Density, in the Eulerian formalism, means mass per unit volume of the current configuration.


2.2 Fluids Physical Properties and Constitutive Equations

Key point of the model is to select suitable constitutive equations for

e, T, q, and s,

in term of the primary dependent variables v, T , and ρ . Of course the selection of theconstitutive equations must be consistent with the physical properties of the fluidswe are considering. For our purposes molten glass can be reasonably described asfollows:

1. The fluid is mechanically incompressible but thermally dilatable (i.e. the fluidcan sustain only isochoric2 motion in isothermal conditions).

2. The fluid behaves as a linear viscous fluid (Newtonian fluid), i.e. the shear stressis proportional to the shear rate.

Focusing on the first aspect (i.e. thermally dilatable fluid), we can say that thermaldilation is described by the thermal expansion coefficient, defined as follows

β =− 1ρ

dρdT

, [β ] = ◦K−1. (7)

The coefficient β may depend on temperature, i.e. β = β (T ), but, since the ma-terial is mechanically incompressible, β does not depend on pressure.

Remark 2.1. Considering a material particle denoted by the Lagrangian coordinateX and β (T ) given, we have

ρ (X, t) = ρo (X)exp

⎧⎪⎨⎪⎩−

T (X,t)∫

To(X)

β (s)ds

⎫⎪⎬⎪⎭

, i.e. ρ = ρ (T ) ,

where ρo (X) and To (X) are density and temperature in the reference configuration,respectively.

We now introduce the Lagrangian density, ρL, defined as mass per reference config-uration unit volume. We have

ρL = ρJ,

with J determinant of the deformation gradient, J = detF. Now, since ρL does notvary, we have

dρL

dT= 0, ⇒ 1

JdJdT

+1ρ

dρdT︸︷︷︸−β

= 0,

2 Volume preserving.


namely

β (T ) =1J

dJdT

, (8)

or, J = exp{T∫

To

β (s)ds}, i.e. J = J (T ).

Equation (1), because of (7), reads as a constraint linking temperature variationswith the divergence of the velocity field

−β DTDt

+ D(v) : I = 0, ⇔ −β DTDt

+ div v = 0. (9)

Once again, if T is uniform and constant, div v = 0, the flow is isochoric.Thus, the constraint ρ = ρ (T ) models the fluid as being incompressible under

isothermal conditions, but with density changing in response to changes in temper-ature. In other words, given a temperature T the fluid is capable to exert any forcefor reaching the corresponding density. This fact is, of course, not physical and aswe shall see in Appendix 1 will introduce a structural instability in the model. Nev-ertheless this has no serious consequences in our case, as we shall see later.

Let us now turn to a discussion of the constraint. We proceed applying classicalprocedure, see [12], which tours out to be feasible in our situation. We modify T, q,ψ and s by adding a term (the so called constraint response) due to the constraintitself. We consider

T = Tc+Tr, q = qc+qr, ψ = ψc+ψr, s=sc + sr, (10)

where the suffix “ c ” denotes the constraint response while the suffix “ r ” denotesthe constitutive part, which is function of the primary state variables.

Concerning the constraint responses, following standard practice, we requirethat:

1. Tc, qc, ψc, and sc have no dependence on the state variables.2. The constraint responses do not dissipate energy, namely recalling (6),

ρ (T )(

Dψc

Dt+ sc

DTDt

)−Tc : D(v)+ qc · ∇T

T= 0, (11)

in any thermo–mechanical process that fulfills (9).

Now, considering various special subsets of the set of all allowable thermo– mechan-

ical processes, the above equation leads to qc = 0 andDψc

Dt= 0, implying psic ≡ 0

if it vanishes in the reference configuration. Hence, (11) reduces to

ρ scDTDt−Tc : D(v) = 0, when −β DT

Dt+ I : D(v) = 0.


The latter is therefore verified if ρ sc and Tc are proportional to −β and to I,respectively. We thus set

⎧⎨⎩ρ (T )sc =−pβ , ⇒ sc =−p

βρ (T )

,

Tc =−pI,

where p is a function of position and time, referred to as mechanical pressure. Recallthat p is not the thermodynamic pressure, defined through an equation of state. Inthis framework there is no constitutive equation for p.

Remark 2.2. We may reach the same conclusions in a more standard way con-sidering the theory of a compressible (i.e. unconstrained) fluid, introducing theHelmholtz free energy ψ = ψ (ρ ,T ), and we develop the standard theory eventu-ally considering the limit ρ = ρ (T ), which gives ψ = ψ (T ) = ψ (ρ (T ) ,T ). Wededuce that:

• There is no equation of state defining the pressure p.

• s = s(T )− pβρ

.

Now, going back to (10), we have

T = Tr−pI, q = qr, ψ = ψr, s=sr− pβ

ρ (T ). (12)

So, as usual, we have now to select appropriate constitutive equations for Tr, qr, ψr,and sr requiring that inequality (6) is fulfilled, namely

ρ(

Dψr

Dt+ sr

DTDt

)−Tr : D(v)+ qr ·

∇TT≤ 0. (13)

The first constitutive assumption is the following:

A1. ψ = ψ(T ).

Inserting the latter into (13), yields

ρ(

dψdT

+ sr

)DTDt−Tr : D(v)+ qr ·

∇TT≤ 0, (14)

which must hold true for all thermo-mechanical processes. Thus, we deduce⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

sr =−dψdT

,

Tr : D(v)≥ 0,

qr ·∇T ≤ 0.

(15)


In particular, (15)2 and (15)3 play the role of restrictions on the constitutiveassumptions. Recalling that we are considering a Newtonian fluid, we assume:

A2. The constitutive part of the Cauchy stress tensor is

Tr = 2μD(v)+η (D(v) : I) I, (16)

where μ = μ (T ) > 0 is the so–called dynamic viscosity and η = η (T ) the so–called bulk viscosity.

Now setting

D(v) = D(v)− 13

(D(v) : I) I, so that I :D(v) = 0, (17)

we may write

Tr = 2μD(v)+(

23μ+η

)(D(v) : I) I. (18)

Hence, inserting (15)1 and (18) into (14), we are left with

−2μD(v) : D(v)−3

(23μ+η

)(D(v) : I)2 + qr ·

∇TT≤ 0. (19)

We thus stipulate:

A3.23μ+η = 0, ⇒ η =−2

3μ , Stokes assumption.

A4. The heat flux vector isqr =−λ∇T, (20)

where λ is the thermal conductivity, λ = λ (T ) > 0.

Then inequality (19) reduces to

−2μD(v) : D(v)− |∇T |2T≤ 0,

which is always fulfilled.Before proceeding further, let us recall the results obtained so far. Starting form

(12) and applying the classical continuum mechanics procedure as well as assump-tions A1.–A4. we have obtained

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

T =−pI+2μD(v)− 23μ (D(v) : I) I,

q =−λ∇T,

ψ = ψ (T ) ,

s =−(

dψdT

+ pβ

ρ (T )

).

(21)


2.3 The General Model

Let us start with the internal energy. From (5), considering (21)3 and (21)4, we have

e = e(T, p) = ψ(T )−T

(dψ(T )

dT+ p

βρ

),

and

DeDt

=

⎛⎜⎜⎜⎜⎜⎝−T

d2ψdT 2 + T p

(βρ2

dρdT

)

︸︷︷︸− β2

ρ

− pTρ

dβdT

⎞⎟⎟⎟⎟⎟⎠

DTDt

−(β pρ

DTDt

)

︸︷︷︸pρ div v

−βTρ

DpDt

.

Concerning the term D(v) : T, recalling also (17), we obtain

T : D(v) =

⎛⎜⎜⎜⎜⎝−pI+2μ

[D(v)− 1

3(D(v) : I) I

]

︸︷︷︸D(v)

⎞⎟⎟⎟⎟⎠

: D(v)

= −p div v+ 2μ D(v) : D(v),

where the term −p div v represents the mechanical work per unit time operated bythe system during expansion or compression. Of course such a term is necessarilycompensated by internal energy gain (or loss) associated with dilation (or compres-sion). Indeed we have developed the theory assuming that the constraint responsedoes not dissipate energy.

Thus, taking (21)2 into account, the energy equation (3) can be rewritten as

ρ(−T

d2ψdT 2 −

β 2

ρT p− pT

ρdβdT

)DTDt

= div(λ∇T

)+βT

DpDt

+ 2μ |D(v)|2, (22)

where the term 2μ |D(v)|2 represents mechanical energy converted into heat by theinternal friction.

We remark that the coefficient in front ofDTDt

is, from the physical point of view,

the isobaric specific heat cp. Actually, (22) is exactly the corresponding general


energy equation in the specific enthalpy formulation from [25], pages 51–85, butwith the particular choice

cp(T, p) = cp1 (T )− pTρ

(β 2 +

dβdT

), with cp1 (T ) =−T

d2ψ (T )dT 2 .

Indeed, the fluid we are modelling admits only the isobaric specific heat, since,because of (8), any change of body’s temperature implies a change in volume. Henceit is not possible to work with the isochoric specific heat cv. Consequently the formof the energy equation here adopted is necessarily different from the theory devel-oped in [29] and [30].

Next, experiments show that the variations of cp with respect to pressure aregenerally very small. We impose that cp is constant with respect to the pressure p,requiring

∂cp

∂ p= 0, ⇒ β 2 +

dβdT

= 0,

that implies

β (T ) =βR

1 +βR (T −TR),

with TR reference temperature and βR = β (TR).As a consequence, from (7), we have the following law for the density

ρ =ρR

1 +βR (T −TR), (23)

with ρR = ρ (TR) reference density. In particular, we will consider the linearizedversion of (23), namely

ρ(T ) = Aρ −BρT.

with

βR =BρρR

and ρR = Aρ −BρTR. (24)

We remark that, from the mathematical point of view such a simplification is notcrucial and it is consistent with the data reported in the experimental literature.

Therefore, once assumptions A1, A2, A3 and A4 are stated, the mathematicalmodel (1)–(3) rewrites as

βDTDt

= div v, (25)

ρDvDt

=−ρge3−∇p + div

{2μD(v)− 2μ

3div vI

}, (26)

ρ cp1 (T )DTDt

= div(λ∇T

)+βT

DpDt

+ 2μ |D(v)|2, (27)


ρ(T ) = Aρ −BρT. (28)

We then introduce the rescaled dimensionless temperature, that is

ϑ =T −TR

Tw−TR, ⇔ T = TR

[1 +ϑ

(Tw−1

)], with Tw =

Tw

TR> 1, (29)

where Tw, Tw > TR, is another characteristic temperature, as we shall see in nextsection. Note that (28) reads now

ρ(ϑ) = ρR−βRρRTR

(Tw−1

)ϑ . (30)

Next, we introduce the hydraulic head

P = p +ρRgx3. (31)

Hence −ρge3−∇p in (26) becomes

−ρge3−∇p = (ρR−ρ) ge3−∇P =[ρRβRTR

(Tw−1

)ϑ]

ge3−∇P.

We note that, after “eliminating” the hydrostatic pressure, the effect of the tempera-ture field on the flow is more easy to observe.

Concerning the shear viscosity μ we assume the well known Vogel–Fulcher–Tamman’s (VFT) formula

logμ(ϑ) =−Cμ +Aμ

TR−Bμ + TR

(Tw−1

)ϑ

, Aμ ,Bμ ,Cμ > 0. (32)

In particular, μ is monotonically decreasing with ϑ . For more details we refer e.g.to [26], Chap. 6. We note that the temperature in the problems we are consideringis such that the denominator in (32) is always positive. Consequently, μ is a givenstrictly positive and C∞ function of the temperature. In the polymer physics, formula(32) is known as Williams–Landau–Ferry (WLF) relation.

Finally, we suppose cp1 and λ to be smooth functions of ϑ , bounded from aboveand from below by positive constants.

2.4 Scaling and Dimensionless Formulation

The scaling of model (25)–(27) will be operated paying particular attention to thespecific problem we are interested in.

We recall that we are considering a gravity driven flow of molten glass through anozzle in the early stage of a fiber manufacturing process. At this stage we considerthe inlet and outlet temperatures of the fluid as prescribed quantities. In particular,referring to Fig. 2, the fluid temperature on Γin is higher than the one on Γout ,


T (Γin) = Tw, T (Γout) = TR < Tw.

We introduce the following dimensionless quantities

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

x =xH

, v =v

VR, t =

ttR

with tR =HVR

,

μ =μμR

, λ =λλR

, cp1 =cp1

cpR,

ρ =ρρR

, P=PPR

, ψ =ψ

TRcpR,

H playing the role of a length scale (see again Fig. 2). The choice of the referencevelocity VR has to be made according to the particular flow conditions we deal with3.We will return to this point later on. As the reference pressure PR we take the pointof view that flows of glass or polymer melts are essentially dominated by viscouseffects. Accordingly we set4

PR =μR

HVR . (33)

Concerning the reference quantities μR, λR and cpR we identify them with the valuestaken by the respective quantities μ , λ and cp for P = PR and T = TR.

