approximate solutions of differential equations of non...

Approximate Solutions of Differential Equations

of Non-Newtonian Fluids Flow Arising in the

Study of Helical Screw Rheometer

By

Muhammad Zeb

CIIT/FA09-PMT-003/ISB

PhD Thesis

In

Mathematics

COMSATS Institute of Information Technology,

Islamabad - Pakistan

Fall, 2013

ii

COMSATS Institute of Information Technology

Approximate Solutions of Differential Equations of

Non-Newtonian Fluids Flow Arising in the Study of

Helical Screw Rheometer

A Thesis Presented to

COMSATS Institute of Information Technology, Islamabad

In partial fulfillment

of the requirement for the degree of

PhD (Mathematics)

By

Muhammad Zeb


Fall, 2013

iii




___________________________________________

A Post Graduate Thesis submitted to the Department of Mathematics as partial

fulfillment of the requirement for the award of Degree of PhD (Mathematics).

Name Registration Number

Muhammad Zeb CIIT/FA09-PMT-003/ISB

Supervisor

Prof. Dr Tahira Haroon

Professor Department of Mathematics Islamabad Campus.

COMSATS Institute of Information Technology (CIIT) Campus.

January, 2014

iv

Final Approval

This thesis titled




By

Muhammad Zeb


Has been approved

For the COMSATS Institute of Information Technology, Islamabad

Examiner 1: ___________________________________

Prof. Dr. Ghulam Shabbir

GIKI, Topi, Khyber Paktunkhwa

Examiner 2: ___________________________________

Dr. Khalid Saifullah Syed

Bahauddin Zakriya University, Multan

Supervisor: _________________________________________

Prof. Dr. Tahira Haroon,

Department of Mathematics CIIT, Islamabad

HoD: ______________________________________________

Prof. Dr. Moiz-ud-Din Khan

Department of Mathematics CIIT, Islamabad

Dean, Faculty of sciences ____________________________________

Prof. Dr. Arshad Saleem Bhatti

v

Declaration

I Muhammad Zeb CIIT/FA09-PMT-003/ISB hereby declare that I have produced the

work presented in this thesis, during the scheduled period of study. I also declare that I

have not taken any material from any source except referred to wherever due that amount

of plagiarism is within acceptable range. If a violation of HEC rules on research has

occurred in this thesis, I shall be liable to punishable action under the plagiarism rules of

the HEC.

Date: _________________ Signature of the student:

___________________

Muhammad Zeb


vi

Certificate

It is certified that Muhammad Zeb CIIT/FA09-PMT-003/ISB has carried out all the work

related to this thesis under my supervision at the Department of Mathematics,

COMSATS Institute of Information Technology, Islamabad and the work fulfills the

requirement for award of PhD degree.

Date: _________________

Supervisor:

______________________

Dr Tahira Haroon

Professor

Head of Department:

_____________________________

Dr Moiz-ud-Din Khan, Professor

Department of Mathematics

vii

Dedicated to

My parents, wife And

Prof. A.M. Siddiqui

ACKNOWLEDGEMENTS

To begin with the name of Almighty ALLAH, the most gracious and the most merciful;

He inculcated in me the consecration to fulfill the requirement of this thesis. Multitudes of

thanks to Him as He blessed us with the Holy Prophet, HAZRAT MUHAMMUD (PBUH)

for whom the whole universe is created and who taught us to worship Allah. He (PBUH)

brought humanity out of abysmal darkness and lit the way to Heaven. I express thanks to

the Holy Prophet Muhammad (Peace be upon Him), Who is forever a touch of guidance

for humanity.

I express my gratitude to all my teachers whose pedagogical assistance and timely presence

have helped me rise to academic zenith. In particular, I wish to express my heartiest

gratitude to my vibrant, affectionate and devoted supervisor, Dr. Tahira Haroon, for

her inspirational discussions, valuable guidance, thereby broadening and developing my

capabilities and so my amateurish potential matured. This work would never have been so

worthwhile without her help.

