Numerical Simulation Of Stratified Flows

And Droplet Deformation In 2D Shear Flow

Of Newtonian And Viscoelastic Fluids

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF

MATHEMATICS AT VIRGINIA POLYTECHNIC INSTITUTE AND STATE

UNIVERSITY

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS

By

T IRIVANHU CHINYOKA

Supervised by: YURIKO & MICHAEL RENARDY

Committee: Jong Kim, Tao Lin, Shu-Ming Sun

DEPARTMENT OF MATHEMATICS

VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY

BLACKSBURG, VIRGINIA

Keywords: Stratified Flows, Droplet Deformation, Viscoelastic fluids

November 15, 2004

Numerical Simulation Of Stratified Flows And Droplet De-

formation In 2D Shear Flow Of Newtonian And Viscoelas-

tic Fluids. By Tirivanhu Chinyoka

Abstract

Analysis of multi-layer fluid flow systems or, in general, flows with interfaces

often leads to mathematical expressions and equations too complicated for pen-

cil and paper hence numerical computation is almost always necessary. In this

dissertation, we develop a numerical code for tracking deformable interfaces. In

particular this code is a viscoelastic version of the volume of fluid algorithm de-

veloped in [11]. The code uses the piecewise linear interface calculation method

to reconstruct the interface and the continuous surface force formulation to model

interfacial tension forces. Our numerical algorithm is primarily designed to sim-

ulate the flow of (i) superposed fluids (herein referred to as fluid-fluid systems)

and (ii) the drop in a flow problem (droplet-matrix systems) in 2D shear flows

of viscoelastic fluids. However by taking the viscoelastic parameters to be zeros,

we in fact can consider cases were either or both of the phases in the fluid-fluid

or droplet-matrix system will be assumed viscoelastic or Newtonian. The ex-

tra stresses governing viscoelasticity will herein be treated with the Oldroyd-B

constitutive equations. The part our work dealing with two-layer flows is in the

same spirit as among others that of Renardy et. al [29], who investigated the

Poiseuille flow counterpart. As mentioned earlier, this part can also be thought

of as a natural extension of the work of Li et. al, [11], to the viscoelastic regime.

Our subsequent work on deformable drops is closely connected to the experimen-

tal investigation of Guido et. al [6], and the numerical works of Sheth et. al.

[32], Pillapakkam et. al [18], and Renardy et. al [28], all of whom considered

the drop in a flow problem in various contexts. As in [11] we employ the volume

of fluid scheme with a semi-implicit Stokes solver (enabling computations at low

Reynolds numbers) in our numerical algorithm. In the first part, the code is

validated against linear theory for the superposed shear flow of well documented

fluid-fluid systems. Numerical validation in the second part will mostly be against

the results of the four major worked cited earlier.

iii

Contents

List of Figures vi

List of Tables vii

Acknowledgements viii

Dedication ix

Declaration x

1 Introduction 1

1.1 Interfacial Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Rayleigh-Taylor Instabilities . . . . . . . . . . . . . . . . . 2

1.1.2 Saffman-Taylor Instabilities . . . . . . . . . . . . . . . . . 2

1.1.3 Viscous Instabilities . . . . . . . . . . . . . . . . . . . . . . 3

1.1.4 Surface Tension Gradient Instabilities . . . . . . . . . . . . 4

1.1.5 Elastic Instabilities . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Governing Equations 8

iv

2.1 Dimensionless Governing Equations . . . . . . . . . . . . . . . . . 9

2.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Bifurcation to traveling wave solution . . . . . . . . . . . . 13

2.2.2 Mean Flow Component . . . . . . . . . . . . . . . . . . . . 16

3 Numerical Implementation 18

3.1 Volume Of Fluid (VOF) Method . . . . . . . . . . . . . . . . . . . 18

3.1.1 Piecewise Linear Interface Calculation (PLIC) . . . . . . . 19

3.1.2 Spartial Discretization and MAC Method . . . . . . . . . . 21

3.2 Temporal Discretization, Projection Method . . . . . . . . . . . . 22

3.2.1 Semi-Implicit Stokes Solver . . . . . . . . . . . . . . . . . 24

3.2.2 Multigrid Method . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Modeling interfacial tension force (CSF) . . . . . . . . . . . . . . 26

3.4 Oldroyd-B Constitutive Equation . . . . . . . . . . . . . . . . . . 27

4 Numerical Validation I 30

4.1 Calibrating The Code . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.1 Velocities and Extra-stresses . . . . . . . . . . . . . . . . . 32

4.1.2 Time derivatives . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Growth rates of amplitudes and velocities . . . . . . . . . . . . . . 36

4.2.1 Growth Rates at the Interface . . . . . . . . . . . . . . . . 36

4.2.2 Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.3 Effects of cell shape . . . . . . . . . . . . . . . . . . . . . . 38

4.2.4 Effects of Surface Tension . . . . . . . . . . . . . . . . . . 40

4.2.5 Growth rates away from Interface . . . . . . . . . . . . . . 43

v

4.2.6 Maximum growth along a horizontal line . . . . . . . . . . 45

4.3 Growth rates of Extra-stresses . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Growth at the interface . . . . . . . . . . . . . . . . . . . . 47

4.3.2 Growth away from Interface . . . . . . . . . . . . . . . . . 48

4.3.3 Maximum growth along a horizontal line . . . . . . . . . . 49

5 Harmonic averaging 51

5.1 Growth rates of amplitudes and velocities . . . . . . . . . . . . . . 52

5.1.1 Growth at the Interface . . . . . . . . . . . . . . . . . . . 52

5.1.2 Maximum growth along a horizontal line . . . . . . . . . . 53

5.2 Growth rates of Extra-stresses . . . . . . . . . . . . . . . . . . . . 55

5.2.1 Growth at the interface . . . . . . . . . . . . . . . . . . . . 55

5.2.2 Maximum growth along a horizontal line . . . . . . . . . . 56

6 Numerical Validation II 57

6.1 Growth rates of amplitudes and velocities . . . . . . . . . . . . . . 58

6.1.1 Growth at the Interface . . . . . . . . . . . . . . . . . . . 58

6.1.2 Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7 Droplet Deformation 63

7.1 Problem formulation and assumptions . . . . . . . . . . . . . . . . 63

7.2 Validation of Results . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.2.1 Small deformation case . . . . . . . . . . . . . . . . . . . . 66

7.2.2 Large deformation case . . . . . . . . . . . . . . . . . . . . 74

7.3 Temporal and spatial convergence . . . . . . . . . . . . . . . . . . 77

vi

7.3.1 Newtonian drops in viscoelastic fluids . . . . . . . . . . . . 79

7.4 Purely Newtonian case and effects of inertia . . . . . . . . . . . . 88

vii

List of Figures

2.1 Flow Schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 MAC cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1 Residual data for velocities and extra-stresses. . . . . . . . . . . . 33

4.2 Residual data for time derivatives. . . . . . . . . . . . . . . . . . . 34

4.3 log(h), log(vmax), log(‖v‖2). respectively . . . . . . . . . . . . . 37

4.4 Wave forms at (a) t = 0.01, (b) t = 0.1, (c) t = 0.3 and (d) t = 1 . 38

4.5 Effects of cell shape for the case α = 2. . . . . . . . . . . . . . . . 39

4.6 <(σ) vs. Tension, T . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.7 Effects of surface tension . . . . . . . . . . . . . . . . . . . . . . . 41

4.8 Spectrum of eigenvalues for α = 6. . . . . . . . . . . . . . . . . . 42

4.9 (a) <(σ)=-0.83306 (b) <(σ)=-7.00952 . . . . . . . . . . . . . 43

4.10 log(v) & log(u) in the Newtonian fluid. . . . . . . . . . . . . . . . 44

4.11 log(v) & log(u) in the Viscoelastic fluid. . . . . . . . . . . . . . . 44

4.12 log(vmax) & log(umax) along a horizontal line in Newtonian fluid. . 45

4.13 log(vmax) & log(umax) along a horizontal line in Viscoelastic fluid. 46

4.14 log(ampl) & log(Tijmax). . . . . . . . . . . . . . . . . . . . . . . . 47

4.15 log(Tij) in the viscoelastic fluid. . . . . . . . . . . . . . . . . . . . 48

viii

4.16 log(Tijmax) in the Viscoelastic fluid. . . . . . . . . . . . . . . . . . 49

5.1 log(h), log(vmax), log(‖v‖2). respectively . . . . . . . . . . . . . 53

5.2 log(vmax) & log(umax) along a horizontal line in Newtonian fluid. . 53

5.3 log(vmax) & log(umax) along a horizontal line in Viscoelastic fluid. 54

5.4 log(ampl) & log(Tijmax). . . . . . . . . . . . . . . . . . . . . . . . 55

5.5 log(Tijmax) in the Viscoelastic fluid. . . . . . . . . . . . . . . . . . 56

6.1 log(h), log(vmax), log(‖v‖2). respectively . . . . . . . . . . . . . 59

6.2 max(amplitude) vs time and wave shapes . . . . . . . . . . . . . . 60

6.3 Wave forms at (a) t = 0.01, (b) t = 0.1, (c) t = 0.3 and (d) t = 1

where R1 = 186, ∆t = 10−3, tmax = 1000 and h = 0.1 . . . . . . . . 61

7.1 Computational domain . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 Temporal evolution of D, when Ca = 0.24 . . . . . . . . . . . . . 68

7.3 Temporal evolution of D when Ca = 0.6 . . . . . . . . . . . . . . 69

7.4 Deformed drops at t = 3 . . . . . . . . . . . . . . . . . . . . . . . 70

7.5 Deformed drops at t = 10 . . . . . . . . . . . . . . . . . . . . . . 70

7.6 Deformed drops at t = 10 . . . . . . . . . . . . . . . . . . . . . . 71

7.7 Contour plots of first normal stress at difference t = 7. . . . . . . 71

7.8 Temporal evolution of D when Ca = 0.24 and zero initial stresses 72

7.9 Evolution of ∆D when Ca = 0.24 . . . . . . . . . . . . . . . . . . 73

7.10 Temporal evolution of D when Ca = 60 . . . . . . . . . . . . . . . 75

7.11 Elongation of drops at t = 10 when Ca = 60, R = 0.3, De = 8. . . 76

ix

7.12 Effect of mesh size on numerical breakup of drops, results shown

for a Newtonian drop in a viscoelastic fluid at t = 10 with R =

0.3, De = 8.0&Ca = 60. . . . . . . . . . . . . . . . . . . . . . . . . 77

7.13 Convergence with mesh resolution, for a Newtonian drop in a vis-

coelastic fluid at t = 10 with R = 0.3, De = 8.0 & Ca = 60,

Red=256× 256, Blue=128× 128 Green=512× 512. . . . . . . . . 77

7.14 Steady state shapes with spatial (a) & spatio-temporal (b) refinement 78

7.15 Transient deformation with temporal refinement . . . . . . . . . . 79

7.16 velocity vector plots for Newtonian drop in a viscoelastic fluid

where Ca = 60, R = 0.3, De = 8 at (a) t = 2, (b) t = 4, (c) t = 5,

& (d) t = 6, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.17 velocity vector plots for Newtonian drop in a viscoelastic fluid

where Ca = 60, R = 0.3 & De = 4, t = 2, De = 4, t = 6, De =

2, t = 2, & De = 2, t = 6, . . . . . . . . . . . . . . . . . . . . . . . 81

7.18 velocity vectors showing linear streamwise profiles at left and right

hand side edges, Ca = 60, R = 0.3, De = 8 . . . . . . . . . . . . . 82

7.19 Surface and contour plots of first normal stress difference at t = 2

where Ca = 60, R = 0.3, De = 8 in (a) & (b) and De = 4 in (c) &

(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.20 Surface and contour plots of first normal stress difference at t = 2

where Ca = 60, R = 0.3, De = 2 in (a) & (b), De = 0.5 in (c) &

De = 0.5 in (d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.21 Surface and contour plots of first normal stress difference at t = 2

where Ca = 60, R = 0.3, De = 0.1 in (a) & (b) and De = 0.01 in

(c) & (d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

x

7.22 Surface and contour plots of pressure at t = 2 where Ca = 60,

R = 0.3, De = 8 in (a), (b) and De = 4 in (c), (d) . . . . . . . . . 87

7.23 Surface and contour plots of pressure at t = 2 where Ca = 60,

R = 0.3, and De = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . 87

7.24 Contour plots of pressure at t = 2 where Ca = 60, R = 0.3,

De = 0.01 in (a) and De = 0 in (b) . . . . . . . . . . . . . . . . . 88

7.25 Evolution of drop deformation in time (a) Ca = 0.2, m = 1, (b)

Ca = 0.2, m = 10, (c) Ca = 0.4, m = 1 & (d) Ca = 0.4, m = 10 90

7.26 Drop shapes at t = 3 (a) Ca = 0.2, m = 1, (b) Ca = 0.2, m = 10,

(c) Ca = 0.4, m = 1 & (d) Ca = 0.4, m = 10 . . . . . . . . . . . 91

xi

List of Tables

4.1 Properties of fluids: . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Conditions of experiments: . . . . . . . . . . . . . . . . . . . . . . 35

7.1 Deformation parameter (D) and Angle (φmax) for four drop-matrix

systems at dimensionless t = 3, Ca = 0.24 . . . . . . . . . . . . . 67

7.2 D and φmax at dimensionless t = 10, Ca = 0.6 . . . . . . . . . . . 69

7.3 D and φmax at t = 3, Ca = 0.24 & zero initial stresses . . . . . . . 72

7.4 ∆D & ∆φmax at t = 3, Ca = 0.24 . . . . . . . . . . . . . . . . . . 73

7.5 D and φmax at t = 10, Ca = 60 . . . . . . . . . . . . . . . . . . . 75

xii

Acknowledgements

This work would not have been possible without the tremendous help

and immense input from two brilliant Mathematicians: my advisors,

Professors Yuriko & Michael Renardy, my acknowledgements should

start with them. By extension I would also like to acknowledge the

help, direct or indirect of Professors Damir Khismatullin and Jie Li

and the Virginia Tech Mathematics department. I should also take

this opportunity to confer a big thank you to two of the finest people

who ever graced this world and who naturally brought me this far,

my parents Tizirai Kunashe and Gelly Gumbo. My siblings Tiani and

Pilia also deserve special mention for bearing all the burden during

my absence, especially following the passing on of our wonderful sister

Piripina Hahani way before her time, we will forever miss her.

Lastly, I extend a hand to the one person I would least think of thank-

ing, the one who has unfolded himself to a higher level, without much

hope from the sidelines... ...self !

tiri

11-15-04

xiii

Dedication

To those who by the simplicity of their deeds, make our

days complicated, without forethought but with uncon-

cious attention.

tiri

11-15-04

xiv

Declaration

No portion of the work referred to in this dissertation has

been submitted in support of an application for another

degree or qualification of this or any other university or

other institution of learning.

xv

Chapter 1

Introduction

In recent years, the stability problem of multi-fluid flows has attracted a consid-

erable amount of research interest among scientists from the fields of aeronau-

tics, applied mathematics, bio-medical and chemical engineering and physics to

name but a few. This can be readily attributed to the ever increasing number

of industrial applications of such problems for example in lubricated pipelining,

manufacture of co-extrusion polymers, [14], photographic film development and

deicing aircraft wings [17].

1.1 Interfacial Instabilities

The earliest significant theory of instability of flow of superposed fluids can be

traced back to Yih [37] with his work on long wave instability of two-layer flows

(Couette & Plane Poiseuille) with viscosity stratification. The years following

the initial publications of Yih saw a ballooning of interest and literature in the

field of multi-fluid flow stability. A concise account can be found say in the book

by Joseph and Renardy [14]. We are here simply going to try to just give a brief

overview of the parallels that can be drawn from most of these works.

1

We should here point out the interesting works of earlier researchers before Yih

for example G. I. Taylor who made the observation that (provided gravity is

set aside) if two fluids, separated by a flat horizontal interface are accelerated

perpendicular to this interface, then there will be an instability of the interface if

the acceleration is directed out of the lighter fluid into the heavier one [34], [13].

1.1.1 Rayleigh-Taylor Instabilities

The observation by Taylor is of course a generalization of what are generally

termed Rayleigh-Taylor instabilities in which under gravitational acceleration

the set-up with a heavy fluid lying above a light fluid is unstable. This can also

be generalized the the case of multi-layer flows with density stratification. Micro-

gravity environments, which simulate the suppression of gravity effects have been

achieved through either density matching or by conducting the relevant experi-

ments in space laboratories aboard space shuttles. An interesting case however

is the case of thermocapillary instabilities under microgravity, in which there

is well-documented evidence of disagreements between results from experiments

conducted in space and theoretical results [39]. The workers in [39] cite Coriolis

forces resulting from the orbiting motion of space shuttles as a possible source

of the discrepancies. Rayleigh-Taylor instabilities have also been investigated in,

[21], with viscous effects in [36], and numerically in [35].

1.1.2 Saffman-Taylor Instabilities

When one fluid displaces another from a porous solid, say a Hele-Shaw cell,

displacements of the interface usually grow into finger-like structures of the pen-

etrating fluid in the compliant fluid. This is an example of what are referred to

as Saffman-Taylor instabilities. Since however, most of the important flows in

this case (over porous solids) do not behave according to the usual Navier-Stokes

2

equations, and are instead governed by Darcy’s law we will therefore not dwell

much on these except to just refer to the (quite interesting as we shall see in the

next paragraph) conclusions of Saffman-Taylor, i.e. the displacement of a more

viscous fluid by a less viscous one in a porous solid may be unstable whereas the

reverse situation is stable, see for example [13] for a complete analysis.