Suppressing tildes to keep notation simple, model (25)–(27) rewrites5

div v =

∣∣Kρ∣∣

ρ(ϑ)(Tw−1)

DϑDt

, (34)

ρ(ϑ)DvDt

=

(∣∣Kρ∣∣(Tw−1)Fr2

)ϑe3−

(PR

ρRV 2R

)∇P

+1

ReDiv

{2μ(ϑ)D(v)− 2μ(ϑ)

3div vI

}, (35)

ρ(ϑ)cp1(ϑ)DϑDt

=

( ∣∣Kρ∣∣PR

ρRcpRTR (Tw−1)

)1 +(Tw−1)ϑ

ρ(ϑ)

[DPDt− ρRgH

PRv3

]

+1

Pediv(λ∇ϑ)+ 2

EcRe(Tw−1)

μ(ϑ)(|D(v)|22−

13( div v)2

),

(36)

3 This makes our approach different from the ones presented in [22] and in [11] where there is novelocity scale defined by exterior conditions.4 Notice that PR→ 0 as VR tends to 0 and, as a consequence p tends to the hydrostatic pressure. Thisis consistent with the fact that P represents the deviation of the pressure from the hydrostatic–onedue to the fluid motion.5 In the phenomena we are considering Tw− 1 is small but not negligible. Typically Tw− 1 is oforder 10−1.


with

cp1(ϑ) =−1 +(Tw−1)ϑ(Tw−1)2

d2ψdϑ 2 . (37)

We list the non-dimensional characteristic numbers appearing in (34)–(36):

Kρ =−βRTR, thermal expansivity number (negative, according to the usual nota-tion) and we may write (30) as

ρ(ϑ) = 1− ∣∣Kρ∣∣(Tw−1)ϑ . (38)

Fr =VR√gH

, Froude’s number.

νR =μR

ρR, kinematic viscosity.

Re =VRHνR

, Reynolds’ number, which, because of (33), can be also written as

Re=ρRV 2

R

PR.

Pe = Re·Pr =VRHρRcpR

λR, Peclet’s number , with cpR = ψR/TR.

Pr =μRcpR

λR, Prandtl’s number.

Ec =V 2

R

cpRTR, Eckert’s number.

As mentioned, we are interested in studying vertical slow flows of very viscousheated fluids (molten glass, polymers, etc.) which are thermally dilatable. So, intro-ducing the so–called expansivity coefficient (or thermal expansion coefficient)

α =∣∣Kρ

∣∣(Tw−1). (39)

we will consider the system (34)–(36) in the realistic situation in which the param-eter α is small. Typically (e.g. for molten glass) 10−3 � α � 10−2. Equation (38)can be rewritten as

ρ(ϑ) = 1−αϑ .

Next, we define the Archimedes’ number

Ar =α

Fr2 =

∣∣Kρ∣∣(Tw−1)

V 2R

gH, ⇒ α = ArFr2.

So, system (34)–(36) rewrites as6

6 We remark that


div v =α

ρ(ϑ)DϑDt

, (40)

ρ(ϑ)DvDt

= Arϑ e3 +1

Re

[−∇P+ Div

(2μ(ϑ)D(v)− 2μ(ϑ)

3div vI

)], (41)

ρ(ϑ)cp1(ϑ)DϑDt

=1 +(Tw−1)ϑ

ρ(ϑ)EcAr

(Tw−1)2

[Fr2

ReDPDt− v3

]

+1

Pediv(λ∇ϑ)+ 2

EcRe(Tw−1)

μ(ϑ)(|D(v)|2− 1

3( div v)2

).

(42)

Notice that the ratiosEcRe

andFr2

Rewhich appear naturally in the equation above can

be interpreted as ratios of characteristic times

EcRe

=(

νR

cpRTR

)1tR

,Fr2

Re=

(νR

gH

)1tR

.

Note also that

EcAr = αgH

cpR TR= α

Ec

Fr2 .

We consider a flow regime such that Ar = O (1), Re = O (1), Pe = O (1) andEc� 1 (e.g. Ec �10−9). The terms in energy equation (42) containing the Eckertare dropped. So (40) and (41) remain unchanged while (42) reads as follow

ρ(ϑ)cp1(ϑ)DϑDt

=1

Pediv

(λ (ϑ)∇ϑ

). (43)

We remark that in the limit α small, while Ar and Re order 1, the buoyancy and theviscous forces cannot be negligible. We will show that in such a case the system (40),(41) and (43) can be approximated by a system similar to the Oberbeck–Boussinesqsystem. Indeed our choice of the parameters takes us close to the conditions of theformal derivation of the Oberbeck–Boussinesq system for bounded domain.

Remark 2.3. Let us note that the model (40)–(42) is criticized in [2] and [3], on thebasis of the fact that giving the density as a function of the temperature contradictsthe Gibbs convexity inequalities thus generating a rest state which is not stable. Inorder to overcome such a difficulty, they propose to replace the constraint ρ = ρ(ϑ)

αρ(ϑ )

=α

1−αϑ .

Hence, if order α2 terms are neglected, we may approximate αρ(ϑ) by α .


with the constraint ρ = ρ (s). However, we emphasize that the system (40), (41),(43) (i.e. with Ec = 0) has a rest state (in absence of any body force) linearly stable.We have also to remark that, when Eckert’s number is very small, the rest stateinstability of the system (40)–(42) would show up after such a long time to be of nointerest in technical problems. Clearly, for some other problems (e.g. atmosphericflows) the presence of instabilities, pointed out in [2] and [3], is a real difficulty andthe meaning of the stationary problems is not clear. Furthermore, we neglected someother terms from the full compressible Navier–Stokes system, whose influence, inthe authors’ knowledge, on linear stability of the rest state has not been investigated.

In Appendix 1 we show a direct analysis of the linear stability of the rest stateaccording to model (25), (26), (27), (28) and model (25), (26), (43), (28).

3 Study of the Stationary Non-Isothermal Molten Glass Flowin a Die

In Sect. 2 a mathematical model which describes the motion of a mechanically in-compressible, but thermally expansible viscous fluid was derived.

Such model can be widely used in industrial simulations of flows of hot meltedglass, polymers etc.

The model for a mechanically incompressible, but thermally expansible viscousfluid could be thought as a particular case of the compressible heat-conductingNavier–Stokes system.

Nevertheless, in the mechanically incompressible but thermally expansible casethe pressure is not linked any more to the density and to the temperature, very muchthe same as in the incompressible case. Consequently, we had to be careful with thethermodynamical modelling.

An important quantity that we have introduced is the thermal expansion coef-ficient β = − d

dT logρ , where ρ is the molten glass density and T is the absolutetemperature, and its typical value βR.

In this section we study the stationary flow within the die, i.e. the stationary flowin stage (b).

The flow domain (see Fig. 2) is the set Ω = {{r < R(x3,φ)}× [0,2π ]× (0,H)},where R : [0,H]× [0,2π ]→ [Rmin,Rmax] is a C∞-map. For the reference problem wehave in mind R and H are lengths of comparable size.

The boundary of Ω contains 3 distinct parts (see again Fig. 2):

• Lateral boundary Γlat = {r = R(x3,φ), x3 ∈ (0,H) , φ ∈ [0,2π ]}.• Inlet boundary or upper boundary Γin = {x3 = H and r ≤ R(H,φ), φ ∈ [0,2π ]}.• Outlet boundary Γout = {x3 = 0 and r ≤ R(0,φ), φ ∈ [0,2π ]}.

The stationary problem in non–dimensional form, for the velocity v, the dimen-sionless hydraulic head P = (p+ρRgHx3)/PR and the rescaled temperature ϑ readsas follows:Problem A. Find (v, P, ϑ) such that


Fig. 2 The domain Ω

Glat

Gout

H

O

R (x3,f)

Gin

x3

div (ρ(ϑ)v) = 0, ρ(ϑ) = 1−αϑ , (44)

ρ(ϑ)(v∇)v=Ar ϑe3− 1Re

{∇(

P +2μ(ϑ)

3div v

)− div (2μ(ϑ)D(v))

}, (45)

cp1(ϑ)ρ(ϑ)v ·∇ϑ =1

Pediv (λ (ϑ)∇ϑ) , (46)

v = v1e3, ϑ = 1, on Γin, v = v2eg, ϑ = 0, on Γout , (47)

v = 0, − 1Peλ (ϑ)∇ϑ ·n = q0ϑ +S , on Γlat , (48)

where the fluid’s thermal expansivity α ≥ 0 is defined by formula (39) and cp1 is afunction of temperature which can be expressed by means of Helmholtz free energy,as shown in (37). We refer the reader to Sect. 2 for the notations which, on the otherhand, are quite standard. In (48) S is a non–negative prescribed function.

The plan of this section is the following:

• In Sect. 3.1 we will explain the basic steps of an existence proof for the boundaryvalue problem (44)–(48), which does not require small data. Furthermore, wewill give a hint on uniqueness for small data and prove that solution is infinitelydifferentiable, if the data are so.

• In Sect. 3.2 we exploite the uniqueness and regularity results concerning system(44)–(48) to obtain a rigorous mathematical justification of the Oberbeck–Boussinesq approximation (in the limit of small expansivity parameter), a longdebated question, and also to compute the next order approximation (the correc-tion). We note that this is another singular limit, not linked at all to the zero Machnumber limit (see [19] or [28]).


3.1 Existence and Uniqueness Result for the Stationary Problem

In order to study the existence of a solution to the problem (44)–(48), we list theassumptions on the data:

We suppose that ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

v1 ∈C10(Γin)∩C∞(Γin),

v2 ∈C10(Γout)∩C∞(Γout),

S ∈C∞(Γlat), S ≥ 0,

(49)

and that∫

Γin

ρ(1)v1 rdrdφ =∫

Γin

(1−α)v1 rdrdφ =∫

Γout

v2 rdrdφ , (50)

since ρ(0) = 1.Concerning the coefficients, it is natural that μ corresponds to Vogel–Fulcher–

Tamman’s VFT formula (32), that we rewrite

logμ(ϑ) =−Cμ +Aμ

Bμ −ϑ , Cμ > 0. (51)

Next, as mentioned we suppose that

cp1,μ ,ρ ,λ ∈C∞(R), (52)

and take values between two positive constants. We note that the coefficients aredefined only on an interval and we may need to extend them on R in some obviousway, not affecting the final estimates.

3.1.1 Existence Result

The existence is proved by constructing an iterative procedure, where we first cal-culate the temperature for given velocity; then for given temperature we calculatethe velocity and the pressure. Procedure is repeated until getting a fixed point. Ouridea is to prove that such iterations give a uniformly bounded sequence in appropri-ate Sobolev spaces. Then elementary Sobolev compactness leads to the result. Weexplain here the main steps of the proof. For details we refer to the article [9].

Step 1. We start by studying the energy equation for a given w =ρv∈H (Ω), whereH (Ω) = {z ∈ L3(Ω)3 | div z = 0 in Ω , z · n = 0 on Γlat , z · n|Γin = ρ(1)v1, z ·n|Γout = ρ(0)v2}.


Our nonlinearities cp1 and λ are defined only for ϑ such that the density is notnegative. We extend it on R by setting

cp1(ϑ) =

{cp1(1)/ϑ 2 for ϑ > 1

(‖S ‖L∞(Γlat )

q0ϑ)2cp1(−‖S ‖L∞(Γlat )

q0) for ϑ <−‖S ‖L∞(Γlat )

q0,

(53)

λ (ϑ) =

{λ (1) for ϑ > 1

λ (−‖S ‖L∞(Γlat )q0

) for ϑ <−‖S ‖L∞(Γlat )q0

.(54)

We will prove that ϑ ∈ [−‖S ‖L∞(Γlat )q0

,1] and the result is independent of the ex-tension. Now for a given

{w ∈H (Ω), S ∈ L∞(Γlat), S ≥ 0λ ,cp1 ∈W 1,∞(R), λ ≥ λ0 and constants q0, Pe > 0,

(55)

we consider the problemProblemΘ : Find ϑ ∈ H1(Ω)∩L∞(Ω) such that

cp1(ϑ)w∇ϑ =1

Pediv

(λ (ϑ)∇ϑ

), (56)

ϑ = 1 on Γin and ϑ = 0 on Γout (57)

− 1Peλ (ϑ)∇ϑ ·n = q0ϑ +S on Γlat (58)

We list some results for ProblemΘ , whose prove is given in [9].