I further, categorically, wish to acknowledge Dr. Muhammad Akram for the invaluable

and intellectual suggestions, beneficial remarks and stunning reformative criticism that he

rendered during the course of this research work. Special thanks are due to Dr. Abdul

Majeed Siddiqui (Pennsylvania State University, USA) who provided me with novel ideas

and enlightened me about the latest literature in the field. I have learned a lot from him and

owe him abundantly for lending a priceless hand through the work.

I also take this opportunity to ink appreciation for the Higher Education Commission of

Pakistan for providing me with full financial support under Indigenous 5000 Scholarship

Batch-IV, which made my P.hD research going. Thanks to COMSATS Institute of

viii

Information Technology, HOD Department of Mathematics and Dr. S. M. Junaid Zaidi,

Rector CIIT, for providing requisite provisions and conducive research environment at CIIT

Islamabad.

Also, my current stars and stature are due to the consistent support, help and most of

all encouragement of my family. Especially I render utmost appreciation to my parents

for their unwavering support and encouragement. I cannot help expressing my apologetic

feelings for my wife and sweet children Shah Zeb, Salar Zeb and Alishba Zeb, who missed

me due to my engagement through the work.

Last but not the least; I would acknowledge the pleasant moments shared with my friends

and well-wishers specially Miss Zarqa Bano. They have been of immense help to me in the

course of the work.

Muhammad Zeb


ix

ABSTRACT


Non-Newtonian Fluids Flow Arising in the Study of Helical

Screw Rheometer

The thesis presents the theoretical analyses of extrusion process inside Helical Screw

Rheometer (HSR). Efforts to obtain better insight into the process must be mainly

theoretical rather than experimental. But the hope, of course, is that better insight

than experimental so gained will provide practical benefits such as better control of the

processing, optimize the processing process and improve the quality of production.

The main objective of the study is to develop mathematical models in order to evaluate the

velocity profiles, shear stresses and volume flow rates for isothermal flow of incompressible

non-Newtonian fluids in HSR. The calculations of these values are of great importance

during the production process. In this thesis, two types of geometries are considered.

• In first geometry the Cartesian co-ordinates system is used to study the flow of

third-grade fluid, co-rotational Maxwell fluid, Eyring fluid, Eyring-Powell fluid and

Oldroyd 8-constant fluid models in HSR. The geometry of the HSR is simplified

by unwrapping or flattening the channel, lands and the outside rotating barrel. A

shallow infinite channel is considered by assuming the width of the channel large

as compared to the depth. We also assumed that the screw surface, the lower plate,

is stationary and the barrel surface, the upper plate, is moving across the top of the

channel with a velocity at an angle to the direction of the channel. The phenomena

x

is same as, the barrel held stationary and the screw rotates. Solutions for velocity

profiles, volume flow rates, average velocity, shear and normal stresses, shear stresses

at barrel surface and shear forces exerted on the fluid are obtained using analytical

techniques. Adomian decomposition method is used to obtain the solutions for

third-grade fluid, Eyring-Powell fluid and Oldroyd 8-constant fluid and perturbation

method for co-rotational Maxwell fluid, where exact solution is obtained for Eyring

fluid model. The effects of the rheological parameters, pressure gradients and flight

angle on the velocity distributions are investigated and discussed. The behavior of

the shear stresses is also discussed with the help of graphs for different values of

non-Newtonian parameters.