1.1.3 Viscous Instabilities

Curiously, when the Navier-Stokes equations do apply, say in industrial scale

pipeline transport of liquids, we usually come to completely opposite conclusions

to those just stated. In this case, the displacement of a more viscous fluid (e.g.

oil) by a less viscous fluid (e.g. water) leads to a stable arrangement where

the less viscous fluid migrates to the walls and acts as a lubricant for the more

viscous fluid [14]. In general for two-layer flows with equal densities and subject

to long-wave disturbances, such long wave instabilities can be stabilized via the

so-called thin-layer effect (i.e. putting the less viscous fluid in a thin layer)

[14], [10]. Short wave instabilities for such flows (which may surprisingly be

due to the viscosity difference and not damped out by it!) will be stabilized by

surface tension [9], [14]. Li, Renardy and Renardy [11], investigated the “viscous

counterpart” of the simulation of an inviscid Kelvin-Helmholtz instability. Their

paper looks at periodic disturbances in two-layer Couette flow with an emphasis

on the effects of viscosity stratification, and its role in fingering instabilities.

Ganpule and Khomami [4] showed the mechanism of instability of short and long

waves to viscosity stratification in both plane Poiseuille and plane Couette flow

by performing a rigorous viscous energy analysis. For short waves, they showed

that the interfacial contribution to the viscous energy term was the mechanism

of instability whereas for long wave disturbances, it is the Reynolds stress term

of the disturbance energy equation.

3

These kinds of discussions are also typical of most stability studies via asymptotic

analysis, in which the instabilities are depicted exclusively as either short or

long wave in nature. However, using a rigorous linear analysis, [21] showed the

existence of unstable regions not described by asymptotic analyses of short or

long waves.

1.1.4 Surface Tension Gradient Instabilities

Even though we just noted that inclusion of surface tension can stabilize short

wave disturbances, it should be noted that surface tension (or interfacial tension

as we should correctly refer to it for multi-layer flows, surface tension being for the

case when a liquid is in equilibrium contact with its vapor) is a thermodynamic

property which depends on both temperature and composition (concentration of

surfactants). Differences in temperature and/or composition in the tangential di-

rection leads to inhomogeneity of surface tension and thus a tangential interfacial

tension gradient which may produce flow or change an existing one and this may

produce instabilities. Surface tension gradient driven convection is also known as

Marangoni driven convection after the Italian physicist or also thermocapillarity

convection in case it arises only from temperature differences. If on the other

hand the temperatures and/or composition gradients are perpendicular to the

interface, then only if such gradients are strong enough can they lead to insta-

bility otherwise since they do not produce any inhomogeneities of the interfacial

tension, one would not expect any interfacial tension related instabilities. For a

thorough treatment of the effects of interfacial tension inhomogeneities on multi-

fluid flow stability we refer the reader to the recent book [16]. In our current

work we will assume that the temperature is constant, the composition is fixed

and hence the interfacial tension is constant.

4

1.1.5 Elastic Instabilities

Our work also deals with viscoelastic fluids hence it is worth reviewing some

of the recent work that has been done by other researchers focusing on the ef-

fects of elasticity on stability of two-layer flows. Y. Renardy in [22] showed that

for short wave disturbances, the order of magnitude of elasticity stratification

in determining growth rates of such disturbances is one less than the stabilizing

effect of surface tension and one more that the destabilizing effects of density

and viscosity stratification, and hence under suitable choices of the surface ten-

sion, densities and viscosities, elasticity stratification may stabilize or destabilize

the flow. Based on this work, Joseph and Renardy [14] showed that under long

wave disturbances, elasticity stratification may also stabilize or destabilize the

flow. Again by performing an energy analysis for long wave as well as short wave

purely elastic instabilities on both plane Poiseuille and Couette flows Ganpule

and Khomami [4] showed the mechanism of instability to be associated with the

coupling between the jump in the base flow normal stresses across the interface

and the perturbation velocity field. The experimental, computational and ana-

lytical works of importance which we did not present here include the work of

Khomami, Renardy, Su & Clark [29], and all the other numerous publications,

far too many to mention, say of K. Chen, V. Coward, D. Joseph, D. Kothe, M.

Renardy, Y. Renardy, S. Zaleski, and their co-workers.

1.2 Scope

This Dissertation is mainly concerned with developing a numerical scheme for

investigating the effects of elasticity stratification on the superposed shear flow

of two liquids and in droplet deformation. This numerical code uses a volume

of fluid (VOF) scheme similar to that of [11] and [5]. The VOF method is a

5

fixed mesh approach that allows for accurate interface advection and handles

changes in interface topology [11]. The interface is reconstructed using using

a piecewise linear interface calculation (PLIC) method where the interface is

assume to be linear in each computational cell [19]. Other algorithms that have

been proposed for interface reconstruction are the SLIC and least squares methods

[19]. To handle the boundary conditions at the interface, we use the continuous

surface force (CSF) technique where the interfacial tension forces are incorporated

as body forces per unit volume in the momentum equations [1], [11]. There

are different variants of the VOF method and other approaches of tracking the

interface between two fluids, for a complete description of how these work and

their relative efficiencies, we refer the reader to [11] and the references there in,

mostly to the same authors. The part of our code that will not be found in [11]

handles the calculation of the extra-stress tensor using a semi-implicit scheme for

the Giesekus equation. This is our viscoelastic contribution.

The dissertation is organized in two parts named Parts I & II respectively. Part

I was originally designed to focus on the numerical validation of the fluid-fluid

systems but for chronological reasons, we found it necessary to also include the

(i) detailed outlines of the equations governing the motion of the flow system

including details of the Giesekus model which governs the behavior of the extra

stresses in the viscoelastic phase and (ii) development of the numerical code for

the simulation of the two flow systems of concern. Part II is mostly devoted to

numerical validations of the droplet-matrix system.

6

PART I

TWO LAYER FLOWS

Chapter 2

Governing Equations

We analyze the linear stability of two-layer shear flow of immiscible fluids, in

which at least one of the fluids is viscoelastic. The flow geometry is sketched

below with the lower fluid denoted Fluid 1 and the upper fluid represented by

Fluid 2.

Figure 2.1: Flow Schematics

Unless otherwise stated, we will herein use the subscript j to represent 1 or 2, with

the understanding that each of these indices describe quantities characteristic to

the corresponding fluid, e.g. ρ2 represents the density of Fluid 2 etc. Densities will

8

be denoted ρj, solvent viscosities ηsj, polymeric viscosities ηpj, total viscosities

µj = ηsj + ηpj and relaxation times λj. We also define βj = ηsj/µj, the viscosity

ratio m = µ1/µ2 and density ratio r = ρ1/ρ2. The lower and upper walls are

located at z∗ = 0 and z∗ = l∗ respectively, where the asterisks are used for

dimensional variables. In the basic flow;

• The upper plate moves with velocity (Up, 0) and the bottom plate is at rest.

• Fluid 1 occupies 0 ≤ z∗ ≤ l∗1 and Fluid 2 occupies l∗1 ≤ z ≤ l∗.

• The interfacial velocity is (U∗(l∗1), 0) and we denote U∗(l∗1) by Ui.

2.1 Dimensionless Governing Equations

The velocity, distance, time and pressure are made dimensionless with respect

to Ui, l∗, l∗/Ui, and ρ1U2i respectively. The extra stress components are scaled

the same as the pressure. Reynolds and Weissenberg numbers in fluid j are

denoted Rj = Uil∗ρj/µj and Wj = Uiλj/l

∗ respectively, where the requirement,

mR1 = rR2, should be satisfied.

For Couette (and Poiseuille) flows, there are 13 dimensionless parameters: a

Reynolds number, say R1, a Weissenberg number W1, the undisturbed depth l1

of fluid 1, a surface tension parameter T =(surface tension coefficient S∗)/(µ2Ui),

a Froude number F given by F 2 = U2i /gl∗ where g is the gravitational acceleration

constant, a dimensionless pressure gradient G = G∗l∗/(ρ1U2i ), the viscosity ratio

m, a density ratio r, the ratio of relaxation times w = λ1/λ2 = W1/W2, β1, β2,

κ1 and κ2. The physically relevant range is κj(1− βj)µj/(µ1R1) < 0.5, [7].

The dimensionless equation of motion is

∂u

∂t+ u · ∇u =

ρ1

ρj

(∇ ·T−∇p) + F +βj

Rj

∇2u, (2.1)

9

where F represents body forces (which includes gravity and interfacial tension

forces) and the total stress tensor is

~τ = −pI + T + (βj

R1

)(µj

µ1

)1

2

(∂uj

∂xi

+∂ui

∂xj

), (2.2)

where T is the extra stress tensor. The Giesekus model has the differential

constitutive relation

∂T

∂t+ (u · ∇)T− (∇u)T−T(∇u)T + κT2 +

T

Wj

= G0j(∇u + (∇u)T ), (2.3)

where the elastic modulus at time t = 0, G0 is

G0j =µj(1− βj)

µ1R1Wj

,

and κ is the Giesekus non-linear parameter. Taking κ ≡ 0 in the Giesekus model

leads to the Oldroyd-B model, which is what we are exclusively going to consider

as governing the viscoelastic phase in this work.

At the interface, the velocity and tangential stress are continuous, the jump in

the normal stress is balanced by surface tension and curvature, and the kinematic

free surface condition holds.

For the combined Couette-Poiseuille flow under the Oldroyd-B model, the dimen-

sionless basic velocity (U(z),0) is the same as the Newtonian case:

U(z) =

−GR1z

2/2 + c1z, 0 ≤ z ≤ l1,

−rGR2(z − 1)2/2 + c2(z − 1) + Up, l1 ≤ z ≤ 1,(2.4)

where

c1 = (1 + GR1l21/2)/l1, l2 = 1− l1, c2 = m(−GR1 + c1),

and the upper plate speed is

Up = 1 +ml2l1

− GR1ml22

.

The basic pressure field P is also the same as the Newtonian case and satisfies

dP/dx = −G and

dP

dz=

−1/F 2, 0 ≤ z ≤ l1,

−1/(rF 2), l1 ≤ z ≤ 1.(2.5)

10

We note that in our shear flow case, we will take G ≡ 0.

The basic extra stress tensor has the form

T =

C1(z) C2(z)

C2(z) 0

, (2.6)

where

C1(z) = 2(1− βj)

(µj

µ1R1

)Wj[U

′(z)]2, C2(z) = (1− βj)

(µj

µ1R1

)U ′(z). (2.7)

The basic shear stress condition is [[C2+(βjµj)U′/(R1µ1)]] = 0 at z = l1. Solutions

that are perturbations of the above basic flow are sought. The perturbations

to the velocity, pressure and interface position are denoted by (u, v), p and h,

respectively. The perturbation to the extra stress tensor is

T11 T12

T12 T22

. (2.8)

The equations of motion in each fluid yield

∂u

∂t+ U

∂u

∂x+ vU ′ +

ρ1

ρj

∂p

∂x− ρ1

ρj

(∂T11

∂x+

∂T12

∂z

)− βj

Rj

(∂2u

∂x2+

∂2u

∂z2

)

= −u∂u

∂x− v

∂u

∂z, (2.9)

∂v

∂t+ U

∂v

∂x+

ρ1

ρj

∂p

∂z− ρ1

ρj

(∂T12

∂x+

∂T22

∂z

)− βj

Rj

(∂2v

∂x2+

∂2v

∂z2)

= −u∂v

∂x− v

∂v

∂z, (2.10)

The constitutive equations for the Oldroyd-B fluid (κ1 = κ2 = 0) yield the

following coupled equations for the extra stress components:

T11 + Wj

(∂T11

∂t+ U

∂T11

∂x+ vC ′

1 − 2C1∂u

∂x− 2C2

∂u

∂z− 2T12U

′)

−(1− βj)

(2µj

µ1R1

)∂u

∂x

= 2Wj

(∂u

∂xT11 +

∂u

∂zT12

)−Wj

(u∂T11

∂x+ v

∂T11

∂z

), (2.11)

11

T12 + Wj

(∂T12

∂t+ U

∂T12

∂x+ vC ′

2 − C1∂v

∂x− T22U

′)

−(1− βj)

(µj

µ1R1

) (∂v

∂x+

∂u

∂z

)

= Wj

(∂v

∂xT11 +

∂u

∂zT22

)−Wj

(u∂T12

∂x+ v

∂T12

∂z

), (2.12)

T22 + Wj

(∂T22

∂t+ U

∂T22

∂x− 2C2

∂v

∂x

)− (1− βj)

(2µj

µ1R1

)∂v

∂x

= 2Wj

(∂v

∂xT12 +

∂v

∂zT22

)−Wj

(u∂T22

∂x+ v

∂T22

∂z

). (2.13)

Continuity is

∂u

∂x+

∂v

∂z= 0. (2.14)

The boundary conditions are u = v = 0 at z = 0, 1. The conditions at the

interface are posed at z = l1 + h(x, t). The unknown h(x, t) is assumed to be

small; the interfacial conditions are expanded as Taylor series about z = l1, and

retaining only the linear order terms.

The shear stress conditions are

[[t · τ · n]] = 0

where the unit tangent vectors are t = (1, hx)/√

1 + h2x, the normal is n =

(−hx, 1)/√

1 + h2x where [[x]] denotes x(fluid 1) - x(fluid 2).

The normal stress condition is

[[n · τ · n]] =T

mR1

hxx

[1 + h2x]

1/2, (2.15)

where the dimensionless interfacial tension parameter is T = S∗/(µ2Ui).

Continuity of velocity yields

h[[U ′]] + [[u]] = −h[[∂u

∂z]],

[[v]] = −h[[∂v

∂z]]. (2.16)

12

Continuity of shear stress yields

[[T12]]− ∂h

∂x[[C1]] + [[

βj

R1

µj

µ1

(uz + vx)]] = 0. (2.17)

Here, [[C2 + (βj/R1)(µj/µ1)U′]] = 0 from the base flow shear stress balance,

[[C ′2 + (βj/R1)(µj/µ1)U

′′]] = 0 from the base flow x-momentum equation, [[C ′2]] =

G(β1 − β2), [[P ]] = 0 in the basic flow, and ux + vz = 0 from incompressibility.

The balance of normal stress yields

[[T22]]− [[p]]T

mR1

hxx + h[[P ′]] + [[2βjµj

µ1R1

vz]] = 0. (2.18)

The kinematic free surface condition holds

2.2 Stability Analysis

For the linear stability analysis, we discard terms that are quadratic or higher in

the perturbations and seek normal mode solutions u, v, w, p, Tij and h which are

proportional to exp(iαx+σt), where σ denotes complex-valued eigenvalues which

are solved with the other parameters given. The details of the discretization, such

as the Chebyshev-tau scheme [?], are given in [25] and are not repeated here.

2.2.1 Bifurcation to traveling wave solution

At the onset of an instability of the base flow, the weakly nonlinear amplitude

equation admits modes proportional to exp(iαx). These modes interact to form

waves that travel in the x-direction. The methodology and notation are identical

to that of [25] and [26]. The details are summarized as follows. Let Φ represent the

set of unknowns (u, v, p, T11, T12, T22, h). The equations and boundary conditions

are represented in the schematic form

LΦ = N2(Φ, Φ) + N3(Φ, Φ, Φ), (2.19)

13

where the real linear operator L has the form A + Bd/dt and L(σ) = A +

σB. N2 contains quadratic terms and N3 contains cubic terms from the right

hand sides of the equations, boundary conditions and interfacial conditions. The

components f1, ..., f15 of N2 + N3 are written in Appendix of [25] for the upper-

convected Maxwell (UCM) liquid case and are modified here for the Oldroyd-

B case. The modifications emanate from the terms involving βj, namely the

base stress components, the Laplacian terms in the momentum equations, terms

from the symmetric part of the velocity gradient in the constitutive equations,

additional terms in the shear stress balance and normal stress balance. As in [25],

we denote the nonlinearities of the momentum equations in fluid 1 by f1, f2 those

of the constitutive equations in fluid 1 by f3, f4, f5, the corresponding notation for

fluid 2 is f6, . . . , f10, f11 for the continuity of u, f12 for the continuity of v, f13 for

the shear stress balance, f14 for the normal stress balance, f15 for the kinematic

condition. We use λ to denote the bifurcation parameter, which can be any of the

parameters, e.g. the Reynolds number or Weissenberg number. At λ = 0, there

is one eigenvalue, the interfacial mode, at σ = −ic, c > 0, for α = αc > 0 and

a corresponding eigenvalue σ = ic for α = −αc, and the rest of the eigenvalues

are stable (Re σ < 0). The eigenfunction with wavenumber α is denoted by

ζ(λ) and that with wavenumber −α by ζ(λ), where the overbar denotes the

complex conjugate. For λ > 0, −ic becomes −s(λ). The eigenfunction ζ satisfies

Aζ(λ) = s(λ)Bζ(λ). The adjoint eigenfunction with wavenumber α is denoted by

b(λ) and is calculated from the discretized matrix representations of the operators

A and B, by using the complex conjugate of the transpose of those matrices. The

normalization condition is (b, Bζ) = 1. ζ and b are proportional to exp iαx. On

the center manifold, the perturbation solution Φ can be decomposed as follows

Φ = Zζ + Zζ + Z2η + ZZχ + ZZχ + Z2η + higher order terms. (2.20)

14

Here, the Z(t) is the complex-valued amplitude function, χ represents the distor-

tion to the mean flow and η is the second harmonic.

(A− 2s(λ)B)η = N2(ζ, ζ), (2.21)

where, in actual computations, η is proportional to exp 2iαcx and λ is set to zero.

Similarly, the equation for the mean flow component χ simplifies to

Aχ = N2(ζ, ζ). (2.22)

The equations governing χ are detailed in Section (2.2.2).

The final equation for the amplitude function is

dZ

dt+ s(λ)Z = κ|Z|2Z, (2.23)

κ = (b, 4N2(ζ, χ) + 2N2(ζ, η) + 3N3(ζ, ζ, ζ)). (2.24)

This is the Suart-Landau equation and κ is the Stuart-Landau coefficient. If the

real part of κ is negative, then the bifurcating solution is supercritical and the

travelling wave solution would be stable for small amplitudes. If the real part of

κ is positive, then the bifurcating solution would be unstable.