Proposition 3.1. Under the stated assumptions, ProblemΘ has at least one varia-tional solution in H1(Ω), satisfying the estimate

λ0

2Pe‖∇ϑ‖2

L2(Ω)3 +q0

4‖ϑ‖2

L2(Γlat)≤ A0 + B0‖w‖L1(Ω)3 , (59)

where B0 and A0 are positive constants.

Corollary 3.2. Under the assumptions of Proposition 3.1, we have the followingestimate

‖ϑ‖2L2(Ω) ≤ 8max

{Peλo

,1qo

∥∥∂x3 R∥∥∞

}(A0 + B0‖w‖L1(Ω)3

). (60)

Lemma 3.3. Any variational solution ϑ ∈ H1(Ω) to Problem Θ , satisfying the apriori estimate (59), satisfies also

−‖S ‖L∞(Γlat)

q0≤ ϑ ≤ 1 a.e. on Ω . (61)


Remark 3.4. Let us now recall a very general result for a class of elliptic problems,includingΘ (see [1]). Under a suitable condition on the interior angle between thelateral boundary r = R(x3,φ) and the upper and lower surfaces, �3(Ω) is the ellipticregularity constant, the solution u for the mixed problem

div (∇u− f) = 0 in Ω (62)

∇u ·n = g on Γlat and u = 0 on Γout ∪Γin (63)

is estimated as

‖∇u‖L3(Ω)3 ≤ �3(Ω){‖g‖L3(Γlat) +‖f‖L3(Ω)3

}. (64)

Hence, applying such a result, we have

‖∇ϑ‖L3(Ω)3 ≤ E00 + E01‖w‖L3(Ω), (65)

where

E00 =|Γlat |1/3 �3(Ω)Pe

λ0

(‖S ‖L∞(Γlat) + qo max

{1,‖S ‖L∞(Γlat)

q0

})

+‖λ‖∞λ0

(|Ω |1/3 + �3(Ω)|Γlat |1/3

), (66)

and

E01 =�3(Ω)λ0

Pe‖cp1‖∞ max


q0

}. (67)

Step 2. Now we turn to the continuity and momentum equations, determining theimpulse w = ρv and the pressure p.

Our system reads as follows:For given ϑ determine {w, p} satisfying

div w = 0 in Ω (68)

(w∇)wρ

=−Arϑe3−∇(p +2μ3Re

divwρ

)

+1

ReDiv

(2μ(ϑ)D(

wρ

))

in Ω (69)

w = 0 on Γlat , w = (1−α)v1e3 on Γin and w = v2e3 on Γout , (70)

Next, there exists

ζ ∈ H2(Ω)3,∇ζ ∈ L3(Ω)9, with curl ζ satisfying (70) (71)


Now we adapt the well-known Hopf construction to our non-standard nonlinearities.we have

Proposition 3.5. (see [9]) Let us suppose that 1/ρ ∈ L∞ (Ω), 1/ρ ≤ 1/ρmin. Then,for every γ > 0 there is a ξ , depending on γ and ρmin, such that

ξ ∈ H1(Ω)3, div ξ = 0 in Ω , ξ = ζ on ∂Ω and (72)

|∫

Ω

1ρ

(φ∇)φ ·ξ dx| ≤ γ‖φ‖2H1(Ω)3 , ∀φ ∈ H1

0 (Ω)3 (73)

Let us introduce now some useful constants:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

γ =2μmin

ρmaxRe, A00 = 4

√6

ρ2min

, B00 =μmin

ρmax, C01 = 2 ·61/6|Ω |5/6

B01 = 2 ·61/3 1

ρ2min

‖ξ‖L6(Ω)3 , B02 =4μmax61/6

ρ3min

C00 =1ρmin

(‖ξ‖2L6(Ω)3 +

2μmax

Re‖D(ξ )‖L2(Ω)9

).

(74)

Our basic result is the following:

Theorem 3.6. (see [9]) Let us suppose that

ΔB =1

Re

(B00−αB02‖∇ϑ‖L3(Ω)3

)−αB01‖∇ϑ‖L2(Ω)3 > 0 (75)

and Δdet = Δ2B−4A00α‖∇ϑ‖L2(Ω)3(C00 + ArC01‖∇ϑ‖L2(Ω)3) > 0, (76)

Then there is a solution w ∈ H1(Ω)3 for the problem (68)–(70), satisfying theestimate

‖D(w)‖L2(Ω)9 ≤ ‖D(ξ )‖L2(Ω)9 +C00 + ArC01‖∇ϑ‖L2(Ω)3√

Δdet(77)

Step 3. Next we define our iterative procedure:

Let γ =μmin

ρmaxReand ξ be the corresponding vector valued function from Hopf’s

construction.For a given wm = Wm +ξ , such that Wm ∈ BR = {z ∈H1

0 (Ω)3 : div z = 0 in Ωand ‖D(Wm)‖L2(Ω)9 ≤ R}, we calculate ϑm, a solution to (56)–(58).

Next, with this ϑm, we determine a solution wm+1 = Wm+1 + ξ for the problem(68)–(70), satisfying the estimate (77).

The natural question arising in the iterative process is does Wm+1 remains inBR ? Of course R has to be selected in a suitable way.


We have the following result

Proposition 3.7. (see [9]) Let the constants A0 and B0 be given as inProposition 3.1 and let E00 and E01 be given by (66)–(67). Let the constantsB00,B01,B02,A00,C00 and C01 be given by formula (74). Let ξ be generalized Hopf’slift, given by Proposition 3.5 and corresponding to γ . Let R be given by

R =√

2Reρmax

μmin

(2Ar|Ω |5/661/6 max


q0

}

+1ρmin

(‖ξ‖2

L6(Ω)3 +2μmax

Re‖D(ξ )‖L2(Ω)9

)). (78)

Then for all α > 0 such that

Δ1 =1

Re

(B00−αB02

(E00 + E01(‖ξ‖L3(Ω)3 + 481/12|Ω |1/6R)

))

−αB01

√Peλ0

(√2A0 +

√2B0

(‖ξ‖1/2L1(Ω)3 + |Ω |1/4

√HR1/2)) > 0 and (79)

Δ2 = Δ21 −4A00α

√2Peλ0

[√A0 +

√B0

(‖ξ ‖1/2L1(Ω)3 + |Ω |1/4

√R)] ·

[C00 + ArC01 max{1,

‖S ‖L∞(Γlat)

q0}]

>B2

00

2Re2 , (80)

Wm ∈ BR implies Wm+1 ∈ BR.

After this invariance result, a simple compactness argument gives the existenceof at least one solution.

The result could be summarize in the following theorem:

Theorem 3.8. (see [9]) There is a weak solution {ϑ ,v} ∈W 1,3(Ω)×H1(Ω)3 forProblem A, such that

⎧⎪⎨⎪⎩‖∇ϑ‖L2(Ω)3 ≤

√2Peλ0

(√A0 + B0‖ξ‖L1(Ω) +

√B0|Ω |1/2R

),

−‖S ‖L∞(Γlat )q0

≤ ϑ ≤ 1, and ‖D(v− ξ)‖L2(Ω)9 ≤ R,

(81)

where ξ is given by Proposition 3.5 with γ = 2μmin/(ρmaxRe).

3.1.2 Regularity and Uniqueness

Regularity of solutions and the uniqueness are also studied in [9]. Getting unique-ness is quite technical. After proving the regularity, one obtains that for small data


there is a unique weak solution {v,ϑ} ∈ H1(Ω)3 × (W 1,3(Ω) ∩C(Ω)) for theproblem (44)–(48) , satisfying the bounds (81). Detailed calculations are in [9].

Let us say a bit more concerning the regularity of solutions:

Lemma 3.9. Let cp0 and λ ∈C∞(R) . Furthermore let Γlat ∈C∞ and S ∈C∞(Γlat).Then ϑ ∈W 2,6(Ω)⊆C1,1/2(Ω).

Lemma 3.10. Let ϑ ∈W 2,6(Ω), let Γlat ∈C∞ and let v j ∈C∞, j = 1,2 satisfy (49).Then {w, p} ∈W 2,q(Ω)×W1,q(Ω), ∀q < +∞.

Theorem 3.11. Let the assumptions on the data from two previous Lemmata holdtrue. Then every weak solution {v, p,ϑ} ∈ H1(Ω)3× L2(Ω)× (H1(Ω)∩L∞(Ω))for the Problem A is an element of W 2,q(Ω)3×W 1,q(Ω)×W 2,q(Ω) , ∀q < ∞. Fur-thermore {v, p,ϑ} ∈C∞(Ω)5.

3.2 Oberbeck–Boussinesq Model

Now we are in position to pass to the limit when the expansivity parameter α tendsto zero.

First we remark that the a priori estimates from the previous section are inde-pendent of α , |α| ≤ α0, where α0 is the maximal positive α satisfying (79)–(80).Consequently, by a simple weak compactness argument, we have

Theorem 3.12. (see [9]). Let {ϑ(α),v(α)}, α ∈ (0,α0), be a sequence of weaksolutions to Problem A, satisfying the bounds (81). Then there exists {ϑOB,vOB} ∈W 1,3(Ω)×H1(Ω)3 and a subsequence {ϑ(αk),v(αk)} such that

⎧⎨⎩ϑ(αk)→ ϑOB, uniformly on Ωϑ(αk) ⇀ ϑOB, weakly in W 1,3(Ω)v(α) ⇀ vOB, weakly in H1(Ω)3.

Furthermore, {ϑOB,vOB} is a weak solution for the equations

div vOB = 0, (82)

(vOB∇)vOB = ArϑOBe3−∇POB +1

Rediv

{2μ(ϑOB)D(vOB)

}, (83)

cp1(ϑOB)vOB∇ϑOB =1

Pediv

(λ (ϑOB)∇ϑOB), (84)

satisfying the boundary conditions (47)–(50) and the bounds (81).

For the mathematical theory of the system (82)–(84), subject to various bound-ary conditions, but with constant viscosity, we refer to [27], pages 129–137. Theexistence and uniqueness for the complete non–stationary problem (82)–(84) is inthe article [8].


3.2.1 Discussion of the Formal Limit

For the glass molt in the die typical values of the thermal expansivity number Kρare of order 0.5 ·10−2, typical temperature oscillation is 10−1. The buoyancy forces

(mixed forced and natural convection) are proportional to1

Fr2 = gH/V 2R and their

typical order of magnitude is 104. Therefore it is important to consider a flow regimesuch that Ar = O (1), Re = O (1) and Pe = O (1). So we have α = Kρ(1−Tw)→ 0with the Archimedes number remaining constant. We have ρ(ϑ)→ 1, as α → 0and the continuity equation (44) becomes the incompressibility condition (82). In(45), the density ρ(ϑ) becomes 1 and we get the (83). Note that extracting the staticpressure and working with the hydraulic head was crucial for correct passing to thelimit. Also, in the energy equation (46) the density ρ(ϑ) becomes equal to 1 and weget the (84). This limit was justified rigorously in Theorem 3.12.

Therefore, in practical situations of industrial importance the system (44), (45)and (46) can be approximated by a system similar to the Oberbeck–Boussinesq sys-tem. Indeed our choice of the parameters takes us close to the conditions of theformal derivation of the Oberbeck–Boussinesq system for bounded domain. Suchconditions (see [25], pages 86–91) are

VR ≈ 0, Ma ≈ 0, Re = O(1) and Ar = O(1), (85)

with Ma being Mach’s number, which in our situation is of order 10−6. Underconditions (85), the non–isothermal compressible Navier–Stokes system can be ap-proximated (formally) by the Oberbeck–Boussinesq system (82)–(84).

We note that the same reasoning applies to the situations in which other boundaryconditions are given, just modifying the definition of ϑ and of Archimedes’ numberAr. For details we refer again to [25], page 89.

In the case of natural convection flows, differently from gravity driven flows,there is no characteristic velocity and one takes VR = μR/(HρR). In this case therelevant characteristic number is the Grashof number, Gr= Ar Re2. Instead of im-posing Ar = O (1), one requires Gr = O (1) and the condition VR ≈ 0 is replacedby μR/ρR ≈ 0.

Another approximation for the natural convection in the case of a layer of fluidof thickness H, with the top and bottom surfaces held at constant temperature, is de-rived in [22]. The authors take the Chandrasekhar’s velocity VR =

√gLβ (Tw−TR)

(linked to buoyancy) as characteristic. Then they introduced the small parameter

ε =(

V 3RρR

gμR

)1/3

, obtaining formally the Oberbeck–Boussinesq equations at order

O(ε3). This approach was developed further in [16] in order to get a modifiedOberbeck–Boussinesq system, with important viscous heating.

A similar derivation of the Oberbeck–Boussinesq approximation for the naturalconvection in the case of a layer of fluid, can be found in [11], pages 43–53. There,for the sake of definiteness, a perfect gas is considered but the scalings and theorders of magnitude involved are similar to [22].