To reconstruct the nonlinear waveform, we refer to the interface perturbation

component in Eq. (2.20), and use the component h in the eigenfunction ζ and

the second harmonic η. We may picture the total interface perturbation as

Φh = 2Re[Z(t)hζ exp(iαx + iImσt) + Z2(t)hη exp(2iαx + 2iImσt)], (2.25)

where we think of the Z(t) as an amplitude factor. The effect of the nonlinearity

can be exaggerated by choosing Z(t) to see the trend of whether the waves steepen

in the front or the back. The second harmonic term η contributes sin 2αx to the

interface shape.

15

2.2.2 Mean Flow Component

Since χ is not periodic in x, its component v satisfies dv/dz = 0 by incompress-

ibility. Since v = 0 at z = 0, 1, v = 0 in the entire domain. Denote the quadratic

terms on the right hand sides of the momentum and constitutive equations as be-

fore by f1, ..., f10 Denote the quadratic terms on the right hand sides of equations

(??)-(2.21)by f11 to f15, respectively. We note that f15 in N2(ζ, ζ) vanishes.

We carry out the formulation for χ keeping the pressure gradient in the x-direction

fixed in the entire nonlinear analysis. Putting d/dx = 0, d/dt = 0, and v = 0 in

the equations of motion, the components in χ satisfy:

−ρ1

ρj

∂T12

∂z− β

Rj

∂2u

∂z2= f1 or f6 of N2(ζ, ζ), (2.26)

ρ1

ρj

(∂p

∂z− ∂T22

∂z) = f2 or f7, (2.27)

T11 + W(− 2C2

∂u

∂z− 2T12U

′) = f3 or f8, (2.28)

T12 −WT22U′ − (1− β)

µ

µ1R1

∂u

∂z= f4 or f9, (2.29)

T22 = f5 or f10. (2.30)

u = 0 at z = 0, 1, (2.31)

h[[U ′]] + [[u]] = f11, (2.32)

[[T12 +βµ

µ1R1

∂u

∂z]] = f13, (2.33)

[[T22 − p]]− h[[P ′]] = f14. (2.34)

To preserve the given volumes of the fluids, we set h = 0 for χ. To fully determine

p requires an additional condition: we set p for fluid 1 equal to zero at z = l1.

The problem for the pressure decouples:

∂p

∂z=

∂f5

∂z+ f2 in fluid 1 (2.35)

∂p

∂z=

∂f10

∂z+

1

rf7 in fluid 2 (2.36)

[[p]] = [f5]z=l1 − [f10]z=l1 − f14, p2(l1) = 0. (2.37)

16

The problem for u and T12 consists of Eqs. (2.26), (2.29), (2.31) - (2.33). T11 is

then calculated from Eq. (2.28).

17

Chapter 3

Numerical Implementation

3.1 Volume Of Fluid (VOF) Method

We have developed a code for 2D simulation of two-layer shear flow of immiscible

fluids, named fluid 1 and 2, where either one or both fluids may be viscoelastic.

The Oldroyd-B model has been used for the viscoelastic phase. The code is based

on the Volume of Fluid (VOF) method and is described in detail in succeeding

sections. In the VOF method, the interface between the two fluids is not tracked

explicitly as in the case of interface markers. Instead, we use a volume fraction

field/function C:

C(t, x, z) =

1 in fluid 1,

0 in fluid 2,(3.1)

(also commonly referred to as the concentration, color or component indicator

function) to track the interface that is transported by the velocity field u = ui+vj:

∂C

∂t+ u · ∇C = 0. (3.2)

The interface, therefore, passes through the computational cells in which 0 < C <

1 and equation (3.2) allows for the calculation of density ρ, solvent and polymeric

viscosities (µs and µp), relaxation time λ and Giesekus non-linear parameter κ.

18

In fact, the average values for these quantities are interpolated as follows:

Ψ = Ψ1C + Ψ2(1− C), Ψ is ρ, µs, µp, λ or κ (3.3)

The VOF method is one of the most popular methods for tracking deformable

interfaces because it provides a simple way of treating the topological changes of

the interface, including viscous fingering between the two fluids.

3.1.1 Piecewise Linear Interface Calculation (PLIC)

In the volume of fluid (VOF) method, the location of the interface is approx-

imately represented by the volume fraction C of fluid 1 in the cell. We have

0 < C < 1 in cells cut by the interface and away from the interface, C = 0 in

fluid 2 and C = 1 in fluid 1.

Since we lose interface information when we represent the interface by a volume

fraction field, the interface needs to be reconstructed approximately in each cell.

In our code, the interface is reconstructed from the volume fraction field by the

Piecewise Linear interface calculation (PLIC) method, [11], [19]. The main idea

behind the interface reconstruction is to calculate the approximate normal ns to

the interface in each cell, since this determines one unique linear interface with the

volume fraction of the cell. At each time step, a plane curve is then constructed

within a cell where the volume fraction C satisfies, 0 < C < 1. This plane curve

divides the cell into two parts, the first corresponding to fluid 1 with volume

equal to the product of the value of C in the cell and the cell volume. The second

part corresponds to fluid 2 and has volume equal to the difference between the

cell volume and the volume of the first part. The coordinate of the intersection

points between the plane curve and the cell boundaries, xn, are calculated from

the equation for the tangent line with normal ns.

19

The normal to the interface ns and hence by extension the total interface cur-

vature κ are calculated from the mollified color function c(t, x, z) that varies

smoothly over a thickness h across the interface (as compared to the discontinu-

ous volume fraction function C used in [11]):

ns =∇c

‖∇c‖ , κ = ∇ · ns − [ns(ns · ∇)] · ns.

In general, the mollified color function is obtained by taking the convolution of

the volume fraction function C with a kernel K as:

c(t,x) =∫

SC(t,x′)K(x′ − x; h)dS′,

where x = (x, z), x′ = (x′, z′), dS′ = (dx′, dz′), S is the area (volume in 3D) of

the interfacial region (of radius h), and the kernel K(x′ − x; h) is the interpola-

tion function (mollifier), which decreases monotonically with increasing ‖x‖, is

equal to zero outside the interfacial region and is differentiable. In our code, the

interpolation function is selected in the form:

S(x′ − x; h) =

A(1− 1

4h2‖x′ − x‖2)4

, if ‖x′ − x‖ < 2h

0, if ‖x′ − x‖ ≥ 2h.(3.4)

Here A is a normalization constant to ensure∫S C(t,x)dS = 1 and h is mesh size.

Once the interface has been reconstructed, its motion by the flow field is modeled

via the topological equation (3.2). In our code, a Lagrangian form of this equa-

tion is used since the interface evolution is governed by a transport equation.

In particular, we calculate the advected coordinates of the intersection points

between the plane curve and cell boundaries, sequentially in x- and z-directions,

i.e. the new position of the interface is calculated using the formula:

xn+1 = xn + u(∆t),

where

u = ul

(1− xn

∆

)+ ur

xn

∆.

20

Here ∆ is ∆x or ∆z and ul and ur are the velocities on the left and right edges

of the cell respectively. Substituting the new coordinates into the equation of the

tangent line gives the updated normal to the interface in the cell.

We then check whether the interface has protruded at the neighboring cells,

and if so, we calculate the volumes moved into those cells using the relationship

between the volume fraction and the coordinates of intersection (if the normal is

known, the volume of fluid 1 is a function of these coordinates, [5]). The advected

volume of fluid 1 in the cell is found by using the same relationship and then by

subtracting the volume moved to the neighboring cells and adding the volume

that comes from them. The details of the PLIC method can be found in [5] and

[19].

3.1.2 Spartial Discretization and MAC Method

v(i,j+1)T12(i+1,j+1)

p, C, T11, T22u(i+1,j)

u(i,j)(i,j)

z

xT12(i+1,j)

v(i,j)

Figure 3.1: MAC cell

The momentum equations are finite differenced on a staggered Marker-and-Cell

21

(MAC) grid Figure (3.1), i.e.pressure p, volume fraction function C, and extra

stress components T11 and T22 are located at the center of a computational cell,

the velocity components u and v (v corresponds to w in 3D) are defined on its

edges and non-diagonal stress components T12 = T21 (or T13 in 3D) are located

on the corners (in 3D case u and w would be defined on the faces parallel to the x

and z directions respectively and T13 is defined on the top front-face edge parallel

to the y-axis).

The spartial derivatives for pressure and velocity in the momentum equations are

calculated using second-order central finite differences over a single mesh spacing,

where possible. For example, ∂p/∂x in (2.1) and ∂u/∂x in (2.3) are evaluated in

cell (i,k) as

∂p

∂x=

p(i, k)− p(i− 1, k)

∆x,

∂u

∂x=

u(i + 1, k)− u(i, k)

∆x.

However, the advective term u∂u/∂x is discretized as

u∂u

∂x= u(i, k)

u(i + 1, k)− u(i− 1, k)

2∆x. (3.5)

The basic idea in equation (3.5) is to weigh the derivatives by cell size such that

the correct order of approximation is maintained in a variable mesh. This type of

approximation is used in our code for all convective terms appearing in equation

(3.8).

3.2 Temporal Discretization, Projection Method

The two-fluid flow is modelled with the Navier-Stokes equations (2.1) which we

write here in the form:

ρ

(∂u

∂t+ u · ∇u

)= −∇p +∇ · (T + 2µsS) + F, (3.6)

where S is the viscous stress tensor:

Sij =1

2

(∂uj

∂xi

+∂ui

∂xj

),

22

and F the source term for the momentum equation, which includes the gravity

and interfacial tension forces. the velocity field is subject to the incompressibility

condition (2.14) which we restate in vector form as

∇ · u = 0. (3.7)

We proceed to solve equations (3.6) and (3.7) by decoupling the solution of (3.6)

from the solution of (3.7) by Chorin’s projection method [3]. In this projec-

tion method, the Navier-Stokes equations are first solved for an approximate u∗

without the pressure gradient:

u∗ − u(n)

∆t= −(un · ∇)un +

1

ρ∇ · (T(n) + 2µsS

(n)) + F(n), (3.8)

where n is the time-step index and u(n) is assumed known. In general, the

resulting flow filed u∗ does not satisfy the continuity equation. However, we

require that ∇ · u(n+1) = 0 and

un+1 − u∗

∆t= −∇p

ρ. (3.9)

Taking the divergence of equation (3.9) then leads to the Poisson equation for

pressure:

∇ ·(∇p

ρ

)=∇ · u∗

∆t, (3.10)

which is solved to find the pressure field. Next, u∗ is corrected by this pressure

field and the updated solution un+1 is found from equation (3.9). This algorithm

is easier to solve than the original fully coupled set of equations. We consider

no-slip boundary conditions in the z-direction and periodic boundary conditions

in the x-direction. It should be noted that in the marker-and-cell (MAC) grid, see

Section (3.1.2) below, the Neumann condition for pressure is automatically in-

volved in the numerical solution, i.e., no numerical boundary condition is needed.

23

3.2.1 Semi-Implicit Stokes Solver

To avoid the problem of viscous diffusion instability, which imposes strict re-

strictions on the time step size in the case of small Reynolds number, we use

the semi-implicit scheme, [11] to calculate the intermediate velocity, u∗, in the

Navier-Stokes equations:

u∗ − un

∆t= −(un · ∇)un +

1

ρ∇ ·T(n) + F(n)

= +1

ρ

∂

∂x(2µs

∂u∗

∂x) +

1

ρ

∂

∂zµs

(∂vn

∂x+

∂u∗

∂z

), (3.11)

v∗ − vn

∆t= −(un · ∇)vn +

1

ρ∇ ·T(n) + F(n)

= +1

ρ

∂

∂z(2µs

∂v∗

∂z) +

1

ρ

∂

∂xµs

(∂v∗

∂x+

∂un

∂z

). (3.12)

The semi-implicit scheme dictates that the terms with asterisks should be im-

plicit. It is precisely these terms that are responsible for viscous diffusion in-

stability, [31]. All other terms in the Navier-Stokes equations (with superscripts

(n)) are left in the explicit part. This can be expressed as

{I− ∆t

ρ

[∂

∂x

(2µs

∂

∂x

)+

∂

∂z

(µs

∂

∂z

)]}u∗ = explicit terms, (3.13)

{I− ∆t

ρ

[∂

∂z

(2µs

∂

∂z

)+

∂

∂x

(µs

∂

∂x

)]}v∗ = explicit terms. (3.14)

This procedure decouples the u component from the parabolic systems (3.11 &

3.12).

The above semi-implicit scheme was demonstrated in [11] to be unconditionally

stable.

As the full explicit scheme, this semi-implicit scheme is first order in precision,

[11]. Although it is easier to solve than the coupled system, it still requires the

24

inversion of a large sparse matrix. What makes the method very efficient is a

factorization technique, [40], that is applied to the left-hand side of equations

(3.13 & 3.14):

{I− ∆t

ρ

∂

∂x

(µs

∂

∂x

)} {I− ∆t

ρ

∂

∂z

(µs

∂

∂z

)}u∗ = explicit terms, (3.15)

{I− ∆t

ρ

∂

∂z

(µs

∂

∂z

)} {I− ∆t

ρ

∂

∂x

(µs

∂

∂x

)}v∗ = explicit terms. (3.16)

This factorization technique is also applied to the Oldroyd-B constitutive equa-

tions and so we will give more details in the next section.

In the VOF method, the boundary conditions at the interface cannot be applied

directly and the only way to take into account the force due to interfacial tension

is to include it into the Navier-Stokes equations as some body force. We use

the Continuous Surface Force (CSF) method, [1] for modeling interfacial tension

effects. In this method, the discontinuity present at the interface is smoothed arti-

ficially, i.e., interfacial tension is assumed to act everywhere within the transition

region (of finite thickness) between fluids 1 and 2.

3.2.2 Multigrid Method

The solution of the discreet counterpart of the Poisson equation (3.10) is the most

time consuming part of our Navier-Stokes solver and, consequently, an efficient

solution is crucial for the performance of the whole method. The performance of

some classic iterative methods, such as Gauss-Seidel, Cholesky incomplete factor-

ization (CIF) and preconditioned conjugate gradient (PCG) methods suffer from

the degradation of the convergence rate when the mesh size increases. More-

over, the system can be very ill-conditioned when a large density ratio of the

two fluids involved causes a sharp variation of the coefficients. Potentially, the

multigrid method is the most efficient method: to reduce the error to a constant

25

factor, the multigrid method needs a fixed number of iterations, whatever the

mesh size. The multigrid method achieves this convergence rate independent of

mesh size by combining two complementary algorithms: one iterative method

to reduce the high-frequency error and one course grid correction step to elimi-

nate low-frequency error [11]. In our code, the Poisson equation for pressure is

solved by the multigrid method as in [11], with the two-color (four-color for 3D

case) Gauss-Seidel method for iteration and the Galerkin method for course grid

correction.

3.3 Modeling interfacial tension force (CSF)

In the VOF method, interfacial tension is posed as a body force over the interfacial

cells. In the continuous surface force (CSF) formulation, which is used in our

algorithm, the body forces in the Navier-Stokes equations include the interfacial

tension force Fs which in turn is approximated by

Fs = σκnsδs,

where as usual, σ is the interfacial tension, κ is the total interface curvature,

ns is the normal to the interface and δs = |∇c| where C is the volume fraction

function.

An alternative way to model the interfacial tension force would be to use is the

continuous surface stress (CSS) formulation, in which

Fs = ∇ ·T = σδsκns,

and

T = [(1− ns ⊗ ns)σδs].

At the continuum level both methods are equivalent.

26

Renardy et. al., [28] recently demonstrated elimination of spurious current by

applying a parabolic reconstruction of the surface tension (PROST) formulation

instead of the above two formulation.

3.4 Oldroyd-B Constitutive Equation

The Oldroyd-B constitutive equation is implemented into the code to capture

viscoelasticity of the fluid(s). This equation is for the extra stress tensor T, which

represents the polymer contribution to the shear stress: the shear stress tensor

τ = 2µsS + T, where 2µsS is the Newtonian part of the stress tensor due to the

solvent. We also recast the Oldroyd-B model in terms of S = [(∇u) + (∇u)T ]/2

and the relaxation time λ = Wl∗/Ui:

λ

[∂T

∂t+ (u · ∇)T− (∇u)T−T(∇u)T

]+ T = 2µpS, (3.17)

where µp is the polymer viscosity. The term (u · ∇)T is the advection part

of the constitutive equation and (∇u)T − T(∇u)T can be considered as the

contravariant part of the constitutive equation because it appears due to the

use of the contravariant time derivative. If µs = 0, this model reduces to the

Upper-Convected Maxwell (UCM) model.

The major problem with numerical integration of differential constitutive equa-

tions is the numerical instability caused by the advection term. This instability

is generated if the advection term is treated explicitly. As a result, the explicit

schemes can only be used for very small relaxation times. Indeed, if we consider

the differential equation

λ

[∂T

∂t+ (u · ∇)T

]+ T = 2µpS,

on the MAC mesh and treat the advection term explicitly (the remaining terms

in the left-hand side being implicit), the Von-Neumann stability analysis will give

27

the following stability condition:

λ ≤ ∆t

−1 +√

1 + 9(CFL)2.

Here CFL is the Courant-Freidrich-Levy number. the right-hand side of this

condition tends to infinity only at negligibly small viscosities or if ∆t is zero, which

is impossible. The advection instability does not disappear if the contravariant

terms are added. It should be noted that the contravariant terms should always

be in the explicit part.

The only way to avoid the advection instability is to treat the advection terms

of the Oldroyd-B constitutive equation implicitly. In this case, the numerical

scheme will be unconditionally stable, i.e., it can be used for any value of the

relaxation time.

We have also developed a semi-implicit scheme for the more general Giesekus

constitutive equations. according to this scheme, the advection terms and the

last term in the left-hand side of equation (3.17) are treated as implicit, but

the contravariant and nonlinear terms are in the explicit part. The Giesekus

constitutive equation in the semi-implicit scheme can then be expressed as

λ

(1 + ∆t + u(n)∆t

∂

∂x+ v(n)∆t

∂

∂z

)T(n+1) = explicit terms. (3.18)

Stability analysis shows that on the MAC mesh the semi-implicit scheme is uncon-

ditionally stable. However, it is extremely time consuming to solve the resulting

discretized equations directly because this would necessarily require the inversion

of a large sparse matrix. An efficient way would be to use a factorization tech-

nique to the left hand side of (3.18) similar to the one described in the previous

section. In this case, we rewrite equation (3.18) as

(λ+∆t)

(1 +

λ

λ + ∆tu(n)∆t

∂

∂x

) (1 +

λ

λ + ∆tv(n)∆t

∂

∂z

)T(n+1) = explicit terms.