Next we mention a number of papers by R. Kh. Zeytounian on the thermocap-illary problems with or without a free boundary. They are reviewed in [29] and[30]. He considered the natural convection in an infinite horizontal layer of viscous,thermally conducting and weakly expansible fluid, heated from below. The smallparameter is α and for

α ≈ 0, Fr≈ 0, Ma ≈ 0, and Gr =αRe2

Fr2 = O(1), (86)

he got the Oberbeck–Boussinesq approximation. In his derivation, the viscous dis-sipation term is absent if

1 mm ≈(μ2

R

gρ2R

)1/3

<< H ≈C0/(g(Tw−TR)) . (87)

The formal analysis of Zeytounian carries over to the free boundary cases.Rigorous justification of the limit when the Mach number tends to zero

was the subject of intense research in recent years. We mention in this di-rection papers [4, 6, 15, 19] and references therein. In the recent paper [10] theOberbeck–Boussinesq system is obtained in the low Mach number limit.

Therefore we established a new singular limit for the isochoric Navier–Stokes-Fourier system. Nevertheless, the expansivity parameter α is small but not zero(mostly of order 0.5 · 10−3. It makes sense to take the approximation with the nextorder correction. In the subsection which follows we undertake the rigorous deriva-tion of the next order correction.

3.2.2 Correction to the Oberbeck–Boussinesq Model

In this section we reconsider the limit when the expansivity parameter α tends tozero. Having justified the Oberbeck–Boussinesq system as the limit equations whenthe expansivity parameterα tends to zero, the next question is: What is the accuracyof the approximation ?

The answer relies on the uniqueness and regularity results from the previoussections.

As usual for the expansions, we need the equations for the derivatives with re-spect to α . For simplicity we suppose that

v2 is independent of α and v1 = V1/(1−α) with V1 independent of α. (88)

Then we have

Proposition 3.13. (see [9]). Let us suppose that data fulfill smallness conditions,from [9], ensuring existence, uniqueness and regularity of a solution lying inside


the ball defined by the bounds (81). Furthermore let the solution {v, p,ϑ} satisfythe inequalities

N =λ0

Pe− C6(Ω)

2(1 +

H√2)(‖cp1‖L∞(R)‖v‖L3(Ω)3ρmax

+‖λ ′‖L∞(R)

Pe‖∇ϑ‖L3(Ω)3

)> 0 (89)

2Re

μmin

ρmax> 4(

32)1/4√

H‖D(v)‖L2(Ω)9 + HLΘ

{2αRe

μmax

ρmin‖v‖L∞(Ω)3

+ArH2√

2+

2Re‖μ ′‖L∞(R)‖D(v)‖L3(Ω)9C6(Ω)+αH‖v‖2

L∞(Ω)3

}, (90)

where

LΘ =H‖cp1‖L∞(R)‖ϑ‖L∞(Ω)√

2N(91)

Then derivatives of the solution, with respect to α , exist at all orders as continuousfunctions of α .

With this result, we are ready to state the error estimate for Boussinesq’s limit.First, we write the 1st order correction, i.e. the system defining the first deriva-

tives {w0, π0,θ 0}=d

dα{v, p,ϑ}|α=0:

div{

w0−ϑOBvOB} = 0 in Ω (92)

−ϑOB(vOB∇)vOB +{(w0∇)vOB +(vOB∇)w0} =−Ar θ 0e3−∇π0

+2

ReDiv

{μ(ϑOB)D(w0)+ μ ′(ϑOB)θ 0D(vOB)

}in Ω (93)

div

{− λ (ϑOB)

Pe∇θ 0 +(vOBcp(ϑOB)− λ ′(ϑOB)

Pe∇ϑOB)θ 0

−ϑOBvOBCp(ϑOB)+ w0Cp(ϑOB)}

= 0 in Ω (94)

θ 0 = 0,w0 = 0 on Γout ; θ 0 = 0,w0 = V2e3 on Γin (95)

w0 = 0 and − 1Pe

(λ (ϑOB)∇θ 0 +λ ′(ϑOB)θ 0∇ϑOB) ·n = q0θ 0 on Γlat , (96)

Under the conditions of the preceding Proposition, with α = 0, the system (92)–(96) has a unique smooth solution. Hence we have established rigorously the O(α2)approximation for Problem A. Clearly, one could continue to any order.

The result is given by the following theorem, which is a straightforward corollaryof Proposition 3.13.


Theorem 3.14. (see [9]). Let us suppose the assumptions of Proposition 3.13. Thenwe have

‖v−vOB−αw0‖W1,∞(Ω)3 +‖ϑ −ϑOB−αθ 0‖W 1,∞(Ω) ≤Cα2 (97)

infC∈R

‖p− pOB−α p0 +C ‖L∞(Ω)3 ≤Cα2 (98)

where p0 = π0−2μ(ϑOB) div w0/(3 Re).

4 Modelling the Viscous Jet at the Exit of the Die

In this section we discuss briefly the stage (c) (recall Sect. 1), that is the free bound-ary flow in the air. Our goal is to derive a mathematical model, making somesimplifying assumptions, in order to discuss the influence of some physical param-eters involved. In this stage one meets several difficulties. The first is that we dealwith the non-isothermal flow of a thermally expansible fluid. Next, when the fluidleaves the die it experiences strong cooling. It is observed that, in a suitable rangeof temperature and of pulling velocity, defining the so called “cold breakdown”, thejet detaches from the solid wall inside the die. Of course, phenomena of wetting ofthe outer surface of the die are also observed if temperature is large enough, makingthe molten glass less viscous, and/or if the pulling velocity is sufficiently small.

Our approach could describe the flow in stage (c) for a general geometry of thedie. Nevertheless, since the main difficulties are not really linked with the form ofthe die and in order to simplify our exposition, we consider cylindrical geometrywith azimuthal symmetry. In particular, x will denote the axial coordinate, r theradial–one and the symmetry axis coincides with the x axis.

The dependence of the fluid viscosity μ , on the temperature T plays a major rolein the process. Therefore, in order to simplify modelling at this stage we neglectvariations of the fluid density ρ and of the surface tension σ . Hence, within theframework of incompressibility, the fluid constitutive model becomes

T =−pI+ 2μD,

and the continuity equation, in the stationary case, writes

∂ (v1)∂x

+1r∂ (r v2)∂ r

= 0, (99)

where v1 is the fluid axial velocity and v2 the radial–one, that is v = v1 (x,r, t)ex +v2 (x,r, t)er.

We consider a die which is a cylinder of radius R and high L1. After entering thedie the molten glass wets only a part of the die walls and then produces a viscous jet.We confine our analysis to the case in which the jet starts at some point xT ∈ (0,L1)and then we have a free boundary S , r = h(x,t), for xT < x < L, separating the fluid


h(x,t)

xT

0

R

x

r

L

L1

Γin

Γwall

Γout RL

Fig. 3 A schematic of stage (c). The x-axis coincides with the cylinder symmetry line. ex, er

denote the unit vectors, x = xex+rer . The velocity field has the form v(x, t) = v1(x, t)ex +v2(x, t)er .The gravity acceleration is parallel to the x axis, g = gex

from the surrounding air (see Fig. 3). The fiber is cooled down by a stream of air,whose interaction with the fibers is also important, but we will not deal with thisaspect here.

Referring to Fig. 3, the flow domain is the set

Ω = Ω die∪Ω jet , with

Ω die = {0 < x < xT , 0 < r < R} ,Ω jet = {xT < x < L, 0 < r < h(x, t)} ,

where, as mentioned, r = h(x,t) is the unknown boundary of the jet. The boundaryof Ω contains four distinct parts

• Inlet boundary Γin = {x = 0, 0≤ r ≤ R}.• Wall boundary Γwall = {0 < x≤ xT , r = R}.• Free boundary S = {r = h(x,t) , x ∈ (xT ,L)}.• Outlet boundary Γout = {x = L, 0≤ r ≤ RL < R}, where RL = h(L), fiber radius

at x = L.

We observe that the whole process is governed by the following easily accessiblequantities (besides the temperature of the cooling stream of air):

• Spool radius, Rsp.• Spool angular velocity, ω .• Power dissipated by the spinning device, W. Indeed, for estimating W it is enough

to measure the intensity of the electric current I entering the electric engine and


the electric potential differenceΔV at the poles of the engine, so that W = IΔV−Wo, where Wo is the power consumption of the engine.

• Discharge, Qout , i.e. rate of increase of the volume of collected fiber.

From these directly measured quantities, one can deduce:

• Spinning speed, Vsp = ωRsp, i.e. fibers velocity entering the spinner.

• Fiber radius, R f = (Qout/πVsp)1/2.

• Torque applied to the spinner, M = W/ω .• Traction force applied to the fiber, Fsp = M/Rsp = W/Vsp.• Mean stress applied to the fiber surface by the spinning device,Φsp = Fsp /πR2

f =W/Qout .

We denote by VL the mean fluid velocity on Γout

VL =2

R2L

∫ RL

0v(L,r) · exr dr. (100)

Now, since the discharge Qout is prescribed, we have

πVL R2L = Qout , ⇒ RL =

√Qout

πVL. (101)

which links together VL and RL.From the analysis of the simple Matovich–Pearson model for stage (d ), that will

be performed in Appendix 2, it will be clear that, once L has been fixed, quantitiessuch VL (the mean fluid velocity on Γout) and the mean traction stress on Γout

ΦL =2

R2L

∫ RL

0T(L,r) ex · ex rdr =

2

R2L

∫ RL

0(−p(L,r)+ 2μ(L)Dex · ex) rdr, (102)

as well as the fiber radius RL can be explicitly related with the terminal quantitiesVsp, Rsp and Qout .

We note that the outlet boundary x = L is an artificial mathematical boundary,because the jet continues beyond that point. In this section, we confine our attentionto the portion of jet in the interval xT < x < L, defining L as a location at which thefiber is cooled down to a point such that the fluid viscosity is large enough to preventany further important fiber radius variations. On the other hand, the temperatureshould still be larger than the glass transition temperature. The value of L is anunknown of the problem and in fact after the point x = L one is likely to use theaveraged equations presented in the next section. Actually, giving a definition of Lis a delicate point of the model and depends on how μ varies with the temperature.

Although there are authors [13] who, on the basis of empirical observations, sug-gest to take L = 12R, here we proceed in a rather different way, analyzing carefullythe jet profile at the end of stage (c). Such an issue will be discussed in Sect. 4.1.


Concerning the boundary conditions to be imposed on the jet boundary S (seeFig. 3), we first assume a purely kinematic condition: the fluid velocity is tangent tothe free surface, namely

v ·n = wS ·n = 0. (103)

where wS is the free boundary velocity. Next, we impose a dynamic condition:equilibrium of the forces acting on S , that is

{−p + 2μ (Dn) ·n =−σκ ,

2μ (Dn) · τ = (τ ·∇)σ ,(104)

where:

• 2κ =1

h(

1 +(h′)2)1/2

− h′′(1 +(h′)2

)3/2.

• The air pressure, p∞ (assumed to be constant), has been rescaled to 0. In otherwords, in place of p we consider p− p∞.

Concerning the boundary conditions on Γin and Γwall , we have:

• On Γin, we prescribe pressure, temperature and no radial velocity

{p(0,r) = Pin, 0≤ r ≤ R,

v2 (0,r) = 0, 0≤ r ≤ R.

• On Γwall , we assume no–slip v(x,R) = 0, 0 < x < xT . Of course, some slippingoccurs near to xT , but it is generally admitted that the slipping length �s is ex-tremely small.

The abscissa of the triple point, xT , is not a priori given and has to be determinedas a part of problem (as well as RL). In most of the references, one imposes thecontact angle (e.g. equal to 0) and uses this additional condition for having a totallydetermined problem.

The boundary conditions that have to be specified on Γout required coupling withthe stage (d). This will be the subject of next section.

In this section we will suppose that the temperature is a given function of x and r.Actually, the main difficulties will remain, but our calculations will be less lengthy.

4.1 Definition of L and Jet’s Profile at the End of Stage (c)

In this section, still considering stationary conditions, we will focus on these issues:

1. To define local boundary conditions on Γout to be used when studying stage (c).2. To formulate a rational procedure allowing to determine L, replacing the heuristic

rule L = 12R.


Table 1 Typical order of magnitude of themain physical quantities at x = L. Notice thatthe traction force Fsp corresponds to a weightof 3gr !