(3.19)

28

The error in the above factorization depends on the term

λ2

λ + ∆tun · v(n)(∆t)2,

and hence if λ is much larger than ∆t, the error of the factorization would be

O[λ(‖un‖max∆t)2]. This error is even smaller (approximately half of that just

given) if λ is of order ∆t.

The inversion of the left-hand side of equation (3.19) requires solving only two

tri-diagonal matrices, thereby greatly reducing computation time and memory.

Hence, the semi-implicit scheme for the Giesekus model used in our code is suf-

ficiently fast and works for any value of the relaxation time.

The remaining chapters in Part I will be devoted to numerical validation of our

code against such works as those of [29] (Chapters 4 - 5) and [38] (Chapter 6).

29

Chapter 4

Numerical Validation I

In the next section, we run a series of basic checks, based mainly on residual

information of the perturbation velocities, extra stresses and their derivatives

against corresponding theoretic values. Based on the magnitude of the relatives

errors obtained from such residual data, we can then determine whether or not our

code is performing as expected. Having successfully completed this calibration

process, the subsequent sections will then focus on computation of perturbation

amplitudes, velocities and extra-stresses using the numerical data of [29], and

matching them against expected linear theoretic results both at and away from

the interface.

4.1 Calibrating The Code

Let Φ represent any of the perturbation quantities u, v, T11, T12 or T22. In physical

terms, we have:

Φ = <{AUmei(αx+σt)}, (4.1)

30

where A is the initial amplitude of the disturbance, U = Ur+iUi and the mapping

m is given by,

m =

c, if Φ is u or v,

1, if Φ is Tij.

Here c = l1/r in the Newtonian fluid (fluid 1) and c = l2/(1−r) in the viscoelastic

fluid, where r = [l1 + A cos(αx)].

The remainder of this section will be devoted to investigating the relative errors

between:

• the numerical values of Φ (ΦN) computed using our code at specific points

(or regions) in the flow domain with the corresponding results expected

from the linear theory (ΦL) and

• numerical time derivatives of Φ with the linear theoretic values of ∂Φ/∂t.

In particular, we need to compute:

relative error in Φ =|ΦN − ΦL||ΦL| and (4.2)

relative derivative error =

∣∣∣∂Φ∂t− [Φ]

Nτ

∣∣∣∣∣∣∂Φ

∂t

∣∣∣, (4.3)

where [Φ] is the difference in corresponding values between N time steps, i.e.

[Φ] = ΦN+i − Φi, i = 1, 2, 3, . . . , size(Φ)−N.

We will mostly consider N = 1.

The derivatives in equation (4.3) given by:

∂Φ

∂t= eσrt{(σrBr − σjBi) cos(σjt + αx)− (σrBi + σjBr) sin(σjt + αx)},

where Br = AUrm, Bi = AUim, σ = σr + iσj denotes complex valued eigenvalues

and α is the wavenumber.

31

The graphs of relative error are plotted in both the Newtonian and viscoelastic

fluids, nodes (130,100) and (130,220) respectively on a 256×256 mesh using the

numerical data on page 92 of [29], except in this case we used wavenumber, α = 6.

Since our code correctly gives zero values for extra-stresses in the Newtonian fluid

but Matlab however computes these to order of O(−21), we will in turn for the

Newtonian fluid part simply graph the more illustrative differences |TijL − TijN |and |∂Tij/dtL − ∂Tij/dtN | instead of the relative errors which would otherwise

be constant graphs at the value one. We also do the same for the error of dv/dt

in viscoelastic fluid since in this case the linear values are very small at some

time-steps.

4.1.1 Velocities and Extra-stresses

a) 0 1000 2000 3000 4000 5000 60000

0.5

1

1.5

2

2.5

3

3.5x 10

−4

time step

relati

ve u

at (13

0,100

)

b) 0 1000 2000 3000 4000 5000 60000

2

4

6

8x 10

−4

time step

relativ

e u at

(130

,220)

c) 0 1000 2000 3000 4000 5000 60000

0.005

0.01

0.015

0.02

0.025

time step

relati

ve v

at (13

0,100

)

d) 0 1000 2000 3000 4000 5000 60000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time step

relati

ve v

at (13

0,220

)

e) 0 1000 2000 3000 4000 5000 60001.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

−22

time step

rel. T

11 at

(130

,100)

f) 0 1000 2000 3000 4000 5000 60000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

time step

rel. T

11 at

(130

,220)

g) 0 1000 2000 3000 4000 5000 60001.85

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35x 10

−21

time step

rel. T

12 at

(130

,100)

h) 0 1000 2000 3000 4000 5000 60000

0.002

0.004

0.006

0.008

0.01

0.012

time step

rel. T

12 at

(130

,220)

32

i) 0 1000 2000 3000 4000 5000 60001.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

−22

time step

rel. T

22 at

(130

,100)

j) 0 1000 2000 3000 4000 5000 60000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

time step

rel. T

22 at

(130

,220)

Figure 4.1: Residual data for velocities and extra-stresses.

Figure (4.1) shows the graphical results of the relative errors obtained via equa-

tions (4.2). The left hand column represents the Newtonian fluid results and the

right hand column, the corresponding viscoelastic quantities. In particular, (a) &

(b) illustrate the relative errors in the perturbation u-velocity whereas (c) & (d);

(e) & (f); (g) & (h) and (i) & (j) show similar computations for the perturbations:

v, T11, T12 and T22 respectively.

4.1.2 Time derivatives

Figure (4.2) is configured similarly to figure (4.1) except it now shows the plots

of the relative errors in the time derivatives as given by equations (4.3).

a) 0 1000 2000 3000 4000 5000 60000

0.05

0.1

0.15

0.2

0.25

time step

relati

ve du

/dt at

(130

,100)

b) 0 1000 2000 3000 4000 5000 60000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time step

relati

ve du

/dt at

(130

,220)

c) 0 1000 2000 3000 4000 5000 60000

0.5

1

1.5

2

2.5

time step

relati

ve dv

/dt at

(130

,100)

d) 0 1000 2000 3000 4000 5000 60000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time step

relati

ve dv

/dt at

(130

,220)

e) 0 1000 2000 3000 4000 5000 60005

5.5

6

6.5

7

7.5

8

8.5

9

9.5x 10

−21

time step

rel. d

T11/d

t at (1

30,10

0)

f) 0 1000 2000 3000 4000 5000 60000

0.01

0.02

0.03

0.04

0.05

0.06

time step

rel. d

T11/d

t at (1

30,22

0)

33

g) 0 1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−20

time step

rel. d

T12/d

t at (1

30,10

0)

h) 0 1000 2000 3000 4000 5000 60000.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

time step

rel. d

T12/d

t at (1

30,22

0)

i) 0 1000 2000 3000 4000 5000 60005

5.5

6

6.5

7

7.5

8

8.5

9

9.5x 10

−21

time step

rel. d

T22/d

t at (1

30,10

0)

j) 0 1000 2000 3000 4000 5000 60000.016

0.018

0.02

0.022

0.024

0.026

0.028

time step

rel. d

T22/d

t at (1

30,22

0)

Figure 4.2: Residual data for time derivatives.

Both figures (4.1)and (4.2) illustrate that the relative errors between the nu-

merical results obtained from our code and the corresponding analytic results

from from the linear theory are well within acceptably small levels. We can thus

conclude from these that our code performs reasonably accurately (i.e. was pro-

grammed correctly) and so we can proceed to use it in computing growth rates

of perturbation quantities.

For the case under consideration, such computations would require require the

following adjustment to the data we will need to use in the stability and VOF

codes.

Based on the data from [29], the Reynolds number for the lower (Newtonian)

fluid calculates as R1 = 0.00084808055. However, such a small R1 value produces

junk in the VOF code with the ratio T/Tvis ≈ 2000, so we decided to increase

R1 a thousand times to 0.84808055. It then naturally follows to also change those

parameters from [29] whose values depend on R1.

The relation R1 = Uil∗ρ1/µ1 implies that, in order to increase R1 as mentioned

above, the viscosity for the lower fluid, µ1, in turn needs to be scaled by a factor

of 0.001 to µ1 = 0.223. We similarly scale the upper fluid viscosity, µ2, to 0.2345

and hence the corresponding reynolds number there, R2, changes to 0.803825.

34

We also had to run the code using a Weissenberg number for fluid 2, that was

10 times less that the original value in [29], hence the need to also reduce the

corresponding relaxation time λ2 to 0.17. The following tables summarize the

changes in the relevant data of [29]:

Table 4.1: Properties of fluids:

Fluid λ ρ µ β

Silicone oil 0 0.908 0.223 1

Boger fluid 0.17 0.905 0.2345 0.84

Table 4.2: Conditions of experiments:

ε U0 d1 λ1U0/d1 ρ1U0d1/µ1

2.700 0.41 0.1373 5.08 0.00022

We write Ui = U0 for the velocity at the interface. Since the depth fraction

ε = d2/d1 then the dimensional depth of the lower fluid d2 = εd1 = 0.37071. Using

the notation in our code, we have l∗1 = d2 = 0.37071, l∗2 = d1 = 0.1373, l∗ =

d1 + d2 = 0.50801, and l1 = hgtliq = l∗1/l∗ = 0.72972973. Furthermore, the

dimensional interfacial tension is S∗ = 8.5 and in the same units (cm, s, etc.),

the gravitational acceleration is g = 980. Using these, we can also calculate the

changes in the other values, most notably (i) the non-dimensional surface tension

parameters ,T and σ, since T depends on µ2 and σ in turn depends on T , see

below. Lastly because the ratio R/F 2 has to remain constant at its value in [29],

F 2 (and hence the non-dimensional gravity) had to be scaled similarly as with

the µ’s. In summary we have:

viscj =βjµj

µ1R1

⇒ visc1 = 1.1791333 & visc2 = 1.0415501, (4.4)

Up = 1 +µ1l

∗2

µ2l∗1= 1.3522072, (4.5)

35

Wj =Ui

l∗λj ⇒ W1 = 0 & W2 = 0.137202, (4.6)

m =µ1

µ2

= 0.9509595, (4.7)

r =ρ1

ρ2

= 1.00331, (check mR1 = rR2 , yes = 0.80649), (4.8)

T =S∗

µ2Ui

= 88.4081, (4.9)

gravity =1

F 2=

gl∗

1000U2i

= 2.9616288, (4.10)

sigma =T

mR1

= 109.621, (4.11)

G0j =(1− βj)µj

µ1R1Wj

⇒ G01 = 0 & G02 = 1.4459736. (4.12)

4.2 Growth rates of amplitudes and velocities

4.2.1 Growth Rates at the Interface

In Figure (4.3), we plot the logarithms of the maximum perturbation amplitude

a)

0 0.01 0.02 0.03 0.04 0.05 0.06−7.4

−7.2

−7

−6.8

log

h

0 0.01 0.02 0.03 0.04 0.05 0.06−4.4

−4.2

−4

−3.8

−3.6

log

vma

x

0 0.01 0.02 0.03 0.04 0.05 0.06−5.4

−5.2

−5

−4.8

−4.6

log

vL2

time b)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.915

−6.91

−6.905

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.885

−4.88

−4.875

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.575

−4.57

−4.565

−4.56

log

vL2

time

c)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.911

−6.91

−6.909

−6.908

−6.907

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.59

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.677

−4.676

−4.675

−4.674

−4.673

log

vL2

time d)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.9105

−6.91

−6.9095

−6.909

−6.9085

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.6495

−6.649

−6.6485

−6.648

−6.6475

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.255

−5.2545

−5.254

−5.2535

−5.253

log

vL2

time

36

e)

0 0.01 0.02 0.03 0.04 0.05 0.06−6.925

−6.92

−6.915

−6.91

−6.905

log

h

0 0.01 0.02 0.03 0.04 0.05 0.06−8.24

−8.23

−8.22

−8.21lo

gvm

ax

0 0.01 0.02 0.03 0.04 0.05 0.06−6.26

−6.24

−6.22

−6.2

log

vL2

time f)

0 0.01 0.02 0.03 0.04 0.05 0.06−6.9135

−6.913

−6.9125

−6.912

logh

0 0.01 0.02 0.03 0.04 0.05 0.06−11.6

−11.4

−11.2

−11

logv

max

0 0.01 0.02 0.03 0.04 0.05 0.06−8.4

−8.2

−8

−7.8

logv

L2

time

Figure 4.3: log(h), log(vmax), log(‖v‖2). respectively

h, the maximum v-velocity and the L2-norm of v. This is done at different values

of the wavenumber α.

Figures (a) - (f) represent the cases α = 6, 3, 2, 1.5, 1 and 0.474 respectively.

In-order to reduce the computational time to acceptable levels there was a need

to use different mesh grids for some of the calculations; a 258 × 258 mesh grid

for the cases (a) - (c), a 66× 66 mesh for (d) & (e) and lastly we used a 34× 34

grid for the case (f).

The graphs of figure (4.3) all correspond to negative decay rates and we expected

all these cases to be stable under the given perturbations. We next show that by

increasing the initial perturbation by about a hundred times, such perturbations

remain stable and evolve to a flat interface.

4.2.2 Waveforms

In figure (6.3) we took the parameters corresponding to figure (4.3a) and instead

put in a perturbation that corresponds to 10% of the fluid depth (i.e. initial per-

turbation h = 0.1) and we let it evolve in time. We notice that the perturbation

evolves to a flat interface, showing that this is a stable case as are all the other

cases in this section.

37

(a)

Wed Nov 10 21:52:53 2004

0 10

1

PLOT

X−Axis

Y−

Axi

s

0 10

1

(b)

Wed Nov 10 21:59:02 2004

0 10

1

PLOT

X−Axis

Y−

Axi

s

0 10

1

(c)

Wed Nov 10 22:02:15 2004

0 10

1

PLOT

X−Axis

Y−

Axi

s

0 10

1

(d)

Mon Nov 15 06:08:31 2004

0 10

1

PLOT

X−Axis

Y−

Axi

s

0 10

1

Figure 4.4: Wave forms at (a) t = 0.01, (b) t = 0.1, (c) t = 0.3 and (d) t = 1

4.2.3 Effects of cell shape

We notice from figure (4.3) that our results regress from excellent (around α = 6)

to awful as the wavenumber, α, gets smaller. This behavior can be explained in

terms of the shape of the computational cells.

A possible explanation for the bad results for the longer wavelength disturbances

(these correspond to the results at very low wave numbers above) maybe due

to the fact that in the linear stability calculations, we redefined the x-direction

length scale Lx to be 2π/α. This means, say for the case α = 0.474, that we have

Lx = 13.25566520502023 which, combined with Lz = 1 and a uniform 34 × 34

mesh makes the computational cell distorted from a square. The finite difference

38

spatial resolution works much better when the cell is square.

The following graphs computed at the wavenumber α = 2 show that we indeed

get better results by forcing the computational cells to be close to squares.

Figure (4.5a) gives a comparison of the expected linear results (shown in red)

with the numerical growth of logarithms of maximum interface height, h (first

row), maximum vertical velocity at the interface, v (second row) and the L2 norm

of maximum v in the third row. The first, second and third columns correspond

to the meshes 258× 66, 258× 130 and 258× 258 respectively.

Figure (4.5b) shows in absolute values the arithmetic difference (error estimates)

between the linear and numerical results. Since all three mesh grids give good

agreement with linear theory for log(hmax), figure (4.5b) only looks at the re-

maining two cases (i.e. the log(v) and log(L2) cases). Of the three mesh grids,

the errors for 258 × 130 (green) are closest to zero, showing that the 258 × 130

mesh, in which the computational cells are closest to square while the mesh is

not too rough, would give the better results.

a)

0 0.5 1

x 10−3

−6.911

−6.91

−6.909

−6.908

−6.907

log

h

258x66

0 0.5 1

x 10−3

−6.911

−6.91

−6.909

−6.908

−6.907258x130

0 0.5 1

x 10−3

−6.911

−6.91

−6.909

−6.908

−6.907258x258

0 0.5 1

x 10−3

−5.592

−5.591

−5.59

−5.589

−5.588

log

vm

ax

0 0.5 1

x 10−3

−5.592

−5.591

−5.59

−5.589

−5.588

0 0.5 1

x 10−3

−5.59

0 0.5 1

x 10−3

−4.654

−4.653

−4.652

−4.651

−4.65

log

vL

2

0 0.5 1

x 10−3

−4.669

−4.668

−4.667

−4.666

−4.665

0 0.5 1

x 10−3

−4.677

−4.676

−4.675

−4.674

−4.673

b)

0 0.2 0.4 0.6 0.8 1

x 10−3

0

0.5

1

1.5

2x 10

−3

log

vm

ax

258x66258x130258x258

0 0.2 0.4 0.6 0.8 1

x 10−3

0

2

4

6

8x 10

−4

log

l2n

orm

258x66258x130258x258

Figure 4.5: Effects of cell shape for the case α = 2.

39

4.2.4 Effects of Surface Tension

We next want to make sure the code works for interfacial distortions. This would

mean that we need to basically get surface tension down very low.

0 10 20 30 40 50 60 70 80 90−7

−6

−5

−4

−3

−2

−1

0

Surface Tension

Re(sig

ma)

Figure 4.6: <(σ) vs. Tension, T .

Figure (4.6) shows a plot of the decay rate (the real part of the most unstable

mode/eigenvalue) on the vertical axis against surface tension (horizontal axis).

The graph shows that the decay rate gets progressively less negative (unstable)

as surface tension decreases.

We next look at the effect of surface tension on the interfacial growth of maximum

amplitude, v-velocity and L2-norm of v.