VL RL ΦL μ σ10 m/s 10−5 m 108 Pa 107 Pa s 10−1 N/m

We point out that, for our purposes, L can be chosen in some range of values. There-fore we look for some criterion which has a physical motivation, but is based just onthe analysis of magnitude orders. So for the purposes we have in mind, i.e. locatingL within a reasonable approximation, we may confound VL and ΦL with their termi-nal values Vsp and Φsp, neglecting the small variations that will occur from x = L tothe collecting spool. Therefore, in place of expressions (188), we set

VL = Vsp and ΦL =Φsp. (105)

A full justification of this approach will be provided in Appendix 2. Indeed we aremainly interested to the orders of magnitude, listed in Table 1, below:

Remark 4.1. After many studies (see e.g. [23] and references therein) and experi-mental observations (see e.g. [13] or [21]) we know that even for x > L fiber couldexhibit a complicated behavior, oscillations and instabilities are possible. Here wesuppose a stabilized behavior when x > L and the asymptotic analysis, which fol-lows in this subsection applies to such situation. In general, one should coupletogether the stages (c) and (d), through an iterative procedure. Here we do not enterthis issue.

For the determination of a physically acceptable value of L we proceed consideringa characterization of L in terms of the dimensionless quantity h′. Hence we definethe endpoint of Ω , i.e. of stage (c), in the following way:

Definition 4.2. Given a sufficiently small number ς , typically ς = O(10−6

), L is

chosen so that ∣∣h′ (L)∣∣ = ς . (106)

We will also make use of the following relationships, that will be checkeda–posteriori

∣∣h′′ (L)∣∣ � ς2

RL⇔ ∣∣h′′ (L)

∣∣RL � ς2. (107)

Our strategy is the following:

(i) To define local boundary conditions such that (100) and (102) are fulfilledwithin a tolerance of order ς .

(ii) To determine L according to Definition 4.2.

Next, we define pL = p(L,RL). Of course, pL is unknown at this stage. In order todetermine pL and L let us write explicitly (103) and (104) on x = L, r = RL


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

v1 (L,RL)h′ (L)− v2 (L,RL) = 0,

pL +2μ (L)

1 +(h′)2

[h′(∂v1

∂ r+∂v2

∂x

)− ∂v2

∂ r− ∂v1

∂x(h′)2

]=

σ

2RL

(1 +(h′)2

)1/2

− σh′′

2(

1 +(h′)2)3/2

,

(∂v1

∂ r+∂v2

∂x

)(1− (h′)2

)−2

(∂v1

∂x− ∂v2

∂ r

)h′ =

(τ ·∇)σμ (L)

(1 +(h′)2

).

(108)

Focusing on equations (108)2 and (108)3 we neglect O(ς2

)terms (recall Definition

4.2) and7 (τ ·∇)σ/μ . Hence, (108)2 and (108)3 can be rewritten as

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

pL + 2μ (L)[

h′(∂v1

∂ r+∂v2

∂x

)− ∂v2

∂ r

]=

σ2RL

,

(∂v1

∂ r+∂v2

∂x

)−2

(∂v1

∂x− ∂v2

∂ r

)h′ = 0.

(109)

Next, (109) can be further simplified. Indeed, neglecting once more O(ς2

)terms,

we have

pL−2μ (L)∂v2 (L,RL)

∂ r=

σ2RL

. (110)

We now assume:

A.1 The horizontal fluid velocity on x = L, r = R is VL, namely

v1 (L,RL) = VL, (111)

Hence, going back to (108)1, we obtain

v2 (L,RL) =− ∣∣h′ (L)∣∣VL. (112)

We also recall that by symmetry v2 (L,0) = 0. Thus, following the results of Sect. 5,we also assume:

A.2 The profile of the radial velocity on Γout is

v2 (L,r) =−|h′ (L)|VL

RLr =−ςVL

RLr, (113)

7 If σ is just a function of temperature, ∇σ =dσdT

∇T and x = L is located in a region where the

temperature variations are very small, i.e. σ ′ (T (L))� 1.


yielding∂v2

∂ r=−|h

′ (L)|VL

RL.

Hence, from (110)

pL =−2μ (L)|h′ (L)|VL

RL+

σ2RL

, (114)

or

pL =−2μ (L)|h′ (L)|VL

RL=−2μ (L)

ςVL

RL, (115)

since the second term on the r.h.s. of (114) is negligible with respect to the first–one.

Indeed σ2RL

/μ(L)ςVL

RL≈ 10−3, according to Table 1.

Of course, we still have to determine L. At this point, consistently with the ap-proximations already introduced, we take

p(L,r) = pL, 0≤ r ≤ RL.

Then we exploit (102) and (99), obtaining

ΦL =−pL +4μ (L)

R2L

∫ RL

0

∂v1

∂x︸︷︷︸− 1

r∂(rv2)∂ r

rdr =−pL− 4μ (L)RL

v2 (L,RL) ,

which, recalling (113) and (115), gives8

ΦL = 6μ (L)ςVL

RL, or ΦL = 6μ (L)

ςVL

π R3L

Qout . (116)

This formula allows us to identify L according to the profile of viscosity, which inturns is known (with a reasonable approximation) in terms of the profile of temper-ature. Thus L is defined via

μ (L) =ΦLRL

6ςVL. (117)

According to Table 1, the r.h.s. of (117) is∼107 Pa s, consistently with the expectedvalue of μ (L). Hence we conclude that L is the distance at which the glass hascooled down to the temperature corresponding to μ ≈ 107 Pas.

We can proceed further obtaining the information on pL and the expected profileof the fiber in the vicinity of x = L. Concerning pL, from (116) and (115) we have

pL =−13ΦL,

8 It is easy to verify that ς ∼ 10−6 is compatible with the order of magnitude of the quantitiesentering in formula (116).


so that the viscous stress on Γout is

4μ (L)R2

L

∫ RL

0

∂v1

∂xrdr =

23ΦL,

Next, we write also v2 (L,r) in terms of ΦL using (113) and (116), namely

v2 (L,r) =− ΦL

6μ (L)r, (118)

Summarizing, the boundary conditions on Γout read as follows

⎧⎪⎨⎪⎩

p =−13ΦL,

v2(r) =− ΦL

6μ (L)r,

on Γout , (119)

with ΦL expressed in terms of Φsp.Going back to (113), we just suppose that it is valid for x from a neighborhood

of x = L:

v2 (x,r)≈−|h′ (x)|

h(x)VLr, 0≤ r ≤ h(x) ,

|L− x|L� 1. (120)

Consistently with this assumption we have that, in the vicinity of x = L, v2 (x,r) canbe also expressed exploiting (118), namely

v2 (x,r)≈− ΦL

6μ (x)r, 0≤ r ≤ h(x) ,

|L− x|L� 1. (121)

In this way, comparing (120) and (121), the position of the free boundary r = h(x)is given by

ΦL

6μ (x)=−h′ (x)

h(x)VL,

so that the following Cauchy problem is obtained

⎧⎪⎪⎨⎪⎪⎩

h′ (x) =− 16μ (x)

ΦL

VLh(x) ,

|L− x|L� 1,

h(L) = RL,

(122)

which provides a sufficiently accurate information on the local profile of the fiber.Problem (122) can be integrated, getting the following profile

h(x) = RL exp

{ΦL

6VL

∫ L

x

1μ (s)

ds

}, (123)

referred to as “transition profile”, in the sequel denoted by hTP (x).


We can now check that hT P fulfills condition (107), i.e. if our result is consistentwith assumption (107). Evaluating h′′ we have

h′′

h=

(ΦL

6μVL

)2(1 +

6VL

ΦLμ ′)

.

Since, close to x = L we have6VL

ΦL|μ ′| � 1, that yields

h′′

h=

(ΦL

6μVL

)2

=(

h′

h

)2

, ⇒ h′′ h =(h′)2 = O

(ς2) . (124)

Hypothesis (107) is thus fulfilled.As final check of our procedure we show that, considering the fiber profile

(123), the assumption (111), the approximation (121) for v2 and boundary condi-tions (119), we can evaluate v1 (L,r), 0 ≤ r ≤ RL, so that the estimated volumetricdischarge differs from the actual one, i.e. Qout , within a tolerance O

(ζ 2

). We thus

use the above mentioned approximations in

⎧⎪⎪⎨⎪⎪⎩

v1 (x,h(x)) h′ (x)− v2 (x,h(x)) = 0,

(∂v1

∂ r+∂v2

∂x

)−2

(∂v1

∂x− ∂v2

∂ r

)h′ =

(τ ·∇)σμ (x)

,

(125)

and look for an estimate of ∂v1/∂ r, since we approximate v1 (L,r) in the follow-ing way

v1 (L,r) = VL− ∂v1

∂ r

∣∣∣∣x=L,r=RL

(RL− r)+O((RL− r)2).

Evaluating (125)2 at x = L, r = RL, we have

[∂v1

∂ r+∂v2

∂x

]

x=L,r=RL

=− ΦL

μ (L)

∣∣h′ (L)∣∣ =−6

h′ (L)2 VL

RL, (126)

where we have used (99) and (118) for expressing ∂v1/∂x, namely

∂v1

∂x=−∂v2

∂ r− v2

r=

13ΦL

μ (L)=−2

|h′|VL

RL,

and where (τ ·∇)σ/μ has been neglected.Differentiating (125)1 w.r.t. x we get

∂v1

∂xh′+

∂v1

∂ rh′2 + v1h′′ − ∂v2

∂x− ∂v2

∂ rh′ = 0.


We now use again (99) and then exploit (118) and (111), obtaining

[∂v1

∂ rh′ (L)2− ∂v2

∂x

]

x=L,r=RL

= 3h′ (L)2 VL

RL−VLh′′ (L) .

Next, using (124) for evaluating h′′ (L), i.e. h′′ (L) = h′ (L)2 /RL and (126) for∂v2/∂x, we have9

(1 + h′ (L)2

) ∂v1

∂ r

∣∣∣∣x=L,r=RL

=−4h′ (L)2 VL

RL.

We are now in position to estimate Δv1 = max |v1 (r,L)−VL|≈∣∣∣∣∂v1

∂ r(L, RL)

∣∣∣∣RL,

namelyΔv1 = O

(ς2VL

),

thus estimating the “relative error” on the outlet discharge

ΔQout

Qout=

1

πR2LVL

∣∣∣∣πR2LVL−2π

∫ RL

0v1 (L,r) rdr

∣∣∣∣ = O(ς2) ,

which is thus shown to respect the desired tolerance.

5 Terminal Phase of the Fiber Drawing

It is considered that the stage (d) starts when the fiber is sufficiently “cooled down.”This means that the viscosity is sufficiently large (larger than 105 Pa s) and the

fiber radius is already rather small (smaller than hundred micrometers). Contrary tothe stage (c), where one should treat the 3D Navier–Stokes system, with the freeboundary, coupled with the nonlinear conduction of heat, here we have a long (sev-eral meters) and tiny filament (radii of hundred micrometers) of molten glass. In thestage (c) one needs an industrial code to solve the corresponding partial differentialequations. Here it is possible to employ the asymptotic analysis and come out witha simplified effective model (Fig. 4).

Fundamental equations, describing the stage (c), are the temperature dependentincompressible Navier–Stokes equations with free boundary coupled with the en-ergy equations. Their derivation from the first principles is in the article [9], wherealso the Oberbeck–Boussinesq approximation is justified rigorously, by passing tothe singular limit when expansivity parameter goes to zero. For more details seeSects. 2 and 3.2.

9 Notice that ∂v1/∂ r is negative, as one would have expected.


Fig. 4 A simulated glass fiber being drawn down

In the engineering literature on the fiber drawing, this system is frequently ap-proximated by a quasi 1D approximation for viscous flows, in which the radiusof the free boundary r = R(z,α,t), axial speed w and the temperature ϑ are in-dependent of the radial variable and depend only on the axial coordinate z andof the time t. This approximation, summarized in the so–called the “equations ofMatovich–Pearson” was introduced in the papers by Kase and Matsuo [17, 18] andMatovich and Pearson [20]. The model was obtained heuristically and reads

∂A∂ t

+∂ (wA)∂ z

= 0;∂∂ z

(3μ(T )A

∂w∂ z

)+∂ (σ(T )

√A)

∂ z= 0, (127)

where A = A(z, t) is the area of fibre section, w = w(z, t) is effective axial veloc-ity, 3μ is Trouton’s viscosity and σ is the surface tension. μ and σ depend on thetemperature, and it is necessary to add the equation for the temperature T = T (z, t).

The equations of Matovich and Pearson were obtained under the assump-tions (H), listed below:

(a) The viscous forces dominate inertia(b) Effects of the surface tension are in balance with the normal stress at the free

boundary(c) The heat conduction is small compared with the heat convection in the fiber(d) All the phenomena are axially symmetrical and the fiber is nearly straight


5.1 Derivation of the Model of Matovich–Pearsonfor the Thermal Case

The derivation presented in [13] and in [14] surprisingly neglects the fact that theviscosity changes over several magnitude orders performing asymptotic expansionsin ε as it was of order one, ε being the ratio of the characteristic thickness RE inthe radial direction with the characteristic axial length of fibre L. The correct formalderivation is in [5] and it confirms the model announced in [14]. We will presenthere the ideas and results. Since we deal with an axially symmetric and long fiber,we slightly change the notation and denote with subscript z the axial component andwith the subscript r the radial one. Instead of x we use z to denote the axial variable.