T = 88.4081 T = 66.3061

0 0.01 0.02 0.03 0.04 0.05 0.06−7.4

−7.2

−7

−6.8

log

h

0 0.01 0.02 0.03 0.04 0.05 0.06−4.4

−4.2

−4

−3.8

−3.6

log

vma

x

0 0.01 0.02 0.03 0.04 0.05 0.06−5.4

−5.2

−5

−4.8

−4.6

log

vL2

time

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.915

−6.91

−6.905

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.25

−4.245

−4.24

−4.235

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.16

−5.155

−5.15

−5.145

log

vL2

time

40

T = 44.204 T = 17.6816

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.915

−6.91

−6.905

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.585

−4.58

−4.575

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.585

−5.58

−5.575

log

vL2

time

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.915

−6.91

−6.905

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.01

−5.005

−5

−4.995

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.855

−5.85

−5.845

log

vL2

time

T = 0.884081 T = 0.0884081

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.909

−6.9085

−6.908

−6.9075

−6.907

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−7.0595

−7.059

−7.0585

−7.058

−7.0575

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−7.9005

−7.9

−7.8995

−7.899

−7.8985

log

vL2

time

0 0.2 0.4 0.6 0.8 1

x 10−3

−9.2105

−9.2105

−9.2104

−9.2104

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−11.731

−11.73

−11.729

−11.728lo

gvm

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−12.5767

−12.5766

−12.5766

−12.5765

−12.5765

log

vL2

time

Figure 4.7: Effects of surface tension

The fact that the VOF scheme is erratic as we make T smaller, see figure (4.7)

could be simply from the fact that we’re simulating a Newtonian interfacial mode

at low Re, see the explanations given under figure (4.8). So we just won’t go to

low Reynolds numbers and Newtonian: that’s the territory we will stay out of.

If it’s Newtonian, we need higher Reynolds numbers for the semi-implicit scheme

to handle the time integration, [11].

However, at low Re, we will show that the viscoelastic modes are well-reproduced.

Hence we won’t need to take Reynolds numbers of order 100 to do viscoelastic

work. Consider the following mode:

30 0 0 |nf,ibk,ipri Couette

41

0.84808055 0.0 |re,grad

1 0.729729 0 0 0 0 0 0 0 0 0 |irds,L1s

1 0.9509595 0 0 0 0 0 0 0 0 0 |mvis,bvis

0.88408 1.00331 2.9616288 0.1372 |t ,r,fr,we2

1 0.0 0 0 0 0 0 0 0 0 0 |mweiss, weiss1

1.0 0.84 |beta(1),beta(2)

With thirty Chebyshev polynomials, the 194th eigenvalue σ = −7.0096−6.53066i

is a converged viscoelastic mode. The computation using twenty Chebyshev

points has 126th eigenvalue as −6.99362 − 6.51317i). The interfacial mode is

−0.8331− 6.009i see fig below:

−9 −8 −7 −6 −5 −4 −3 −2 −1 0−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5

Re(sigma)

Im(sigm

a) converged mode

interfacial mode

nchp=30nchp=20

Figure 4.8: Spectrum of eigenvalues for α = 6.

Figure (4.8) gives the plot of the imaginary part of the eigenvalue, =(σ), (ver-

tical axis) against the real part of σ. We have the continuous spectrum along

<(σ) = −1/(βWe) = −8.6768 and a second continuous spectrum along <(σ) =

−1/We = −7.2885. Figure (4.8) also shows that there has been a ’cross-over’ of

the least stable mode from a viscoelastic mode to a Newtonian interfacial mode

between the T = 88.4081 and 0.884081 cases. i.e. at T = 88.4081, the least stable

mode has a flat interface and the mode that distorts the interface is much more

stable than that. At T = 0.884081, the least stable mode is associated with one

42

that can distort the interface with waves, while the viscoelastic modes are much

more stable.

Using the viscoelastic mode in our VOF code produces results which closely match

the linear decay. These results prove to be much better than those obtained using

the interfacial mode, see fig below with T = 0.884081 for both cases.

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.909

−6.9085

−6.908

−6.9075

−6.907

logh

0 0.2 0.4 0.6 0.8 1

x 10−3

−7.0595

−7.059

−7.0585

−7.058

−7.0575

logv

max

0 0.2 0.4 0.6 0.8 1

x 10−3

−7.9005

−7.9

−7.8995

−7.899

−7.8985

logv

L2

time

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.915

−6.91

−6.905

logh

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.575

−4.57

−4.565

−4.56

logv

max

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.43

−5.425

−5.42

−5.415

logv

L2

time

Figure 4.9: (a) <(σ)=-0.83306 (b) <(σ)=-7.00952

Figs. (4.9) (a) & (b) illustrate the behavior of the growth rate when using the

interfacial and viscoelastic modes respectively. The viscoelastic mode clearly

gives better results.

However, this viscoelastic mode doesn’t distort the interface so we still can’t

check the code for the coupling of interface distortion to the (probably correct)

viscoelastic equations!

4.2.5 Growth rates away from Interface

a) 0 1000 2000 3000 4000 5000 6000−5

−4.98

−4.96

−4.94

−4.92

−4.9

−4.88

−4.86

−4.84

−4.82

time step

log(

v) a

t (13

0,10

0)

NumericalLinear

d) 0 1000 2000 3000 4000 5000 6000−0.638

−0.637

−0.636

−0.635

−0.634

−0.633

−0.632

−0.631

−0.63

time step

log(

u) a

t (13

0,10

0)

NumericalLinear

43

b) 0 10 20 30 40 50 60 70 80 90 100−5.294

−5.292

−5.29

−5.288

−5.286

−5.284

−5.282

time step

log(

v) a

t (13

0,10

0)

NumericalLinear

e) 0 10 20 30 40 50 60 70 80 90 100−0.6385

−0.6385

−0.6385

−0.6385

−0.6385

−0.6384

−0.6384

−0.6384

−0.6384

time step

log(

u) a

t (13

0,10

0)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−5.945

−5.944

−5.943

−5.942

−5.941

−5.94

−5.939

−5.938

time step

log(

v) a

t (13

0,10

0)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 100−0.6401

−0.6401

−0.6401

−0.6401

−0.6401

−0.6401

−0.6401

time step

log(

u) a

t (13

0,10

0)

NumericalLinear

Figure 4.10: log(v) & log(u) in the Newtonian fluid.

a) 0 1000 2000 3000 4000 5000 6000−4.6

−4.55

−4.5

−4.45

−4.4

−4.35

time step

log(v)

at (1

30,22

0)

NumericalLinear

d) 0 1000 2000 3000 4000 5000 6000

0.15

0.151

0.152

0.153

0.154

time step

log(u)

at (1

30,22

0)

NumericalLinear

b) 0 10 20 30 40 50 60 70 80 90 100−5.77

−5.768

−5.766

−5.764

−5.762

−5.76

−5.758

time step

log(v)

at (1

30,22

0)

NumericalLinear

e) 0 10 20 30 40 50 60 70 80 90 1000.1501

0.1501

0.1501

0.1501

0.1501

0.1501

0.1501

0.1501

0.1501

0.1501

0.1502

time step

log(u)

at (1

30,22

0)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−6.611

−6.61

−6.609

−6.608

−6.607

−6.606

−6.605

−6.604

−6.603

−6.602

time step

log(v)

at (1

30,10

0)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 1000.1496

0.1496

0.1496

0.1496

0.1496

0.1496

0.1496

0.1496

0.1496

0.1496

0.1496

time step

log(u)

at (1

30,22

0)

NumericalLinear

Figure 4.11: log(v) & log(u) in the Viscoelastic fluid.

Figures (4.10) and (4.11) show the growth rates of perturbation velocities in the

Newtonian and viscoelastic fluids respectively at different wavenumbers. These

44

where computed away from both the interface and the walls at the points (130,100)

and (130,220) respectively on a 258× 258 grid. In particular figures labelled (a),

(b) and (c) are plots of the logarithm of the perturbation velocity v obtained

using α = 6, 3, 2 respectively. Those labelled (d), (e) and (f) are similarly defined

for u. we notice that even though the case α = 6 remains the most appealing,

the overall results have deteriorated in accuracy as compared to those at the in-

terface, see figure (4.3). In the subsequent sections, we show that this is not the

case at all when we consider the growth rates of the extra-stresses.

4.2.6 Maximum growth along a horizontal line

a) 0 20 40 60 80 100 120 140 160 180 200−6

−4

−2

0

2

4

6

8

time step

log(

vmax

) alo

ng (i

,100

)

NumericalLinear

d) 0 20 40 60 80 100 120 140 160 180 200−2

0

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,100

)

NumericalLinear

b) 0 20 40 60 80 100 120 140 160 180 200−6

−4

−2

0

2

4

6

8

time step

log(

vmax

) alo

ng (i

,100

)

NumericalLinear

e) 0 20 40 60 80 100 120 140 160 180 200−2

0

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,100

)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−6

−4

−2

0

2

4

6

time step

log(

vmax

) alo

ng (i

,100

)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 100−2

0

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,100

)

NumericalLinear

Figure 4.12: log(vmax) & log(umax) along a horizontal line in Newtonian fluid.

45

a) 0 20 40 60 80 100 120 140 160 180 200−6

−4

−2

0

2

4

6

8

time step

log(

vmax

) alo

ng (i

,220

)NumericalLinear

d) 0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,220

)

NumericalLinear

b) 0 20 40 60 80 100 120 140 160 180 200−6

−4

−2

0

2

4

6

time step

log(

vmax

) alo

ng (i

,220

)

NumericalLinear

e) 0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,220

)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

time step

log(

vmax

) alo

ng (i

,220

)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,220

)NumericalLinear

Figure 4.13: log(vmax) & log(umax) along a horizontal line in Viscoelastic fluid.

Figures (4.12) and (4.13) show the maximum growth rates of perturbation veloc-

ities in the Newtonian and viscoelastic fluids respectively at different wavenum-

bers. These where computed away from the interface along the horizontal lines

(i, 100) and (i, 220), for the Newtonian and Viscoelastic fluids respectively, and

using a 258 × 258 grid, where i = 1, 2, 3, · · · , 258. In particular figures labelled

(a), (b) and (c) are plots of the logarithm of the perturbation velocity v obtained

using α = 6, 3, 2 respectively. Those labelled (d), (e) and (f) are similarly defined

for u. we will later show, as mentioned in the previous sub-section, that the

results for the growth rates of the extra-stresses are much better.

46

4.3 Growth rates of Extra-stresses

4.3.1 Growth at the interface

As in figure (4.3), figure (4.14) below shows the growth rates of maximum pertur-

bation amplitudes and extra-stresses (instead of velocities).The behavioral pat-

terns illustrated here are similar to those in figure (4.3) and can thus be similarly

explained in terms of the connections between the wavenumbers and the compu-

tational cell shapes etc.

a)

0 0.5 1 1.5 2 2.5

x 10−3

−6.94

−6.92

−6.9

logh

0 0.5 1 1.5 2 2.5

x 10−3

3.05

3.06

3.07

3.08

logT

11m

ax

0 0.5 1 1.5 2 2.5

x 10−3

1.3

1.32

logT

12m

ax

0 0.5 1 1.5 2 2.5

x 10−3

0.55

0.56

0.57

logT

22m

ax

time b)

0 0.5 1 1.5 2 2.5

x 10−3

−6.94

−6.92

−6.9

logh

0 0.5 1 1.5 2 2.5

x 10−3

−0.5

−0.49

−0.48

logT

11m

ax

0 0.5 1 1.5 2 2.5

x 10−3

−0.84

−0.83

−0.82

logT

12m

ax

0 0.5 1 1.5 2 2.5

x 10−3

−2.13

−2.12

−2.11

−2.1

logT

22m

ax

time

c)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.915

−6.91

−6.905

logh

0 0.2 0.4 0.6 0.8 1

x 10−3

−2.12

−2.118

−2.116

−2.114

logT

11m

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−1.255

−1.25

−1.245

logT

12m

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.572

−4.57

−4.568

logT

22m

ax

time d)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.909

−6.908

−6.907

logh

0 0.2 0.4 0.6 0.8 1

x 10−3

−2.308

−2.307

−2.306

logT

11m

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−1.32

−1.319

−1.318

logT

12m

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.015

−6.014

−6.013

logT

22m

ax

time

e)

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−6.912

−6.91

−6.908

logh

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−2.366

−2.364

−2.362

logT

11m

ax

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

−1.345

logT

12m

ax

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−7.72

−7.715

−7.71

logT

22m

ax

time f)

0 0.01 0.02 0.03 0.04 0.05 0.06

−6.913

−6.912

logh

0 0.01 0.02 0.03 0.04 0.05 0.06−2.38

−2.3795

−2.379

logT

11m

ax

0 0.01 0.02 0.03 0.04 0.05 0.06−1.353

−1.352

−1.351

logT

12m

ax

0 0.01 0.02 0.03 0.04 0.05 0.06−11.2

−11

−10.8

−10.6

logT

22m

ax

time

Figure 4.14: log(ampl) & log(Tijmax).

47

4.3.2 Growth away from Interface

The following results computed away from the interface (at similar regions as

we did with velocities) are quite remarkable compared to the corresponding ones

obtained at the end of the preceding section where we were investigating the

growth of velocities.

a) 0 1000 2000 3000 4000 5000 6000−1.5

−1

−0.5

0

time step

log

(T1

1)

at

(13

0,2

20

)

NumericalLinear

d) 0 1000 2000 3000 4000 5000 6000−0.8

−0.75

−0.7

−0.65

−0.6

−0.55

−0.5

−0.45

time step

log

(T1

2)

at

(13

0,2

20

)

NumericalLinear

g) 0 1000 2000 3000 4000 5000 6000−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

time step

log

(T2

2)

at

(13

0,2

20

)

NumericalLinear

b) 0 10 20 30 40 50 60 70 80 90 100−0.6765

−0.676

−0.6755

−0.675

−0.6745

−0.674

−0.6735

−0.673

−0.6725

−0.672

time step

log

(T1

1)

at

(13

0,2

20

)

NumericalLinear

e) 0 10 20 30 40 50 60 70 80 90 100−1.334

−1.3335

−1.333

−1.3325

−1.332

time step

log(T

12)

at (1

30,2

20)

NumericalLinear

h) 0 10 20 30 40 50 60 70 80 90 100−2.422

−2.421

−2.42

−2.419

−2.418

−2.417

−2.416

−2.415

−2.414

−2.413

−2.412

time step

log

(T2

2)

at

(13

0,2

20

)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−2.2827

−2.2827

−2.2827

−2.2826

−2.2826

−2.2826

−2.2826

−2.2826

−2.2825

−2.2825

−2.2825

time step

log

(T1

1)

at

(13

0,2

20

)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 100−1.3624

−1.3624

−1.3623

−1.3623

−1.3623

−1.3623

−1.3623

−1.3623

−1.3623

−1.3623

time step

log

(T1

2)

at

(13

0,2

20

)

NumericalLinear

i) 0 10 20 30 40 50 60 70 80 90 100−4.607

−4.6065

−4.606

−4.6055

−4.605

−4.6045

−4.604

−4.6035

time step

log

(T2

2)

at

(13

0,2

20

)

NumericalLinear

Figure 4.15: log(Tij) in the viscoelastic fluid.

Figure (4.15) shows the growth rates of the perturbation stresses (T11, T12 & T22)

in the viscoelastic fluid (the point (130,220) on a 258× 258 grid) away from both

the interface and the walls and using different wavenumbers. The cases (a), (d) &

(g) represent log(T11) for α = 6, 3 & 2 respectively. Similarly (b), (e) & (h) show

the graphs of log(T12) and (c), (f) & (i) are the plots of log(T22). Unlike in the

48

case of growth at the interface, figure (4.14) where the numerical results decrease

in accuracy with decreasing wavenumbers, here they are in excellent agreement

with the linear results for all the three wavenumbers used.

Since the stresses calculate as machine zeros in the Newtonian fluid for both the

numerical and linear cases, the differences in their growth rates between these

two cases is for all practical purposes insignificant and shown in figures (4.1 (e),

(g) & (i)), so these plots will not be repeated here.

4.3.3 Maximum growth along a horizontal line

a) 0 20 40 60 80 100 120 140 160 180 2000.565

0.57

0.575

0.58

0.585

0.59

time step

log(T

11m

ax)

alo

ng (

i,220)

NumericalLinear

d) 0 20 40 60 80 100 120 140 160 180 200−0.52

−0.515

−0.51

−0.505

−0.5

−0.495

−0.49

−0.485

−0.48

time step

log

(T1

2m

ax)

alo

ng

(i,2

20

)

NumericalLinear

g) 0 20 40 60 80 100 120 140 160 180 200−1.134

−1.132

−1.13

−1.128

−1.126

−1.124

−1.122

−1.12

−1.118

−1.116

time step

log(T

22m

ax)

alo

ng (

i,220)

NumericalLinear

b) 0 20 40 60 80 100 120 140 160 180 200−0.52

−0.515

−0.51

−0.505

−0.5

−0.495

−0.49

−0.485

time step

log(T

11m

ax)

alo

ng (

i,220)

NumericalLinear

e) 0 20 40 60 80 100 120 140 160 180 200−0.915

−0.91

−0.905

−0.9

−0.895

−0.89

−0.885

−0.88

−0.875

−0.87

time step

log(T

12m

ax)

alo

ng (

i,220)

NumericalLinear

h) 0 20 40 60 80 100 120 140 160 180 200−2.22

−2.218

−2.216

−2.214

−2.212

−2.21

−2.208

−2.206

−2.204

−2.202

time step

log

(T2

2m

ax)

alo

ng

(i,2

20

)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−2.36

−2.34

−2.32

−2.3

−2.28

−2.26

−2.24

time step

log(T

11m

ax)

alo

ng (

i,220)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 100−1.38

−1.37

−1.36

−1.35

−1.34

−1.33

−1.32

−1.31

time step

log(T

12m

ax)

alo

ng (

i,220)

NumericalLinear

i) 0 10 20 30 40 50 60 70 80 90 100−4.602

−4.6015

−4.601

−4.6005

−4.6

−4.5995

−4.599

−4.5985

−4.598

time step

log

(T2

2m

ax)

alo

ng

(i,2

20

)

NumericalLinear

Figure 4.16: log(Tijmax) in the Viscoelastic fluid.

49

Figure (4.16) shows the maximum growth rates of the perturbation stresses

(T11, T12 & T22) in the viscoelastic fluid along the horizontal line (i, 220) on a

258× 258 grid, where as before i = 1, 2, · · · , 258. These are computed away from

the interface using the three wavenumbers α = 6, α = 3 and α = 2. The cases

(a), (d) & (g) represent log(T11) for α = 6, 3 & 2 respectively. Similarly (b), (e)

& (h) show the graphs of log(T12) and (c), (f) & (i) are the plots of log(T22).

Again, since the stresses calculate as machine zeros in the Newtonian fluid for

both the numerical and linear cases, the differences in their growth rates between

these two cases is for all practical purposes insignificant and shown in figures (4.1

(e), (g) & (i)), hence will not be repeated here.

50

Chapter 5

Harmonic averaging

Let u′ represent the shear rate, η be the total viscosity, Newtonian plus viscoelas-

tic, and let µ = βη be the viscoelastic part of the viscosity and λ the relaxation

time.

Note that u is the integral of u′, and integration corresponds to arithmetic averag-

ing. Since the total shear stress is continuous, we should use harmonic averaging

for λ:

1

η= C

1

η1

+ (1− C)1

η2

,

where C is the volume fraction function. For the extra-stresses, we have, in the

base flow:

T12 = βηu′,

T11 = 2λT12u′ = 2

(λβ

η

)(ηu′)2.

Since we are interpolating T11 and T12 linearly, we want to view them as arithmetic

averages. To be consistent, we must first make β an arithmetic average (note that

the shear stress (ηu′) is continuous):

β = Cβ1 + (1− C)β2.

51

From β and η, we can then calculate the Newtonian and polymer viscosities.

Finally, we want to make λβ/η an arithmetic average:

λ =η

β

[C

λ1β1

η1

+ (1− C)λ2β2

η2

].

5.1 Growth rates of amplitudes and velocities

5.1.1 Growth at the Interface

In Figure (5.1), we plot the logarithms of the perturbation amplitude, maxi-

mum v-velocity and the L2-norm of v. This is done at different values of the

wavenumber α.

Figures (a) - (f) represent the cases α = 6, 3, 2, 1.5, 1 and 0.474 respectively.

a)

0 0.5 1 1.5 2 2.5

x 10−3

−6.925

−6.92

−6.915

−6.91

−6.905

log

h

0 0.5 1 1.5 2 2.5

x 10−3

−4

−3.99

−3.98

−3.97

log

vma

x

0 0.5 1 1.5 2 2.5

x 10−3

−4.815

−4.81

−4.805

−4.8

−4.795

log

vL2

time b)

0 0.5 1 1.5 2 2.5

x 10−3

−6.925

−6.92

−6.915

−6.91

−6.905

log

h

0 0.5 1 1.5 2 2.5

x 10−3

−4.895

−4.89

−4.885

−4.88

−4.875

log

vma

x

0 0.5 1 1.5 2 2.5

x 10−3

−4.58

−4.575

−4.57

−4.565

log

vL2

time

c)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.911

−6.91

−6.909

−6.908

−6.907

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.59

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.677

−4.676

−4.675

−4.674

−4.673

log

vL2

time d)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.909

−6.9085

−6.908

−6.9075

−6.907

log

h

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.7

−6.65

−6.6

−6.55

log

vma

x

0 0.2 0.4 0.6 0.8 1

x 10−3

−5.3005

−5.3

−5.2995

−5.299

−5.2985

log

vL2

time

52

e)

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−6.912

−6.911

−6.91

−6.909

−6.908

log

h

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−8.22

−8.218

−8.216

−8.214

−8.212lo

gvm

ax

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−6.23

−6.225

−6.22

−6.215

−6.21

log

vL2

time f)

0 0.01 0.02 0.03 0.04 0.05 0.06−6.9135

−6.913

−6.9125

−6.912

logh

0 0.01 0.02 0.03 0.04 0.05 0.06−11.6

−11.4

−11.2

−11

logv

max

0 0.01 0.02 0.03 0.04 0.05 0.06−8.4

−8.2

−8

−7.8

logv

L2

time

Figure 5.1: log(h), log(vmax), log(‖v‖2). respectively

5.1.2 Maximum growth along a horizontal line

a) 0 10 20 30 40 50 60 70 80 90 100−6

−4

−2

0

2

4

6

time step

log(

vmax

) alo

ng (i

,100

)

NumericalLinear

d) 0 20 40 60 80 100 120 140 160 180 200−2

0

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,100

)

NumericalLinear

b) 0 20 40 60 80 100 120 140 160 180 200−6

−4

−2

0

2

4

6

8

time step

log(

vmax

) alo

ng (i

,100

)

NumericalLinear

e) 0 20 40 60 80 100 120 140 160 180 200−2

0

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,100

)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−6

−4

−2

0

2

4

6

time step

log(

vmax

) alo

ng (i

,100

)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 100−2

0

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,100

)

NumericalLinear

Figure 5.2: log(vmax) & log(umax) along a horizontal line in Newtonian fluid.

Figures (5.2) and (5.3) show the maximum growth rates of perturbation velocities

53

a) 0 10 20 30 40 50 60 70 80 90 100−5

−4

−3

−2

−1

0

1

2

3

4

5

time step

log(

vmax

) alo

ng (i

,220

)

NumericalLinear

d) 0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,220

)

NumericalLinear

b) 0 20 40 60 80 100 120 140 160 180 200−6

−4

−2

0

2

4

6

time step

log(

vmax

) alo

ng (i

,220

)

NumericalLinear

e) 0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,220

)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

time step

log(

vmax

) alo

ng (i

,220

)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

time step

log(

umax

) alo

ng (i

,220

)NumericalLinear

Figure 5.3: log(vmax) & log(umax) along a horizontal line in Viscoelastic fluid.

in the Newtonian and viscoelastic fluids respectively at different wavenumbers.

These where computed away from the interface along the horizontal lines (i, 100)

and (i, 220), for the Newtonian and Viscoelastic fluids respectively, and using a

258 × 258 grid, where i = 1, 2, 3, · · · , 258. In particular figures labelled (a), (b)

and (c) are plots of the logarithm of the perturbation velocity v obtained using

α = 6, 3, 2 respectively. Those labelled (d), (e) and (f) are similarly defined for

u.

54

5.2 Growth rates of Extra-stresses

5.2.1 Growth at the interface

a)

0 0.5 1 1.5 2 2.5

x 10−3

−6.94

−6.92

−6.9

logh

0 0.5 1 1.5 2 2.5

x 10−3

3.05

3.06

3.07

3.08

logT

11m

ax

0 0.5 1 1.5 2 2.5

x 10−3

1.3

1.32

logT

12m

ax

0 0.5 1 1.5 2 2.5

x 10−3

0.55

0.56

0.57

logT

22m

ax

time b)

0 0.5 1 1.5 2 2.5

x 10−3

−6.94

−6.92

−6.9

logh

0 0.5 1 1.5 2 2.5

x 10−3

−0.5

−0.48

logT

11m

ax

0 0.5 1 1.5 2 2.5

x 10−3

−0.84

−0.83

−0.82

logT

12m

ax

0 0.5 1 1.5 2 2.5

x 10−3

−2.13

−2.12

−2.11

−2.1

logT

22m

ax

time

c)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.915

−6.91

−6.905

logh

0 0.2 0.4 0.6 0.8 1

x 10−3

−2.12

−2.118

−2.116

−2.114

logT

11m

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−1.255

−1.25

−1.245

logT

12m

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−4.572

−4.57

−4.568

logT

22m

ax

time d)

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.909

−6.908

−6.907

logh

0 0.2 0.4 0.6 0.8 1

x 10−3

−2.308

−2.307

−2.306

logT

11m

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−1.32

−1.319

−1.318

logT

12m

ax

0 0.2 0.4 0.6 0.8 1

x 10−3

−6.015

−6.014

−6.013

logT

22m

ax

time

e)

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−6.912

−6.91

−6.908

logh

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−2.366

−2.364

−2.362

logT

11m

ax

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

−1.345

logT

12m

ax

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−7.72

−7.715

−7.71

logT

22m

ax

time f)

0 0.01 0.02 0.03 0.04 0.05 0.06

−6.913

−6.912

logh

0 0.01 0.02 0.03 0.04 0.05 0.06−2.38

−2.3795

−2.379

logT

11m

ax

0 0.01 0.02 0.03 0.04 0.05 0.06−1.353

−1.352

−1.351

logT

12m

ax

0 0.01 0.02 0.03 0.04 0.05 0.06−11.2

−11

−10.8

−10.6

logT

22m

ax

time

Figure 5.4: log(ampl) & log(Tijmax).

Figure (5.4) shows graphs of the logarithms of the maximum growth at the in-

terface of perturbation extra-stresses and also the logarithms of the amplitudes.

This is done at different values of the wavenumber α. Figures (a) - (f) represent

the cases α = 6, 3, 2, 1.5, 1 and 0.474 respectively. The results displayed here are

show better behavior in the log(h) plot even at smaller wavenumbers as compared

to those in figures (4.3), (??) and (5.1).

55

5.2.2 Maximum growth along a horizontal line

a) 0 20 40 60 80 100 120 140 160 180 2000.572

0.574

0.576

0.578

0.58

0.582

0.584

0.586

0.588

0.59

time step

log(T

11m

ax)

alo

ng (

i,220)

NumericalLinear

d)0 20 40 60 80 100 120 140 160 180 200

−0.494

−0.492

−0.49

−0.488

−0.486

time step

log(T

12m

ax)

alo

ng (

i,220)

NumericalLinear

g) 0 20 40 60 80 100 120 140 160 180 200−1.134

−1.132

−1.13

−1.128

−1.126

−1.124

−1.122

−1.12

−1.118

−1.116

time step

log(T

22m

ax)

alo

ng (

i,220)

NumericalLinear

b) 0 20 40 60 80 100 120 140 160 180 200−0.502

−0.5

−0.498

−0.496

−0.494

−0.492

−0.49

−0.488

time step

log(T

11m

ax)

alo

ng (

i,220)

NumericalLinear

e) 0 20 40 60 80 100 120 140 160 180 200−0.882

−0.881

−0.88

−0.879

−0.878

−0.877

−0.876

−0.875

−0.874

time step

log(T

12m

ax)

alo

ng (

i,220)

NumericalLinear

h) 0 20 40 60 80 100 120 140 160 180 200−2.22

−2.218

−2.216

−2.214

−2.212

−2.21

−2.208

−2.206

−2.204

−2.202

time step

log

(T2

2m

ax)

alo

ng

(i,2

20

)

NumericalLinear

c) 0 10 20 30 40 50 60 70 80 90 100−2.2696

−2.2695

−2.2694

−2.2693

−2.2692

−2.2691

−2.269

time step

log

(T1

1m

ax)

alo

ng

(i,2

20

)

NumericalLinear

f) 0 10 20 30 40 50 60 70 80 90 100−1.3278

−1.3277

−1.3276

−1.3275

−1.3274

−1.3273

−1.3272

time step

log

(T1

2m

ax)

alo

ng

(i,2

20

)

NumericalLinear

i) 0 10 20 30 40 50 60 70 80 90 100−4.602

−4.6015

−4.601

−4.6005

−4.6

−4.5995

−4.599

−4.5985

−4.598

time step

log

(T2

2m

ax)

alo

ng

(i,2

20

)

NumericalLinear

Figure 5.5: log(Tijmax) in the Viscoelastic fluid.

Figure (5.5) shows the maximum growth rates of the perturbation stresses (T11, T12

& T22) in the viscoelastic fluid along the horizontal line (i, 220) on a 258 × 258

grid, where as before i = 1, 2, · · · , 258. These are computed away from the inter-

face using the three wavenumbers α = 6, α = 3 and α = 2. The cases (a), (d) &

(g) represent log(T11) for α = 6, 3 & 2 respectively. Similarly (b), (e) & (h) show

the graphs of log(T12) and (c), (f) & (i) are the plots of log(T22). These results

are no better or worse than those obtained without harmonic averaging, so we

will not pursue the idea of harmonic averaging further.

56

Chapter 6

Numerical Validation II

This chapter is based on the results and data from the experiments of [38]. In

this paper, they considered Poiseuille flow in a rectangular channel, and since

the authors did not provide a concise tabulation of their experimental data, we

found it necessary to re-construct such data from the relevant governing fluid

dynamical relations. In particular we had to re-compute their interfacial shear

rate so as to match ours to that. To do this, consider the basic velocity profile

for Poiseuille flow

U(z) =

−GR1z

2/2 + c1z, 0 ≤ z ≤ l1,

−rGR2(z − 1)2/2 + c2(z − 1), l1 ≤ z ≤ 1.(6.1)

Since the pressure gradient, G, was not provided in [38], we have three unknowns,

c1, c2 and G. We first found c1 and c2 in terms of G by using the continuity of

the velocities Uj(z) and shear stresses ηjU′j(z) at the interface where j = 1, 2 and

U1, U2 are as given in (6.1). This leads to

c1 = − [2η1Gl1R1(1− l21) + η2Gl21R1 + η2GrR2(1− l1)2]

2(η1(l1 − 1)− l1η2), (6.2)

c2 = − 1

2(η1(1− l1) + l1η2)[η1Gl21R1 − η2Grl1R2(l

21 − 1)]. (6.3)

The superficial Reynolds number is given by (Re)s = Vsd∗/ν, where Vs is the

57

superficial velocity in the rectangular channel, d∗ is the dimensional channel width

and ν = η/ρ is the kinematic viscosity. Taking d∗ = 2cm we can use this

expression to calculate Vs in terms of the other known quantities, R1, ν and d∗.

However we also have

Vs =1

d

∫ l1

0U(z) dz, (6.4)

where d is the dimensionless channel width. Substituting the now known value

of Vs in (6.4) leads to a linear equation in the G. This can easily be solved for

G which can then be back-substituted to yield the numerical values of c1 and c2.

These can in turn be used to find the interfacial shear rate U ′(l1) and velocity

Ui = U(l1) which we can then use in the VOF code to change the interfacial shear

rate for our Couette flow so it matches this computed U ′(l1).

6.1 Growth rates of amplitudes and velocities

6.1.1 Growth at the Interface

a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−4

−13.818

−13.817

−13.816

−13.815

−13.814

log

h

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−4

−1.072

−1.071

−1.07

−1.069

log

vma

x

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−4

−3.092

−3.09

−3.088

log

vL2

b)

0 1 2 3 4 5 6 7 8

x 10−5

−13.8158

−13.8157

−13.8156

−13.8155

log

h

0 1 2 3 4 5 6 7 8

x 10−5

−6.4905

−6.4904

−6.4903

−6.4902

log

vma

x

0 1 2 3 4 5 6 7 8

x 10−5

−7.2996

−7.2995

−7.2994

−7.2993

log

vL2

cont.

58

c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−13.8157

−13.8157

−13.8156

−13.8156

log

h

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−7.8765

−7.8764

−7.8763

−7.8762lo

gvm

ax

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−8.3648

−8.3647

−8.3647

−8.3646

log

vL2

Figure 6.1: log(h), log(vmax), log(‖v‖2). respectively

In Figure (6.1), we plot the logarithms of the perturbation amplitude, maxi-

mum v-velocity and the L2-norm of v. This is done at different values of the

wavenumber α. Figures (a) - (c) represent the cases:

a); R1 = 186, ∆t = 10−8, tmax = 2× 104.

b); R1 = 177, ∆t = 10−8, tmax = 7× 103.

c); R1 = 167, ∆t = 10−6, tmax = 1× 102.

We should point out that we had to change the viscosity ratio of [38] which

from an order of almost 3600 to an order of 10. This amounted to reducing the

viscosity of Boger fluid (fluid 2) to 10 times that of water (fluid 1). This was

necessitated by the fact that keeping such a large viscosity ratio would not only

be incompatible with the designs of our code but would mean that we would be

mostly looking at the effects of viscosity stratification instead of concentrating

on elastic effects.

6.1.2 Waveforms

Below, figure (6.2), we show both the spatial and temporal evolution of the

maximum interface amplitude. For the graphs of A(t) vs t this amounts to the

following: for each time T ∈ {0, ∆t, 2∆t, . . . , tmax∆t}, we find the corresponding

59

amplitude A(T ) = maxx,z∈D h(T, x, z) where D represents the spatial mesh grid

covering the interfacial region.

R1 = 186, ∆t = 10−8, tmax = 2× 104 R1 = 177, ∆t = 10−8, tmax = 7× 103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−4

9.98

9.985

9.99

9.995

10x 10

−7

am

plit

ud

e

time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−1

−0.5

0

0.5

1x 10

−6

am

plit

ude

x

0 1 2 3 4 5 6 7 8

x 10−5

9.9975

9.998

9.9985

9.999

9.9995

10x 10

−7

am

plit

ud

e

time

0 0.2 0.4 0.6 0.8 1 1.2 1.4−1

−0.5

0

0.5

1x 10

−6

am

plit

ude

x

R1 = 167, ∆t = 10−6, tmax = 1× 102

0 1 2 3 4 5 6 7 8

x 10−3

9.88

9.9

9.92

9.94

9.96

9.98

10

10.02x 10

−7

am

plit

ud

e

time

0 0.2 0.4 0.6 0.8 1 1.2 1.4−1

−0.5

0

0.5

1x 10

−6

am

plit

ude

x

Figure 6.2: max(amplitude) vs time and wave shapes

We notice from figure (6.2) A(t) doesn’t reach saturation for the particular choices

of ∆t and tmax that we used, meaning that we couldn’t in turn obtain the satura-

tion wave forms (unless we performed a weakly nonlinear analysis leading to the

Stuart-Landau equation). We thus simply graphed the wave shapes by fixing the

time at T = tmax∆t as follows: for each point x ∈ {0, ∆x, 2∆x, 3∆x, . . . , (nx −2)∆x} where nx represents the total number of mesh points in the x-direction,

∆x is configured such that for any mesh, (nx − 2)∆x = Lx and the x-direction

60

length scale is redefined for the stability code as Lx = 2π/α. We then plot the

corresponding amplitude H(x) = maxz∈D h(T, x, z). These wave forms still have

the sinusoidal cosine wave shape of the initial disturbance wave showing that we

are still in the linear regime. This should be expected since these waveforms were

obtained at very small times.