Our unknowns are the following:

• Velocity v = vzez + vrer

• Hydrodynamic pressure p• Temperature T• Fiber radius (being the distance from the symmetry axis ) R = R(t,z)

Moreover the following physical quantities are given function of T :

• Specific heat cp = cp(T )• Density ρ = ρ(T )• Surface tension σ = σ(T )• Viscosity μ = μ(T )• Thermal conductivity λ = λ (T )

We will suppose that the fibre is long and thin, so that the principal flow is directedalong the axis z and the velocity is basically one dimensional. In particular, theproperty of “small thickness” leads to the “lubrication approximation” of Reynolds.The strategy of the “lubrication approximation” is to expand the velocity field withrespect to ε . The zero order terms should be sufficient to describe the motion.

Effective equations are then derived starting from the compatibility conditions forthe solvability of the problems. This idea is traditionally applied to flows throughthin domains. Treating the flows with a free boundary is much more complicated.

Initially, the radius r describes the position of the free boundary r = R(x, t). Thesmallness of the expansion parameter ε = maxx,t R(x, t)/{ fiber length }will dependon the solution itself.

In the second place, it is not obvious which forces in the equations must be takeninto account (for example torsion can be important or negligible).

In order to correctly model the heat exchange at the free boundary, we adopt forthe heat transfer coefficient the empirical formula of Kase–Matsuo

h =λ∞

R(z,t)C

(2ρ∞vz(z,t)R(z, t)

μ∞

)m

, (128)

where the subscript ∞ denotes the parameters of the surrounding air and C > 0 andm, 0 < m < 0.4, are constants determined from the experimental data.


Concerning characteristic length, in [13] the authors take L =ρEREcpEvE

2hE(1−Tg/TE), where Tg is the glass transition temperature and TE is temperature at theorifice. Other possibility is to take the distance between the spooler and extrusiondie. This will be our choice.

We will restrict our considerations to the stationary case, even if the generaliza-tion to the non-stationary case is straightforward. Our first difficulty is that viscositychanges over several orders of magnitude. This motivates us to write the Oberbeck–Boussinesq equations in Ω = {z ∈ (0,L); 0≤ r < R(z)}, in the following form:

∇(

pμ(T )

+23

div v)−∇2v +

((p

μ(T )+

23

div v)

I−2D(v))

×∇ logμ(T )−∇ div v =ρ(T )μ(T )

gez− ρ(T )μ(T )

{(v∇)v +

∂v∂ t

}in Ω (129)

∂ρ(T )∂ t

+ div (ρ(T )v) = 0 in Ω (130)

ρ(T )cp(T )(∂T∂ t

+ v∇T

)= div (λ (T )∇T ) in Ω . (131)

The stress tensor is now

Σ =−μ(T )((

pμ(T )

+23

div v)

I−2D(v))

. (132)

The natural small parameter is ε =RE

L. We introduce the following dimensionless

quantities:

r = RE r, t =LvE

t, vz = vE vz, vr = εvE vr,T −T∞ΔT

= T ,

μ = μE μ,σ = σE σ , h = hEh,p

μ(T )+

23

div v =vE p

L, λ = λE λ .

Equations in non-dimensional form will contain the following dimensionlessnumbers:

Re =ρEvELμE

is Reynolds number; Ca =μEvE

σEis capillary number;

Pe =cpEρvEL

λEis Peclet’s number; Bi =

hERE

λEis Biot’s number;

Fr =vE√gL

is Froude’s number; Bo =Re

Fr2 is Bond’s number

and draw ratio is E =Vf

vE, where Vf is the outlet fiber velocity. α = logE is usually

known as Hencky strain.


In the text, which follows, we will omit the wiggles. The axially symmetricgeneralized Oberbeck–Boussinesq system takes the following form:

∂vr

∂ r+

vr

r+∂vz

∂ z=−vz∂z logρ(T )− vr∂r logρ(T )− ∂t logρ(T ), in Ω ; (133)

ε2Reρ(T )μ(T )

(∂tvr + vr

∂vr

∂ r+ vz

∂vr

∂ z

)=−∂ p

∂ r+(∂ 2vr

∂ r2 +1r∂vr

∂ r

− vr

r2 + ε2 ∂ 2vr

∂ z2

)− ∂ logμ

∂ r

(p−2

∂vr

∂ r

)+∂ logμ∂ z

(ε2 ∂vr

∂ z+∂vz

∂ r

)

− ∂∂ r

(vz∂z logρ(T )+ vr∂r logρ(T )+ ∂t logρ(T )), in Ω ; (134)

ε2Reρ(T )μ(T )

(∂tvz + vr

∂vz

∂ r+ vz

∂vz

∂ z

)=−ε2 ∂ p

∂ z+(∂ 2vz

∂ r2 +1r∂vz

∂ r+ ε2 ∂ 2vz

∂ z2

)

− ε2 ∂ logμ∂ z

(p−2

∂vz

∂ z

)+∂ logμ∂ r

(ε2 ∂vr

∂ z+∂vz

∂ r

)+ ε2 Re

Fr2

ρ(T )μ(T )

− ε2 ∂∂ z

(vz∂z logρ(T )+ vr∂r logρ(T )∂t logρ(T )), in Ω ; (135)

ε2Peρ(T )cp(T )(∂tT + vr

∂T∂ r

+ vz∂T∂ z

)=

1r∂∂ r

(rλ (T )

∂T∂ r

)

+ ε2 ∂∂ z

(λ (T )

∂T∂ z

), in Ω . (136)

Next we have

∂R(z, t)∂ t

+ vz∂R(z)∂ z

= vr on r = R(z) (the kinematic condition) (137)

and the dynamic conditions at the free boundary read:

μ(T )Ca

(2ε2 ∂R(z)

∂ z

(∂vr

∂ r− ∂vz

∂ z

)+

(1− ε2

(∂R∂ z

)2)

(ε2 ∂vr

∂ z+∂vz

∂ r)

)

= ε(∂R(z)∂ z

∂σ(T )∂ r

+∂σ(T )∂ z

)√1 + ε2

(∂R(z)∂ z

)2

on r = R(z); (138)

εCaμ(T )

((1 + ε2

(∂R(z)∂ z

)2)

p−2

(∂vr

∂ r+ ε2

(∂R(z)∂ z

)2 ∂vz

∂ z

− ε2 ∂R(z)∂ z

∂vr

∂ z− ∂R(z)

∂ z∂vz

∂ r

))=σ(T )R(z)

√1 + ε2

(∂R(z)∂ z

)2

− ε2σ(T )∂ 2R(z)∂ z2 /

√1 + ε2

(∂R(z)∂ z

)2

on r = R(z). (139)


Finally, the heat transfer to the environment is given by Newton’s cooling law

∂T∂ r− ε2 ∂R

∂ z∂T∂ z

=−Bih

λ (T )T

√1 + ε2

(∂R(z)∂ z

)2

on r = R(z); (140)

Before starting the two-scale expansion it is essential to relate with ε the magni-tude of all non–dimensional coefficients entering the system. The experimental datafrom [13] imply that

Re∼ ε, Bo∼ 1, Ca∼ 1ε, Pe∼ 1

ε, Bi∼ ε as ε → 0. (141)

In order to perform the asymptotic analysis of our equations, we expand all un-known functions with respect to ε , i.e. for an arbitrary function f = f (r,z) we setf = f (0) + ε2Pe f (1) + ε2 f (2) + . . . .

Let Ω0 = { 0 ≤ r < R(0)(z,t), 0 < z < 1 }. After inserting the expansions forunknowns into the system (133)–(140), we find out that the zero order terms in thesystem (133)–(136) are:

∂v(0)r

∂ r+

v(0)r

r+∂v(0)

z

∂ z=−v(0)

z ∂z logρ(T (0))− v(0)r ∂r logρ(T (0))

− ∂tρ(T (0)) in Ω0; (142)

∂∂ r

(∂tv

(0)z + v(0)

z ∂z logρ(T (0))+ v(0)r ∂r logρ(T (0))

)=−∂ p(0)

∂ r+∂ 2v(0)

r

∂ r2 +1r∂v(0)

r

∂ r

−v(0)r

r2 +∂ logμ(T (0))

∂ r

(2∂v(0)

r

∂ r− p

)+∂ logμ(T (0))

∂ z∂v(0)

z

∂ rin Ω0; (143)

0 =∂ 2v(0)

z

∂ r2 +1r∂v(0)

z

∂ r+∂ logμ(T (0))

∂ r∂v(0)

z

∂ rin Ω0; (144)

0 =1r∂T (0)

∂ r+∂ 2T (0)

∂ r2 in Ω0, (145)

with the boundary conditions

v(0)r = ∂tR

(0) + v(0)z∂R(0)

∂ zon r = R(0)(z); (146)

∂v(0)z

∂ r= 0 on r = R(0)(z, t); (147)

p(0) =1

εCaσ(T (0))

μ(T (0))R(0) + 2

(∂v(0)

r

∂ r− ∂zR

(0) ∂v(0)z

∂ r

)on r = R(0)(z, t); (148)

∂T (0)

∂ r= 0 on r = R(0)(z, t). (149)


We note that it is advantageous to have logμ , in the differential equations, since,contrary to μ , its variation is not dramatic. Using (145) we obtain T (0) = T (0)(z, t).Next, (144) yields v(0)

z = v(0)z (z,t).

Radial component of the velocity is calculated using the (142). We integrate itand get

∂v(0)z

∂ r= 0 =

∂T (0)

∂ r, v(0)

r (r,z) =− r2

(∂v(0)

z

∂ z+ ∂z logρ(T (0))v(0)

z + ∂t logρ(T (0))

)

(150)

Consequently, (143) reads∂ p(0)

∂ r= 0 and, after using the boundary value (148), we

obtain

p(0) =1

εCaσ(T (0))

μ(T (0))R(0) + 2∂v(0)

r

∂ r=

1εCa

σ(T (0))μ(T (0))R(0)

− ∂v(0)z

∂ z− ∂z logρ(T (0))v(0)

z − ∂t logρ(T (0))). (151)

The kinematic condition (146) now transforms to

0 =∂∂ t

(ρ(T (0))(R(0))2)+

∂∂ z

(ρ(T (0))v(0)

z (R(0))2). (152)

At the order O(ε2 Pe ), we have the following boundary value problem for thetemperature correction T (1):

ρ(T (0))cp(T (0))

(∂T (0)

∂ t+ v(0)

z∂T (0)

∂ z

)=

1r∂∂ r

(rλ (T (0))∂T (1)

∂ r) in Ω0, (153)

ε2Peλ (T (0))∂T (1)

∂ r+ Bi

(v(0)z R(0))mT (0)

ε2R(0) = 0, on r = R(0)(z), (154)

where m is the exponent from Kase–Matsuo’s formula. The Neumann problem(153)–(154) for T (1) with respect to r has solution if and only if the usual com-patibility condition from Fredholm’s alternative is satisfied. It reads

ρ(T (0))(R(0))2cp(T (0))

(∂T (0)

∂ t+ v(0)

z∂T (0)

∂ z

)+

2Biε2Pe

(v(0)

z R(0))m

T (0) = 0 on (0,1).

(155)

Next, at the order O(ε2), (135) and condition (138) give

Reρ(T (0))μ(T (0))

(∂t v

(0)z + v(0)

z ∂zv(0)z

)+∂ p(0)

∂ z+∂ logμ(T (0))

∂ z

(p(0)−2

∂v(0)z

∂ z

)


=∂ 2v(0)

z

∂ z2 +∂ 2v(2)

z

∂ r2 +1r∂v(2)

z

∂ r+ Bo

ρ(T (0))μ(T (0))

− ∂∂ z

(v(0)

z ∂z logρ(T (0))+ ∂t logρ(T (0)))

in Ω0. (156)

2∂R(0)(z)∂ z

(∂v(0)

r

∂ r− ∂v(0)

z

∂ z

)+∂v(0)

r

∂ z+∂v(2)

z

∂ r

=1

εCaμ(T (0))∂σ(T (0))

∂ zon r = R(0)(z, t). (157)

Now (150) yields

∂z(μ(T 0)(p0−2∂zv

0z ))

= ∂z

(1

εCaσ(T (0))

R(0) −3μ(T0)∂v(0)

z

∂ z

−μ(T 0)∂z logρ(T (0))v(0)z − μ(T0)∂t logρ(T (0))

)

(158)

and integration of the (156) yields

∂rv(2)z |r=R(0) =

R(0)

2

(Re

ρ(T (0))μ(T (0))

(∂tv

(0)z + v(0)

z ∂zv(0)z

)−Bo

ρ(T (0))μ(T (0))

− 1

μ(T (0))∂∂ z

(μ(T (0))

(3∂v(0)

z

∂ z+∂z logρ(T (0))v(0)

z

+ ∂t logρ(T (0))

))+

1

μ(T (0))∂z

(1

εCaσ(T (0))

R(0)

)). (159)

On the other hand, after inserting (150) into the boundary condition (157) we get

∂rv(2)z |r=R(0) =

1εCa

∂zσ(T (0))μ(T (0))

+R(0)

2

(∂zzv

0z + ∂z

(v0

z∂z logρ(T (0))+∂t logρ(T (0))))

+∂zR(0)

(3∂v(0)

z

∂ z+ v0

z∂z logρ(T (0))+ ∂t logρ(T (0))

). (160)

After comparing (159)–(160), we obtain the effective momentum equation:

∂∂ z

(3μ(T (0))(R(0))2 ∂v(0)

z

∂ z+ μ(T (0))(R(0))2

(v(0)

z ∂z logρ(T (0))+ ∂t logρ(T (0)))

+1

εCaσ(T (0))R(0)

)= ρ(T (0))(R(0))2

(Re

(∂t v

(0)z + v(0)

z ∂zv(0)z

)−Bo

). (161)


Although Re∼ ε , we retain the inertia term, just for sake of completeness. Onecan imagine situations corresponding to large Reynolds numbers. As conclusion wesummarize our results in dimensional form:

Proposition 5.1. Let veff = vEv(0)z be the effective axial velocity, Reff = RER(0) the

effective fiber radius and Teff = TET (0) the effective temperature. Let us suppose thatthe quantities Q0 (the mass flow), R f (the final fiber radius),Vf (the pulling velocity),FL (the traction force) and TE (the extrusion temperature) are given positive con-stants. Then all other relevant physical quantities are determined by {veff ,Reff ,Teff }and given by:

effective radial velocity:

veffr (r,z) =− r

2

(∂zveff (z)+ veff∂z logρ(Teff (z))+ ∂t logρ(Teff (z))

)(162)

effective pressure: peff (z) =σ(Teff (z))

Reff (z)− μ(Teff (z))∂zveff (z)

−μ(Teff (z))3

(veff (z)∂z logρ(Teff (z))+ ∂t logρ(Teff (z))); (163)

effective axial stress:

Σeff (r,z)ez = μ(Teff (z))(3∂zveff (z)+ ∂t logρ(Teff (z))

+veff (z)∂z logρ(Teff (z)))ez− σ(Teff (z))

Reff (z)ez + μ(Teff (z))

(3∂zveff (z)

+∂t logρ(Teff (z))+ veff (z)∂z logρ(Teff (z)))r∂zReff (z)Reff (z)

er + r∂zσ(Teff (z))

Reff (z)er. (164)

effective traction : Feff =πR2

eff

g

(μ(Teff (z))

(3∂zveff (z)+ ∂t logρ(Teff (z))

+veff (z)∂z logρ(Teff (z)))− σ(Teff (z))

Reff (z)

). (165)

Functions {veff ,Reff ,Teff } are given by the Cauchy problem

∂∂ t

(ρ(Teff )(Reff )2)+

∂∂ z

(ρ(Teff )(Reff )2veff

)= 0, 0 < z < L; (166)

∂∂ z

{μ(Teff (z))(Reff (z))2

[3∂veff

∂ z+

DDt

logρ(Teff (z))]

+σ(Teff (z))Reff (z)}

=(

DDt

veff −g

)ρ(Teff (z))(Reff (z))2, 0 < z < L; (167)

cp(Teff )ρ(Teff )R2eff

DTeff

Dt

= −Cλ∞(

2ρ∞veff Reff

μ∞

)m

(Teff −T∞), 0 < z < L; (168)

Reff (L) = R f , veff (L) = Vf , Teff (L) = TE , (169)


whereDDt

=∂∂ t

+ veff∂∂ z

denotes the total derivative. Finally, we have

vz(r,z) = veff (z)+O(ε2Pe ); vr(r,z) = veffr (r,z)+O(ε2Pe );

R(r,z) = Reff +O(ε2Pe ); p(r,z) = peff (z)+O(ε2Pe );

Σ(r,z) = Σeff (r,z)+O(ε2Pe ).

Remark 5.2. It is important to recall that in (167) the termDDt

veff is negligible if

compared to g. Such a term, namely the fluid acceleration, is present in (167) be-cause we decided to retain inertia although O(ε).

5.2 Solvability of the Boundary Value Problems for the StationaryEffective Equations

Clearly, the values at the extrusion boundary could be replaced by the values at theinterface SE between the stages (c) and (d) of the fiber drawing process.

In the industrial simulations it makes sense to solve the full 3D Navier–Stokessystem in the stage (c), to solve the (166)–(168) corresponding to the stage (d) andto couple them at the interface SE . Coupling at the interface requires construction ofthe boundary layer.

Following ideas from [24], a typical iterative procedure for the Navier–Stokesequations with free boundary is the following: for a given free boundary we solve theNavier–Stokes equations with normal stress given at the lateral boundary. Then weupdate position of the free boundary using the kinematic free boundary condition.Iterations are repeated until the stabilization. Such procedure requires solving the(166)–(168) with the boundary conditions (169) replaced by

veff (L) = Vf , veff (0) = vE , Teff (0) = TE . (170)

In this section we present the study of the stationary version of the boundary valueproblem (166)–(168), (170), which was undertaken in [5]. We simply drop the indexeff and set Q = Q0/π .

In the absence of the gravity and inertia effects, with constant density and withthe heat transfer coefficient depending only on the temperature, the problem wassolved in the reference [14]. Using the temperature as variable, it was possible towrite an explicit solution for the radius and prove existence and uniqueness. In thegeneral situation, the approach from [14] is not possible any more. Nevertheless,their change of the unknown function will be useful in our existence proof. We provean existence result under the following physical properties on the coefficients:


(H1) Functions μ , ρ andσρ1/3

(T ) are defined on R, bounded from above and

from below by positive constants and decreasing. We suppose them infinitelydifferentiable. ρ f = minT∞≤T≤TE ρ(T ).

(H2) 0 < vE = v|z=0 < Vf = v|z=L and TE > T∞.(H3) cp is an infinitely differentiable strictly positive function.

We introduce the new unknown w by

w = logVfρ

1/3f

vρ1/3(T ). (171)

Let G = Vfρ1/3g and C1 =

Cλ∞Q

(2ρ∞√

Gμ∞

)m

, m being the Kase–Matsuo exponent,

see(128). Then the boundary value problem (166)–(168), (170) transforms to

∂∂ z

(−3

μ(T )ρ(T )

∂w∂ z

+1√QG

σ(T )ρ1/3(T )

ew/2)

= − gGρ1/3(T )ew−Gρ−1/3e−w∂zw− 5G

6ρ−4/3e−w∂zρ , 0 < z < L; (172)

cp(T )∂T∂ z

+C1ρ−2m/3(T )e−mw/2(T −T∞) = 0, 0 < z < L; (173)

w(0) = w0 = logVfρ

1/3f

vEρ1/3> 0, w(L) = 0, T (0) = TE . (174)

We will obtain existence of C∞-solutions to problem (172)–(174), such that w≤ w0.

Then the velocity v is calculated using the formula v = Vf

(ρ f

ρ(T )

)1/3

e−w. It is a

C∞-function and satisfies v(z) ≥ vE on [0,L]. We note as well that ρ f = ρ(T (L)) isnot given. For simplicity, we suppose w0 known. Otherwise, we should do one morefixed point calculation for ρ f , which does not pose problems.

Definition 5.3. The corresponding variational formulation for problem (172)–(174)is: Find functions w ∈ H1(0,L) and T ∈ H1(0,L), ∂zT ≤ 0, such that the boundaryconditions (174) are satisfied and

∫ L

03μ(T )ρ(T )

∂w∂ z

∂ϕ∂ z

dz−∫ L

0

1√QG

σ(T )ρ1/3(T )

emin{w,w0}/2 ∂ϕ∂ z

dz

+∫ L

0

gGρ1/3(T )emin{w,w0}ϕ dz+

∫ L

0Gρ−1/3(T )e−w ∂w

∂ zϕ dz

+∫ L

0

5G6ρ−4/3(T )e−w∂zρϕ dz = 0, ∀ϕ ∈H1

0 (0,L). (175)

∂T∂ z

=− C1

cp(T )ρ−2m/3(T )e−mmin{w,w0}/2(T −T∞), 0 < z < L. (176)


Proposition 5.4. (see [5]). Let μ , σ and ρ satisfy (H1)–(H2) and let {w,T} bea variational solution to (174), (175) and (176). Then we have w ≤ w0 and TE ≥T ≥ T∞.

Corollary 5.5. (see [5]). Under hypothesis (H1)–(H2), any variational solution{w,T} to (174), (175) and (176) solves equations (172)–(173).

Theorem 5.6. (see [5]). Let us suppose hypotheses (H1)–(H3) hold true. Let T bethe solution for

∂ T∂ z

=− C1

cp(T )ρ−2m/3(T )e−mw0/2(T −T∞), 0 < z < L; T (0) = TE . (177)

Let κ = maxT∞≤T≤TE |∂Tρ(T )| and let A =∫ L

0dz

μ(T (z)) . Then there is δ0 > 0 such

that for κA < δ0 , problem (172)–(174) admits a solution {w,T} ∈C∞[0,L]2, suchthat ∂zT ≤ 0 and w(z)≤ w0.

Remark 5.7. We see that in the case of constant density, the necessary conditionfrom the Theorem is always fulfilled. Furthermore, since viscosity takes large valuesas temperature decreases, A is a very small quantity. Consequently, the Theoremcovers all situations of practical interest. Proof uses Brouwer’s fixed point theorem.

Appendix 1

Let us consider the state

S = {v = 0, ϑ = 1, ρ = 1, P = 0} ,

corresponding to the fluid at rest subject only to a uniform pressure and to isothermalconditions. We now study the linear stability of S , in absence of any body forces10

and external heat sources, in two cases:

1. The system dynamics is governed by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ (ϑ) div v = αDϑDt

,

ρ (ϑ)DvDt

=1

Re

[−∇P+Δv+

13∇( div v)

],

ρ (ϑ)DϑDt

=1

PeΔϑ +

1 +ϑ (Tw−1)(Tw−1)2

αEcRe

DPDt

+2

(Tw−1)EcRe|D(v)|2,

(178)

10 We consider g = 0, and, as a consequence, p = P from (31).


where, for the sake of simplicity, we have considered β , cp1, μ and λ constant,i.e. β = βR, cp1 = cpR, μ = μR and λ = λR.

2. The system dynamics is governed by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ (ϑ) div v = αDϑDt

,

ρ (ϑ)DvDt

=1

Re

[−∇P+Δv+

13∇( div v)

],

ρ (ϑ)DϑDt

=1

PeΔϑ ,

(179)

corresponding to Ec = 0, i.e. to the fact that the kinetic energy of the fluid isreally negligible when compared to its thermal energy.

Let us first linearize the equations considering small (dimensionless) perturbationsof the state S , namely

v = v∗ (x,t) , ϑ = 1 +ϑ ∗ (x,t) , ρ = 1 +ρ∗ (x, t) , P = P∗ (x, t) .

In particular, we have also

ρ = 1−αϑ ∗, ⇒ ρ∗ =−αϑ ∗.

Case 1. Dynamics governed by (178). We will see, as put in evidence in [2], that,in such a case, the rest state is linearly unstable.The linearized form of system (178) is

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

div v∗ = α∂ϑ ∗

∂ t,

∂v∗

∂ t=

1Re

[−∇P∗+Δv∗+

13∇( div v∗)

],

∂ϑ ∗

∂ t=

1PeΔϑ ∗+β

∂P∗

∂ t,

(180)

where

β =Tw

(Tw−1)2

αEcRe

.

We now take the Fourier transform11 with respect to the spatial variable x of thedependent variable v∗, P∗ and ϑ ∗, namely

11 The Fourier transform, with respect to the spatial variable x, of a function f (x, t) is defined by

f (k, t) =1

(2π)3/2

∫

R3

f (x, t)exp(−ik ·x)dx.


F.T.

v∗ (x,t) −→ v(k,t) ,

ϑ ∗ (x,t) −→ ϑ (k,t) ,

P∗ (x,t) −→ P(k,t) ,

getting ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

ik · v = α∂ϑ∂ t

,

∂ v∂ t

=− ikRe

P− k2

Rev− 1

3Rek (k · v) ,

∂ϑ∂ t

=− k2

Peϑ +β

∂ P∂ t

,

(181)

where k = |k|. Decomposing v as follows

v = v‖+ v⊥ ,

where ⎧⎪⎨⎪⎩

v‖ =(

v · kk

)kk

= v‖kk,

v⊥ = v− v‖ ,

system (181) rewrites

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂ v⊥∂ t

=− k2

Rev⊥ ,

∂ v‖∂ t

=− ikRe

P− 43Re

k2 v‖,

∂ϑ∂ t

= ik

Kρv‖ ,

∂ P∂ t

=k2

βPeϑ + i

kαβ

v‖ .