Below we increase both the times at which we will plot the wave shapes and also

the initial amplitudes (to about 10% of the fluid depth i.e. initial perturbation

h = 0.1) and we let these evolve in the new time scale. We notice as before that

the perturbations evolves to a flat interface, showing stability.

(a)

Wed Nov 10 22:27:02 2004

0 0.1 0.2 0.3 0.4 0.50

1

PLOT

X−Axis

Y−Ax

is

0 0.1 0.2 0.3 0.4 0.50

1

(b)

Wed Nov 10 22:28:40 2004

0 0.1 0.2 0.3 0.4 0.50

1

PLOT

X−Axis

Y−Ax

is

0 0.1 0.2 0.3 0.4 0.50

1

(c)

Wed Nov 10 22:30:12 2004

0 0.1 0.2 0.3 0.4 0.50

1

PLOT

X−Axis

Y−Ax

is

0 0.1 0.2 0.3 0.4 0.50

1

(d)

Wed Nov 10 22:31:28 2004

0 0.1 0.2 0.3 0.4 0.50

1

PLOT

X−Axis

Y−Ax

is

0 0.1 0.2 0.3 0.4 0.50

1

Figure 6.3: Wave forms at (a) t = 0.01, (b) t = 0.1, (c) t = 0.3 and (d) t = 1

where R1 = 186, ∆t = 10−3, tmax = 1000 and h = 0.1

61

PART II

DEFORMABLE DROPS

Chapter 7

Droplet Deformation

7.1 Problem formulation and assumptions

In this part, we focus attention on the behavior of deformable drops in simple

shear flows where either or both the droplet or matrix fluid may be considered

viscoelastic.

Figure 7.1: Computational domain

In fact, we will investigate all four possible combinations for the droplet-matrix

63

fluid system; (i) newtonian drop in a viscoelastic fluid (ii) viscoelastic drop in

a newtonian fluid (iii) newtonian drop in a newtonian fluid and (iv) viscoelastic

drop in a viscoelastic fluid.

As in [18] an initially circular drop of radius a is placed at the center of a rect-

angular domain and subjected to a simple shear flow by moving the top and

bottom walls in opposite directions, see Figure 7.1. We assume that the top wall

is located at z = Lz and moves to the right with speed U(Lz) = UT and that the

bottom wall is located at z = 0 and moves to the left with speed U(0) = UB.

Since the flow is essentially unidirectional and parallel to the x-axis, the velocity

profile is given by

U(z) = UB + (UT − UB)z

Lz

, (7.1)

leading to a shear rate of

γ = U ′(z) =(UT − UB)

Lz

. (7.2)

Following the notation of Chapter 2, in each component fluid we denote the (non-

dimensional) solvent viscosity by ηs, polymeric viscosity by ηp, total viscosity by

η = ηs + ηp, density by ρ and relaxation time by λ. If we let the drop radius be

a and write σ for the interfacial tension, then besides the Reynolds number R,

R =ργa2

ηs

, (7.3)

the droplet deformation will also be characterized using two additional dimen-

sionless numbers namely the capillary (Ca) and Deborah (De) numbers defined

respectively as

Ca =γηa

σ, (7.4)

64

De = λγ. (7.5)

Instead of using the Deborah number, [28] and [6] utilize the dimensionless ratio

of elastic to interfacial stress, N , to describe non-Newtonian effects, recall Ca

being the ratio of viscous to interfacial stress governs the Newtonian case. Here

N =N1a

2σ, (7.6)

where N1 is the so called first normal stress difference, N1 = Ψ1γ2 with Ψ1 the first

normal stress difference coefficient. The reasoning behind using N is that non-

Newtonian effects become observable in the drop deformation when Ca2 ∼ N .

In other words, effects of fluid elasticity on drop deformation appear when the

dimensionless ratio p = N/Ca2 = Ψ1σ/2aη2 is close to one.

Also as in Chapter 2, the two fluid system is governed by equations which arise

from incompressibility and the principle of balance of linear momentum:

∇ · u = 0, (7.7)

ρ(∂u

∂t+ u · ∇u) = ∇ · ~τ + F, (7.8)

where the total stress tensor ~τ is

~τ = −pI + T + η[∇u + (∇u)T ], (7.9)

I is the unit tensor and T is the extra stress tensor.

The extra stress tensor, T, is governed by the Oldroyd-B constitutive equations

(i.e. the Giesekus model with the nonlinear parameter κ ≡ 0)

λ∇T +T = ηp[∇u + (∇u)T ], (7.10)

where the upper convected time derivative∇T is defined as,

∇T=

∂T

∂t+ (u · ∇)T− (∇u)T−T(∇u)T .

65

The algorithm for computing solutions of equations (7.7), (7.8) and (7.10) is

identical to that illustrated in Chapter 3 with fluids 1 and 2 denoting the drop

and the matrix fluid respectively.

7.2 Validation of Results

In this section, we will compare our results to those of other recent similar works.

These include the works of Guido et. al. [6] who performed an experimental

validation of the small deformation theory for a Newtonian drop immersed in a

viscoelastic fluid, the numerical work of Pillapakkam & Singh [18] who considered

both small and large deformation cases for the drop in a fluid problem, and lastly

the three dimensional numerical investigations of Renardy et. al. [28].

Since interfacial tension tends to resist drop deformation, high interfacial ten-

sions and thus low capillary number systems (say Ca << 1) lead to the small

deformation cases. In such instances, provided Ca is below certain critical val-

ues, the drop usually deforms into a steady state elliptical shape. The droplet

deformation in this case can then be measured in terms of a parameter D called

the deformation parameter,

D =Rmax −Rmin

Rmax + Rmin

,

where Rmax and Rmin will in this work represent the major and minor axes of the

ellipse respectively.

7.2.1 Small deformation case

In the same spirit with our work a level set method is used in [18] to compute

among other things the drop deformation in simple shear flow. Therein, they

consider the cases of (i) a circular Newtonian drop in a viscoelastic matrix and

66

(ii) a viscoelastic drop in a Newtonian liquid, where the viscoelastic fluid in either

cases is the Oldroyd-B with ηs = ηp.

With R = 0.0003, Ca = 0.24, De = 0.4, ηs = ηp and equal viscosities (η1/η2 =

1), they report in [18] that a Newtonian drop in a viscoelastic liquid reaches a

steady state deformation D = 0.48. At these parameters [28] find that a three

dimensional evolution settles to D ≈ 0.3 at a deformation angle φmax = 27deg.

Table 7.1 shows our results for two dimensional evolution using a = 0.125 and

γ = 1. For the Newtonian drop, viscoelastic matrix system, we notice that the

steady state value is D ≈ 0.3 and φmax ≈ 28deg, in close agreement with the

results of [28] just cited.

Table 7.1: Deformation parameter (D) and Angle (φmax) for four drop-matrix

systems at dimensionless t = 3, Ca = 0.24

System Rmax Rmin D φmax

NN 0.165016 0.091254 0.287831 32.255300

VN 0.163570 0.091864 0.280724 31.146500

NV 0.161379 0.093850 0.264583 28.164600

VV 0.160449 0.094302 0.259656 28.213100

For the two-letter shorthand notation in both Table 7.1 and figure (7.2), the first

letter gives the droplet type and the second represents the matrix fluid, so NV

stands for a Newtonian drop in a viscoelastic matrix etc.

67

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time

D

NNVNNVVV

Figure 7.2: Temporal evolution of D, when Ca = 0.24

Figure 7.2 shows the transient evolution of D up to t = 3.

In general we notice from Table 7.1 and figure (7.2) that when the matrix fluid is

viscoelastic, the droplet reaches a steady state with D ≈ 0.3 and φmax ≈ 28deg

whereas if the matrix fluid is Newtonian then steady state is achieved when

D ≈ 0.3 and φmax ≈ 32deg. This is in agreement with small deformation theory

for second order fluids, that ”the orientation of the drop towards the direction

the flow is significantly enhanced in the viscoelastic case (i.e φmax is lower there)

than in the Newtonian case” which is independently verified by the experimental

work of [6] and the three dimensional numerical results of [28]. [18] report a

steady state deformation D ≈ 0.5 for the NV case ath these parameters.

When Ca is increased to 0.6 and all the other parameter are kept the same

as before, [18] report that a Newtonian drop in shear flow of a viscoelastic fluid

continues to deform and eventually breaks up whereas a viscoelastic drop in shear

68

flow of a Newtonian fluid settles to a steady state deformation D ≈ 0.47.

However, as illustrated by the data in the Table 7.2 and the graphs of figure (7.3)

our results and conclusions are quite the contrary. We notice that with Ca = 0.6,

if the matrix fluid is Newtonian then the drop continues to deform into a long

thin sheet (and maybe eventually break up say in a 3D investigation), but not

when the matrix fluid is viscoelastic.

Table 7.2: D and φmax at dimensionless t = 10, Ca = 0.6

System Rmax Rmin D φmax

NN 0.306991 0.039363 0.772700 14.746900

VN 0.282036 0.044982 0.724896 15.668800

NV 0.239284 0.058797 0.605497 14.657000

VV 0.222299 0.065064 0.547163 15.805100

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time

D

NNVNNVVV

Figure 7.3: Temporal evolution of D when Ca = 0.6

69

(a)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(b)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.4: Deformed drops at t = 3

Figures (7.4a) and (7.4b) show the deformed drops at time t = 3 for the capillary

numbers Ca = 0.24 & Ca = 0.6 respectively. In each of these figures the different

colors represent the four cases NN, VN, NV and VV.

(a)

Thu Nov 18 13:03:32 2004

0 1 20

1

PLOT

X−Axis

Y−

Axis

0 1 20

1

(b)

Thu Nov 18 13:05:20 2004

0 1 20

1

PLOT

X−Axis

Y−

Axis

0 1 20

1

Figure 7.5: Deformed drops at t = 10

Figures (7.5a) and (7.5b) show the deformed drops at time t = 10 for the large

capillary number Ca = 0.6 for the NN and VV cases respectively.

We notice that the VV case has the steady state elliptical shape whereas the NN

case displays the characteristic dumbbell shape showing continued deformation.

This confirms our earlier observations about the continued deformation at Ca =

0.6 when the bulk fluid is Newtonian, and the attainment of steady state when

70

the bulk fluid is viscoelastic. This is also verified in Figures (7.6a) and (7.6v)

below for VN & NV cases respectively.

(a)

Thu Nov 18 13:06:43 2004

0 1 20

1

PLOT

X−Axis

Y−

Axis

0 1 20

1

(b)

Thu Nov 18 13:00:55 2004

0 1 20

1

PLOT

X−AxisY

−A

xis

0 1 20

1

Figure 7.6: Deformed drops at t = 10

(a)

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(b)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.7: Contour plots of first normal stress at difference t = 7.

Figure (7.7) shows the plot of the first normal stress difference N1 = T11−T22 for

the cases (a) VN and (b) NV. We notice that in both cases N1 is highest near the

tips of the drops. However the maximum of N1 is higher in the NV than in the

VN case. In the NV case, N1 acts to resist deformation of the Newtonian drop

whereas it enhances deformation of the viscoelastic drop in the VN case. This

would explain why the viscoelastic drop in the VN case is more deformed than

the Newtonian drop in the NV case, in contradiction to the results of [18], where

71

they get higher deformation for a Newtonian drop in NV than for a viscoelastic

drop in VN and explain this in terms of the “viscoelastic stresses inside the drop

tend to reduce drop deformation” and since “the viscoelastic stresses of the bulk

fluid tend to increase drop deformation.”

All the results given so far are based on using, in our VOF code, the initial extra

stress tensor given by equation (2.6):

Table 7.3 and figure (7.8) show the results obtained by using zero initial extra

stresses in our algorithm.

Table 7.3: D and φmax at t = 3, Ca = 0.24 & zero initial stresses

System Rmax Rmin D φmax

NN 0.165016 0.091254 0.287831 32.255300

VN 0.163428 0.091945 0.279915 32.533400

NV 0.161600 0.093765 0.265638 28.122500

VV 0.160473 0.094285 0.259809 28.220100

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time

D

NNVNNVVV

Figure 7.8: Temporal evolution of D when Ca = 0.24 and zero initial stresses

72

Shown in Table (7.4 and figure (7.9)are the differences in the values obtained

using zero and non-zero initial stresses. As expected, the NN system remains

unchanged at exactly the same values as before. For the remaining cases, the

deformation parameter and angle both show exhibit very small differences.

Table 7.4: ∆D & ∆φmax at t = 3, Ca = 0.24

System ∆Rmax ∆Rmin ∆D ∆φmax

NN 0.000000 0.000000 0.000000 0.000000

VN 0.000142 0.000081 0.000809 1.386900

NV 0.000221 0.000084 0.001055 0.042100

VV 0.000024 0.000017 0.000153 0.007000

Table 7.4 shows comparisons of stationary states for zero and non-zero initial

stress cases. Arithmetic differences in the deformation parameter are denoted

∆D and those for the angle as ∆φmax.

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

Time

|∆D|

NNVNNVVV

Figure 7.9: Evolution of ∆D when Ca = 0.24

These results show that the initial stresses are virtually inconsequential. In other

73

words we can always start with zero initial stresses and still converge with rea-

sonable accuracy to the correct solutions. We next look at what happens when

we reduce the interfacial tension (i.e. increase the capillary numbers).

7.2.2 Large deformation case

Capillary numbers way beyond critical values imply correspondingly small inter-

facial tensions which in a 3D set-up would most likely lead to drop break up. For

the 2D case, the drops keep deforming into long thin sheets (and any break-up

that might be noticed would only be numerical artifacts resulting from the drop

widths falling within the range of the minimum dimensions of computational

cells). In theory, a drop would be at the onset of pinch off and hence of break up

when Rmin = 0 or in other words D = 1. However, for finite difference/element

based numerical investigations, the smallest Rmin we can get would be within

the bounds of the minimum dimensions of the respective cell or element size. In

such cases numerical break up of the drop for the 2D case will be noticed when

D ≈ 1−.

Pillapakkam and Singh [18] numerically investigated the case of a viscoelastic

drop subjected to a Newtonian shear flow with R = 0.0003, Ca = 60, De =

8.0, ηs = ηp and equal viscosities. At t=10, [18] had D = 0.8152 and they

conclude that the drop continues to deform and will eventually break up.

From the data of Table 7.5 and the graphs in figure (7.10), our results generally

support the conclusions of [18]. We notice that at t = 10, R = 0.3, De = 8.0,

Ca = 60, ηs = ηp and m = r = 1, the deformation parameter D ≈ 1, and that the

drops deform into long and thin sheets. As expected, we also notice numerical

break-up of the drop for lower mesh resolutions as illustrated in figures (7.11)

and 7.12). Notice how these numerical artifacts disappear for finer meshes.

The quantities appearing in Table 7.5 need to be interpreted in the context of

74

Table 7.5: D and φmax at t = 10, Ca = 60

System Rmax Rmin D φmax

NV 1.002910 0.004066 0.991925 6.684540

VN 1.002410 0.008171 0.983829 6.057800

NN 1.003580 0.011877 0.976608 6.040730

VV 1.001860 0.012205 0.975928 6.150100

figures (7.11). These figures illustrate the deformation of the drops into long

thin sheets and the quantities just cited are in reference to part of the drop that

appears in the middle of the computational domain.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

D

Plot of Deformation parameter (D) for Re=0.3, De=8.0, Ca=60

NVVNNNVV

Figure 7.10: Temporal evolution of D when Ca = 60

Figures (7.11(a) - (d)) show the large deformation of drops at high capillary and

Deborah numbers. In particular figures (7.11(a) - (d)) respectively correspond to

the cases of a Newtonian drop immersed in a Newtonian fluid, a viscoelastic drop

in a viscoelastic fluid, a viscoelastic drop in a Newtonian fluid and a Newtonian

drop in a viscoelastic fluid.

Two interesting observations are immediate from Figure (7.11). The first is the

75

(a)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(b)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(c)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(d)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.11: Elongation of drops at t = 10 when Ca = 60, R = 0.3, De = 8.

hook shaped tips of the Newtonian drop immersed in a viscoelastic fluid, this will

be the subject of the following section. The second observation is the apparent

break-up of the drop in figure (7.11(c) & (d)). As shown in figure (7.12) this is

simply a numerical artifact and can be eliminated by refining the mesh.

Figure (7.12(a)) shows numerical break-up of a Newtonian drop in a viscoelastic

fluid on a coarse mesh (128× 128). In figure (7.12(b)), this numerical artifact is

eliminated on a finer (512× 512) mesh resolution.

Before we look into the details of the peculiar hooks appearing in the large de-

formation (Ca >> 1) cases for the Newtonian drop in a viscoelastic fluid, we

proceed first to verify that out results temporally and spatially convergent.

76

(a)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(b)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.12: Effect of mesh size on numerical breakup of drops, results shown for a

Newtonian drop in a viscoelastic fluid at t = 10 with R = 0.3, De = 8.0&Ca = 60.

7.3 Temporal and spatial convergence

In this section, we perform the necessary verifications that our results are inde-

pendent of time step size and/or mesh resolution

(a)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(b)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.13: Convergence with mesh resolution, for a Newtonian drop in a vis-

coelastic fluid at t = 10 with R = 0.3, De = 8.0 & Ca = 60, Red=256 × 256,

Blue=128× 128 Green=512× 512.