(182)

So, we immediately realize that v⊥ vanishes as t → ∞. Therefore we are left with

∂∂ t

⎛⎜⎜⎜⎜⎜⎝

ϑ

v‖

P

⎞⎟⎟⎟⎟⎟⎠

= A

⎛⎜⎜⎜⎜⎜⎝

ϑ

v‖

P

⎞⎟⎟⎟⎟⎟⎠

, A=

⎛⎜⎜⎜⎜⎜⎝

0 ikα

0

0 − 43Re

k2 − ikRe

k2

βPei

kαβ

0

⎞⎟⎟⎟⎟⎟⎠

.


We have to evaluate the eigenvalues of the matrix A. Denoting them by λi, i = 1,2,3we have

detA = λ1λ2λ2 =k4

βPeRe≥ 0.

Now, since the secular equation is a third degree equation in λ whose coefficient arereal, at least one eigenvalue is real. The above relations therefore implies that thereexists at least a positive eigenvalue. We thus conclude that the state S is instable.

Case 2. Dynamics governed by (179).

Considering, as before, the Fourier transform of the linearized form of system (179)we obtain ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂ v⊥∂ t

=− k2

Rev⊥ ,

∂ v‖∂ t

=− ikRe

P− 43Re

k2 v‖,

∂ϑ∂ t

= ikα

v‖ ,

∂ϑ∂ t

=− k2

Peϑ .

(183)

Form (183)4 and (183)1 we deduce that ϑ and v⊥ vanishes as t → ∞. In particular,

ϑ = ϑo exp

(− k2

Pet

)and

∂ϑ∂ t

=− ϑok2

Peexp

(− k2

Pet

), ⇒ ∂ϑ

∂ t→ 0, as t→ ∞.

So, by (183)3, we have that also v‖ vanishes as t → ∞. Finally, (183)2 implies that

P→ 0, as t→ ∞. We therefore conclude that the state S is linearly stable.

Appendix 2

Referring to Fig. 5, let us consider the simplest version of the Matovich–Pearsonmodel for the stage (d). We have

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∂∂x

(v1A) = 0,

∂∂x

(A∂v1

∂x

)= 0,

(184)

where, as in formula (127), A = A(x) is the fiber cross section area, A = πR2.


0 xT L Lf

x

Stage (c) Stage (d)

Fig. 5 A schematic of stages (c) and (d). The spinneret is located at x = L f . x = 0 is the die inflowsurface. Picture not in scale

As boundary conditions, following [23], we consider

v1(x = Lf

)= Vsp, and Φ

(x = Lf

)=Φsp =

Fsp

πR2sp

, (185)

where Φ denotes the longitudinal stress

Φ = 3μ f∂v1

∂x,

with μ f fluid viscosity (considered constant along stage (d)). Solving (184), (185)we obtain

v1 (x) = Vsp exp

{− Φsp

3μ fVsp

(Lf − x

)}, (186)

R(x) = Rsp exp

{Φsp

6μ fVsp

(Lf − x

)}. (187)

Hence the explicit expressions of VL and ΦL, in terms of the terminal quantities arethe following

VL = Vsp exp

{− Φsp

3μ fVsp

(Lf −L

)}, ΦL =−Φsp exp

{− Φsp

3μ fVsp

(Lf −L

)}.

(188)

As we have seen in Sect. 5.1, the Matovich–Pearson model can be generalizedconsidering the residual spatial dependence of viscosity, still neglecting inertia, sur-face tension and density variation. In that framework, according to (127) and (167)(see also [7]), (184)2 modifies to

∂∂x

(μ (x) A(x)

∂v1

∂x

)= 0.


The expressions replacing (186) and (187) are

v1 (x) = Vsp exp

{− Φsp

3Vsp

∫ Lf

x

dx′

μ (x′)

}, (189)

R(x) = Rsp exp

{Φsp

6Vsp

∫ Lf

x

dx′

μ (x′)

}. (190)

Thus, coming back to Sect. 4.1, namely to the definition of L, on the basis of (189),(190) we may confirm the consistency of our approach, taking x = L and observing

that Φsp6Vsp

∫ LfL

dx′μ(x′) = O (1) if

∫ LfL

dx′μ(x′) = Lf−L

μ . Since Lf −L≈ 10 m, this requires μ =

O(107 Pa s

), consistently with the thermal field experimentally known in stage (d).

References

1. Antontsev, S.N., Kazhikhov, A.V., Monakhov, V.N.: Boundary Value Problems in Mechanicsof Nonhomogeneous Fluids, North-Holland, Amsterdam (1990)

2. Bechtel, S.E., Forest, M.G., Rooney, F.J., Wang, Q.: Thermal expansion models of viscousfluids based on limits of free energy. Phys. Fluids 15, 2681–2693 (2003)

3. Bechtel, S.E., Rooney, F.J., Forest, M.G: Internal constraint theories for the thermal expansionof viscous fluids. Int. J. Eng. Sci. 42, 43–64 (2004)

4. Beirao da Veiga, H.: An Lp-Theory for the n−Dimensional, stationary, compressible Navier-Stokes Equations, and the Incompressible Limit for Compressible Limit for CompressibleFluids. The Equilibrium Solutions. Commun. Math. Phys. 109, 229–248 (1987)

5. Clopeau, T., Farina, A., Fasano, A., Mikelic, A. : Asymptotic equations for the terminal phaseof glass fiber drawing and their analysis, accepted for publication in Nonlinear Analysis TMA:Real World Applications, 2009, http://dx.doi.org/10.1016/j.nonrwa.2008.09.017.

6. Desjardins, B., Grenier, E., Lions, P.-L., Masmoudi, N.: Incompressible limit for solutionsof the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. PuresAppl. 78, 461–471 (1999)

7. Dewynne, J.N., Ockendon, J.R., Wilmott, P.: On a mathematical model for fibre tapering,SIAM J. Appl. Math. 49, 983–990 (1989)

8. Diaz, J.I., Galiano, G.: Existence and uniqueness of solutions of the Boussinesq system withnonlinear thermal diffusion. Topol. Methods Nonlin. Anal. 11, 59–82 (1998)

9. Farina, A., Fasano, A., Mikelic, A.: On the equations governing the flow of mechanically in-compressible, but thermally expansible, viscous fluids, M3AS : Math. Models Methods Appl.Sci. 18, 813–858 (2008)

10. Feireisl, E., Novotny, A.: The low Mach number limit for the full Navier-Stokes-Fourier sys-tem. Arch. Ration. Mech. Anal. 186, 77–107 (2007)

11. Gallavotti, G.: Foundations of Fluid Dynamics. Springer, Berlin (2002)12. Green, A.E., Naghdi, P.M., Trapp, J.A., Thermodynamics of a continuum with internal

constraints. Int. J. Eng. Sci. 8, 891–908 (1970)13. Gupta, G., Schultz, W.W.: Non-isothermal flows of Newtonian slender glass fibers. Int. J. Non-

Linear Mech. 33, 151–163 (1998)14. Hagen, T.: On the Effects of Spinline Cooling and Surface Tension in Fiber Spinnning, ZAMM

Z. Angew. Math. Mech. 82, 545–558 (2002)15. Hoff, D.: The zero Mach number limit of compressible flows. Comm. Math. Phys. 192,

543–554 (1998)16. Kagei, Y., Ruzicka, M., Thater, G.: Natural Convection with Dissipative Heating. Commun.

Math. Phys. 214, 287–313 (2000)


17. Kase, S., Matsuo, T.: Studies of melt spinning I. Fundamental Equations on the Dynamics ofMelt Spinning, J. Polym. Sci. A 3, 2541–2554 (1965)

18. Kase, S., Matsuo, T.: Studies of melt spinning. II, Steady state and transient solutions of fun-damental equations compared with experimental results. J. Polym. Sci. 11, 251–287 (1967)

19. Lions, P.L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math.Pures Appl. 77, 585–627 (1998)

20. Matovich, M.A., Pearson, J.R.A.: Spinning a molten threadline-steady state isothermal viscousflows. Ind. Engrg. Chem. Fundam. 8, 512–520 (1969)

21. von der Ohe, R.: Simulation of glass fiber forming processes. Thesis PhD (2005) – AUCImprint: ISBN: 8791200253. – Aalborg : Department of Production, Aalborg University (2005)

22. Rajagopal, K.R., Ruzicka, M., Srinivasa, A.R.: On the Oberbeck-Boussinesq approximation.Math. Models Methods Appl. Sci. 6, 1157–1167 (1996)

23. Renardy, M.: Draw resonance revisited, SIAM J. Appl. Math. 66, 1261–1269 (2006)24. Renardy, M.: An existence theorem for a free surface flow problem with open boundaries.

Commun. Partial Differ. Equ. 17, 1387–1405 (1992)25. Schlichting, H., Gersten, K.: Boundary-Layer Theory, 8th edn. Springer, Heidelberg (2000)26. Shelby, J.E.: Introduction to Glass Science and Technology, 2nd edn. RSCP Publishing,

London (2005)27. Temam, R.: Infinite-Dimensional Dynamical systems in Mechanics and Physics. Springer,

New York (1988)28. Zeytounian, R.Kh.: Modelisation asymptotique en mecanique des fluides newtoniens. Springer,

New York (1994)29. Zeytounian, R.Kh.: The Benard-Marangoni thermocapillary instability problem. Phys.

Uspekhis 41, 241–267 (1998)30. Zeytounian, R.Kh.: Joseph Boussinesq and his approximation: A contemporary view. C.R.

Mecanique 331, 575–586 (2003)

List of Participants

1. Aionicesei ElenaUniv. of Maribor, [email protected]

2. Antonietti Paola FrancescaPolitecnico of Milano, [email protected]

3. Benedetti IreneUniv. of Firenze, [email protected]

4. Borsi IacopoUniv. of Firenze, [email protected]

5. Butt Azhar Iqbal KashifTU Kaiserslautern, [email protected]

6. Ceseri MaurizioUniv. of Firenze, [email protected]

7. Curkovic AndrijanaUniv. of Split, [email protected]

8. Dierich FrankFreiberg Univ. of Mining and Technology, [email protected]

9. Doschoris MichaelUniv. of Patras, [email protected]

10. Ebert SvendFreiberg Univ. of Mining and Technology, [email protected]

11. Faienza LoredanaUniv. of Firenze, [email protected]

A. Fasano (ed.), Mathematical Models in the Manufacturing of Glass,Lecture Notes in Mathematics 2010, DOI 10.1007/978-3-642-15967-1,c© Springer-Verlag Berlin Heidelberg 2011

225

226 List of Participants

12. Fusi LorenzoUniv. of Firenze, [email protected]

13. Geyer AnnaUniv. of Vienna, [email protected]

14. Hadjloizi DemetraUniv. Patras, [email protected]

15. Lebedyanskaya ElenaVoronezh State agricolture Univ., [email protected]

16. Nagwanshi RekhaVikram Univ. Ujjain MP, [email protected]

17. Niedziela MaciejUniv. of Zielona Gora, [email protected]

18. Nouri Fatma ZohraUniv. Badji Mokhtar, Francefz [email protected]

19. Olech MichalThe Univ. of Wroclaw, [email protected]

20. Pazanin IgorUniv. of Zagreb, [email protected]

21. Peter CristianUniv. of Vienna, [email protected]

22. Ricci RiccardoUniv. of Firenze, [email protected]

23. Rosso FabioUniv. of Firenze, [email protected]

24. Senger BennoTechnische Univ. Bergakandemie Freiberg, [email protected]

25. Serpa CristinaUniversity of Lisbon, [email protected]

26. Speranza AlessandroUniv. of Firenze, [email protected]

List of Participants 227

27. Starcevic MajaUniv. of Zagreb, [email protected]

28. Sulkowski TomaszUniv. of Zielona Gora, [email protected]

29. Togobytska NataliyaWeierstrass Inst. for Appl Anals and Stoch, [email protected]

30. Turbin MikhailVoronezh State Univ., [email protected]

31. Verani MarcoPolitecnico of Milano, [email protected]

32. Vorotnikov DmitryVoronezh State Univ., [email protected]

33. Zubkov VladimirUniv. of Limerick, [email protected]

34. Zvyagin AndreyVoronezh State Univ., [email protected]

mathematical models in the manufacturing of glass: c.i.m.e. summer school, montecatini terme, italy...

Documents