From the numerical breakup cases, figures (7.11d) and (7.12), we already have

77

three plots performed at the same parameters but with three different mesh res-

olutions, 256× 256, 128× 128 and 512× 512. It should therefore be instructive

to first start with these three figures, plot them on the same axes as shown figure

(7.13) and then check if they match, in which case we will have spatial conver-

gence. The results of figure (7.13) verify that our results are indeed spatially

convergent, in other words, they are independent of the mesh resolution.

a)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

b)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.14: Steady state shapes with spatial (a) & spatio-temporal (b) refine-

ment

We performed a similar convergence tests as those done in figure (7.13) but for

the low capillary number cases. Shown in figures (7.14a) and (7.14b) are the

steady state drop shapes for the viscoelastic drop in a viscoelastic fluid plotted

at time t = 3 and using the common parameters values, R = 0.3, De = 0.4 and

Ca = 0.24.

Figure (7.14a) shows convergence with mesh resolution. In particular we used

the three following mesh resolutions: 128× 128, 256× 256 and 512× 512.

Figure (7.14b) similarly gives convergence with spatio-temporal refinement. We

used the following mesh resolutions and time steps:

128× 128, ∆t = 0.001, 128× 128, ∆t = 0.0005, 256× 256, ∆t = 0.001,

256× 256, ∆t = 0.0005.

78

(a)0 0.5 1 1.5 2 2.5 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time

D

(b)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.15: Transient deformation with temporal refinement

Figures (7.15a) and (7.15b) show convergence of the transient deformation of a

viscoelastic drop in a viscoelastic fluid using two different time steps. The figure

have the common parameter values, R = 0.3, De = 0.4 & Ca = 0.24. Figures

(7.15a) and (7.15b) use 128 × 128 and 256 × 256 mesh resolutions respectively.

The time step sizes in each figure are ∆t = 0.001 and ∆t = 0.0005.

From figures (7.13), (7.14) and (7.15) we conclude that our results are indepen-

dent of mesh resolution as well as time step size.

7.3.1 Newtonian drops in viscoelastic fluids

As illustrated in figures (7.11) and (7.12), the large deformation of a Newtonian

drop in a viscoelastic fluid displays peculiar hook shapes at tips. This phe-

nomenon doesn’t seem to have received much attention and investigation if at all

any in the current literature, and hopefully the investigation of these herein will

serve as a new contribution.

In order to explain these features we will proceed to construct among other things,

(i) velocity vector plots that illustrate the intensity of the velocity field inside the

drops and over the entire computational domain with particular focus on the

79

vicinity of the hooked parts of the interface and near the left and right edges of

the domain and (ii) the first normal stress difference and pressures, and how they

change as the viscoelasticity of the matrix fluid decreases to zero, so that we can

separate out the viscoelastic features of the flow.

(a)

Mon Sep 13 13:22:01 2004

0 1 20

1

PLOT

X−Axis

Y−Ax

is

0 1 20

1

(b)

Mon Sep 13 13:25:57 2004

0 1 20

1

PLOT

X−Axis

Y−Ax

is

0 1 20

1

(c)

Mon Sep 13 13:28:07 2004

0 1 20

1

PLOT

X−Axis

Y−Ax

is

0 1 20

1

(d)

Mon Sep 13 13:31:18 2004

0 1 20

1

PLOT

X−Axis

Y−Ax

is

0 1 20

1

Figure 7.16: velocity vector plots for Newtonian drop in a viscoelastic fluid where

Ca = 60, R = 0.3, De = 8 at (a) t = 2, (b) t = 4, (c) t = 5, & (d) t = 6,

Figure (7.16) shows the intensity of velocity vectors and the corresponding hooked

droplet shapes at various times. We will show in subsequent figures the relation-

ship between some of the the dimensionless parameters with the above shapes and

velocity fields and that these velocity fields and hook shapes for any particular

drop are independent of the velocity fields of neighboring drops.

80

We notice from figure (7.17) that both the intensity of the velocity vectors and

the droplet shapes are directly affected by the viscoelasticity. In particular as the

Deborah number decreases to zero both the hooks and the vortices around them

gradually disappear.

(a)

Mon Sep 20 09:30:48 2004

0 1 20

1

PLOT

X−Axis

Y−Ax

is

0 1 20

1

(b)

Mon Sep 20 09:37:16 2004

0 1 20

1

PLOT

X−Axis

Y−Ax

is

0 1 20

1

(c)

Tue Sep 21 10:56:34 2004

0 1 20

1

PLOT

X−Axis

Y−Ax

is

0 1 20

1

(d)

Tue Sep 21 11:27:00 2004

0 1 20

1

PLOT

X−Axis

Y−Ax

is

0 1 20

1

Figure 7.17: velocity vector plots for Newtonian drop in a viscoelastic fluid where

Ca = 60, R = 0.3 & De = 4, t = 2, De = 4, t = 6, De = 2, t = 2, & De = 2, t = 6,

The above figures also stress the point that a decrease in viscoelasticity of the

matrix fluid leads to a corresponding decrease in the drop deformation rate.

However due to the high capillary number, the droplet will keep deforming into

long sheets, only not as fast they would at low Deborah numbers.

To illustrate that the velocity field of the neighboring drops are not interacting, we

81

double the horizontal length scale from 2 to 4 and hence also the mesh width along

the x-direction from 256 to 512. We see from figure (7.18 that the streamwise

velocity profile appears to be linear near the left and right hand edges of the

computational cell. Magnifying these velocity vectors on the boundary confirms

that they are indeed linear (horizontal). This shows that the circulation or vortex

motion around the hooks of each drop is not transferred to neighboring drops

but that neighboring drops inherit the regular linear velocity profiles, or in other

words, the velocity profiles of neighboring drops do not interact in any unusual

way.

Tue Nov 9 19:27:42 2004

0 1 2 3 40

1

PLOT

X−Axis

Y−Axis

0 1 2 3 40

1

Figure 7.18: velocity vectors showing linear streamwise profiles at left and right

hand side edges, Ca = 60, R = 0.3, De = 8

From figures (7.16) and (7.17) we notice genuine hooks even at time t = 2. To

confirm that the hooks are independent of velocity fields from neighboring drops,

we plot the velocity vector field at t = 2 using a longer x-direction length scale

and hence use a correspondingly finer mesh in this direction. As depicted in

figure (7.18), the velocity profile is linear at the left and right hand boundaries

thus confirming no unusual interaction of neighboring drop velocities.

82

As mentioned earlier, we next investigate the first normal stress difference (T11−T22) and pressure (p), and how they change as the viscoelasticity of the matrix

fluid decreases to zero, so that we can separate out the viscoelastic features of

the flow. These investigations will be carried out at time t = 2 at which as just

noted, the droplet shows genuinely hooked tips.

From the velocity vector plots in figures (7.16 - 7.18), we notice that the velocity

at the tips of the drops is linear and that the stagnation points of the flow lie well

behind these hooked tips. Figure (7.19) correspondingly illustrates that these are

the same regions where the first normal stress difference is highest.

This observation is contrary to usual expectations for low Deborah numbers

De <∼ O(1), say for example as described in Pillapakkam and Singh [18], in

which they conjecture that the trace of the viscoelastic stresses in the bulk fluid

being highest near the tips and along the major axis of the drop should lead to

tip streaming in such a way as will increase the drop deformation.

On the other hand, our results above performed with a relatively high Deborah

number (De = 8) however show that the first normal stress difference being

highest (and positive) behind the top hook, on the upper surface of the drop,

and in front of the lower hook, at the lower surface of the drop, will instead

push down on the drops at the top end and up at the lower end. This offers one

explanation as to why the highly elastic cases have lower deformation angles. As

we reduce the Deborah number we notice agreement with the expectations of [18]

as shown in figure (7.21).

Figure (7.19) shows the surface and contour plots of the first normal stress differ-

ence (N1 = T11−T22) at time t = 2 for fixed capillary and Reynolds numbers but

different Deborah numbers. We make two observations as the Deborah number

is decreased from 8 in (7.19a) and (7.19b) to 4 in (7.19c) and (7.19d). The first

is that the hooks are no longer as pronounced and the second is that the highest

83

(a) (b)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(c) (d)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.19: Surface and contour plots of first normal stress difference at t = 2

where Ca = 60, R = 0.3, De = 8 in (a) & (b) and De = 4 in (c) & (d)

and positive N1 regions have migrated closer to the tips and at the same time

have correspondingly decreased in magnitude.

This trend continues as we further decrease the Deborah number in the bulk

fluid as shown in figures (7.20) and (7.21). In figure (7.20) we used the Deborah

numbers De = 2, 1 & 5. Figure (7.21) uses De = 0.1 & De = 0.01.

84

(a) (b)

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(c)

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(d)

0.005

0.01

0.015

0.02

0.025

0.03

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.20: Surface and contour plots of first normal stress difference at t = 2

where Ca = 60, R = 0.3, De = 2 in (a) & (b), De = 0.5 in (c) & De = 0.5 in (d)

(a) (b)

1

2

3

4

5

6x 10

−3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

85

(c) (d)

−1

−0.5

0

0.5

1

1.5

x 10−3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.21: Surface and contour plots of first normal stress difference at t = 2

where Ca = 60, R = 0.3, De = 0.1 in (a) & (b) and De = 0.01 in (c) & (d)

Figure (7.22) shows the surface and contour plots of the pressure for the parame-

ters, Ca = 60, R = 0.3, De = 8 in (a) & (b) and De = 4 in (c) & (d) all performed

at time t = 2.

We notice that the high (and positive) pressure regions lie below the tip of the

front hook and above the tip of the lower-end hook. The lowest (and negative)

pressure regions lie inside the “neck” regions of the hooks. Comparing figure

(7.22) to the velocity vector plots for the same parameters (say figure 7.18) we

notice as expected that the high pressure region is pushing the fluid back towards

the low pressure zones, creating the vortices in say figure (7.18).

(a) (b)

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

86

(c) (d)

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.22: Surface and contour plots of pressure at t = 2 where Ca = 60,

R = 0.3, De = 8 in (a), (b) and De = 4 in (c), (d)

Figures (7.23) and (7.24) show that as the matrix fluid becomes more and more

Newtonian, the high pressure regions migrate inside the tips of the drop along

the major axis of the drop and also, as with N1, correspondingly decrease in

magnitude as De → 0.

As expected, we also notice that the inclination of the drop increases as the

elasticity decreases and that the hooks become less and less pronounced until

they vanish altogether.

(a) (b)−1

−0.5

0

0.5

1

1.5

2

x 10−3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.23: Surface and contour plots of pressure at t = 2 where Ca = 60,

R = 0.3, and De = 0.1

87

(a)−0.5

0

0.5

1

1.5

2

2.5

3

x 10−3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

(b)−6

−4

−2

0

2

4

6

8

10

12

14

x 10−4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

z

Figure 7.24: Contour plots of pressure at t = 2 where Ca = 60, R = 0.3,

De = 0.01 in (a) and De = 0 in (b)

7.4 Purely Newtonian case and effects of inertia

We have found it necessary to include this section for reasons largely to do with

completion. For otherwise without checking the performance of our code against

well documented Newtonian results (a goal achieved by taking our viscoelastic

parameters as identically zero) we feel the thesis will lack the required complete-

ness.

Sheth and Pozrikidis [32] considered the deformation of liquid drops subject to

a simple shear flow, similar to that described in the preceding sections. For all

their computations [32] exclusively used Newtonian fluids for both the drop and

bulk phase. We intend to show that by taking the viscoelastic parameters to be

zero in our codes, we can recover the Newtonian results of [32].

In accordance with [32] we consider circular drops of radius a and viscosity mη

immersed in a fluid of viscosity η. Here m is the viscosity ratio and η = ηs since

the polymeric viscosity ηp ≡ 0. The drops are separated by a distance L, hence

we will only consider the dynamics of a single drop in a computational domain

of width L and then use periodic boundary conditions at the left and right hand

88

edges.

The height of the computational domain is 2H, the top wall moves to the right

with velocity U and the lower wall moves to the left with velocity −U . Lengths

are non-dimensionalized using the half width H hence the height of the domain

becomes 2 in non-dimensional terms. For square the computational domain

considered in [32] this leads to the non-dimensional quantities, L/H = 2 and

drop radius = a/H.

In [6], [28] and [18], the Reynolds and Capillary numbers were defined in terms of

the shear rate. In this section, we find it necessary to define these dimensionless

parameter in terms of the velocity U as done in [32]. Hence for the Reynolds

number we have

R =ρU 2a

η, (7.11)

and the capillary is given by

Ca =Uη

σ(7.12)

where σ is the surface tension of the interface.

In their numerical studies, [32] used the square computational box L/H = 2,

and the drop radius a/H = 0.5 (the fraction of the drop area π/4 to that of the

domain (L/H)× (2H) = 4 is then π/16 and is commonly referred to as the areal

fraction of the drop [32]). The authors in [32] then proceeded with computations

using mainly two values of the Capillary number Ca = 0.2 and Ca = 0.4, two

values of the viscosity ratio m = 1 and m = 2 and performed these computations

at four different values of the Reynolds number R = 1, 10, 50 and R = 100.

The figure below illustrate our results for the evolution of drop deformation in

time for the same parameters as those of [32] given above.

The results of figure (7.25) accurately reproduce those given in figures (3) and

(4) of [32]. Since the authors of [32] did not actually provide the values of the

deformation parameter and corresponding times, we had to show this agreement

89

(a)0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time

D

Re=1Re=10Re=50Re=100

(b)0 0.5 1 1.5 2 2.5 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

time

D

Re=1Re=10Re=50Re=100

(a)0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time

D

Re=1Re=10Re=50Re=100

(b)0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time

D

Re=1Re=10Re=50Re=100

Figure 7.25: Evolution of drop deformation in time (a) Ca = 0.2, m = 1, (b)

Ca = 0.2, m = 10, (c) Ca = 0.4, m = 1 & (d) Ca = 0.4, m = 10

by superposing their graphs onto ours using equal scales with the help of relevant

imaging software and ordinary transparencies!

We next show the plots of drop shapes at say time t = 3 and compare with the

qualitative behavior with that illustrated in figure (5) of [32].

We notice that the shapes in figure (7.26) give the correct qualitative features

of the corresponding shapes in figure (5) of [32]. Had they provided the times

at which they plotted those shapes we are confident that our code would have

accurately reproduced these quantitatively as well.

90

(a)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

z

Re=1Re=10Re=50Re=100

(b)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

z

Re=1Re=10Re=50Re=100

(a)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

z

Re=1Re=10Re=50Re=100

(b)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

z

Re=1Re=10Re=50Re=100

Figure 7.26: Drop shapes at t = 3 (a) Ca = 0.2, m = 1, (b) Ca = 0.2, m = 10,

(c) Ca = 0.4, m = 1 & (d) Ca = 0.4, m = 10

In conclusion, we have successfully produced a two-dimensional viscoelastic ver-

sion of the volume of fluid code with piecewise linear interface calculation and

utilizing the continuous surface force technique (VE-VOF(PLIC)-CSF). This al-

gorithm we primarily designed for the numerical simulation of two-layer flows as

well as droplet deformation in two-dimensional shear flows of viscoelastic fluids.

We have verified that the code works accurately in the Newtonian case and also

that our results are independent of mesh resolution and time step size.

91

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96

Index

97

Index

advection instability, 27

advective term, 21

amplitude equation, 14

amplitude function, 15

bifurcation, 14

bifurcation parameter, 15

bifurcation solution, 16

boundary conditions, 12, 14

capillary number, 73

CFL, 26

Chebyshev-tau scheme, 14

CIF, 21

convective, 21

Couette-Poiseuille flow, 10

CSF, 25

Deborah number, 73

deformable drops, 71

deformable interfaces, 19

deformation angle, 76, 78, 80

deformation parameter, 75, 80

density, 8

dimensionless equations, 9

drop break up, 78, 81

droplet deformation, 72, 75

eigenfunction, 15, 16

eigenvalues, 14, 15

elastic instabilities, 5

explicit scheme, 25

extra stress tensor, 10, 11, 26, 74, 79

finite differences, 20

first normal coefficient, 74

first normal difference, 74

Galerkin method, 22

Gauss-Siedel method, 21, 22

Giesekus model, 10, 74

Giesekus nonlinear parameter, 10

Hele-Shaw cell, 2

high-frequency error, 21

ill-conditioned, 21

interface markers, 18

interface perturbation, 16

interface shape, 16

interfacial conditions, 12

98

interfacial instabilities, 1

interfacial tension, 25

interfacial tension parameter, 13

kinematic condition, 13, 15

Lagrangian, 23

linear stability analysis, 14

low-frequency error, 21

MAC, 20

mean flow component, 15, 16

microgravity, 2

multigrid method, 21

Navier-Stokes equations, 19

Neumann condition, 20

no-slip conditions, 20

nonlinear analysis, 16

normal mode, 14

normal modes, 14

normal stress, 10, 13, 15

normal stress condition, 12

numerical instability, 26

Oldroyd-B model, 10, 14, 18, 26, 74

PCG, 21

periodic conditions, 20

PLIC, 22

Poisson equation, 20, 21

polymeric viscosity, 8

Projection method, 19

Rayleigh-Taylor instabilities, 2

relaxation time, 9

Reynolds number, 9, 15, 73

Saffman-Taylor instabilites, 2

second harmonic, 15, 16

second order fluids, 78

semi-implicit scheme, 24, 25, 27

shear stress, 13, 15

shear stress condition, 11, 12

shear stress tensor, 26

short, long waves, 4

small deformation theory, 75, 78

solvent viscosity, 8

stability analysis, 14

steady state deformation, 75, 78

Stokes solver, 24

stress tensor, 74

Stuart-Landau equation, 16

supercritical, 16

surface tension, 10

tangential stress, 10

thermocapillary instabilities, 2

thin-layer effect, 3

topological equation, 23

total stress tensor, 10

99

UCM, 14, 26

unit tensor, 74

upper convected derivative, 74

viscous diffusion instability, 24

viscous fingering, 19

viscous instabilities, 3

viscous stress tensor, 19

VOF, 18, 22, 25

volume fraction field, 18, 20, 22

Von-Neumann stability, 26

wavenumber, 15

weakly nonlinear, 14

Weissenberg number, 9, 15

100