• For better analysis cylindrical co-ordinates system is taken in second geometry,

assuming that the outer barrel of radius r2 is stationary and the screw of radius

r1 rotates with angular velocity Ω. Here we have used third-grade fluid model

with and without flight angle and co-rotational Maxwell fluid model with nonzero

flight angle in HSR. The analytical expressions for the velocities, shear and normal

stresses and the shear stresses exerted by the fluid on the screw, volume flow rates

and average velocity are derived using analytical techniques and the outcomes have

been presented with the help of graphs. The effects of the rheological parameters and

pressure gradients on the velocity distribution are investigated.

xi

TABLE OF CONTENTS

Preface 1

1 Introduction 7

1.1 Types of Viscometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.1 Standard Viscometers . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.2 Process Viscometers . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Helical Screw Rheometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Inviscid Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.1 Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.2 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6.2 Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . 15

1.7 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.7.1 Third Grade Fluid Model . . . . . . . . . . . . . . . . . . . . . . . 18

1.7.2 Co-rotational Maxwell Fluid Model . . . . . . . . . . . . . . . . . 19

1.7.3 Eyring Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.7.4 Eyring-Powell Fluid Model . . . . . . . . . . . . . . . . . . . . . 20

1.7.5 Oldroyd 8−Constant Fluid Model . . . . . . . . . . . . . . . . . . 20

1.8 Methods of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

xii

1.8.1 Perturbation Method (PM) . . . . . . . . . . . . . . . . . . . . . . 22

1.8.2 Homotopy Perturbation Method (HPM) . . . . . . . . . . . . . . . 23

1.8.3 Adomian Decomposition Method (ADM) . . . . . . . . . . . . . . 25

2 Analysis of Third-Grade Fluid in Helical Screw Rheometer: Adomian Decom-

position Method 27

2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Solution of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.1 Zeroth Component Solution . . . . . . . . . . . . . . . . . . . . . 38

2.2.2 First Component Solution . . . . . . . . . . . . . . . . . . . . . . 39

2.2.3 Second Component Solution . . . . . . . . . . . . . . . . . . . . . 40

2.2.4 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.5 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.6 Volume flow rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.7 Average velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Study of Co-rotational Maxwell Fluid in Helical Screw Rheometer: Perturba-

tion Method 56



3.2.1 Zeroth order problem . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.2 First order problem . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.3 Second order problem . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.4 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.5 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2.6 Volume flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 68



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3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Analysis of Eyring Fluid in Helical Screw Rheometer: Exact Solution 77



4.2.1 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2.2 Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82




4.4 conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Analysis of Eyring-Powell Fluid in Helical Screw Rheometer: Adomian

Decomposition Method 91



5.2.1 Zeroth order Solution . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.2 First order Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.3 Second order Solution . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.4 Third order Solution . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.5 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.6 Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101




5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Analytical Solution For the Flow of Oldroyd 8-Constant Fluid in Helical Screw

Rheometer 114


xiv





6.2.4 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.2.5 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124




6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7 Homotopy Perturbation Method for Flow of a Third-Grade Fluid Through

Helical Screw Rheometer with Zero Flight Angle 141






7.2.4 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.2.5 Volume flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 156


7.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8 Homotopy Perturbation Solution for Flow of a Third-Grade Fluid in Helical

Screw Rheometer 161





xv


8.2.4 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.2.5 Volume flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 174



8.4 conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9 Co-rotational Maxwell Fluid Analysis in Helical Screw Rheometer Using

Adomian Decomposition Method 184






9.2.4 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.5 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.2.6 Volume flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 198



9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

10 Conclusion 205

11 Appendices 209

11.1 Appendix I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

11.2 Appendix II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

11.3 Appendix III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

11.4 Appendix IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

11.5 Appendix V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

11.6 Appendix VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

xvi

11.7 Appendix VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

12 References 227

xvii

LIST OF FIGURES

1.1 Geometry of Helical Screw Rheometer. . . . . . . . . . . . . . . . . . . . 10

2.1 The geometry of the “unwrapped” screw channel and barrel surface. . . . . 29

2.2 Velocity profile u(y) for different values of β, keeping P,x = −2.0, P,z =

−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3 Velocity profile w(y) for different values of β, keeping P,x = −2.0, P,z =

−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4 Velocity profile s(y) for different values of β, keeping P,x = −2.0, P,z =

−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Velocity profile u(y) for different values of P,x, keeping β = 0.3, P,z =−2.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6 Velocity profile u(y) for different values of P,z, keeping β = 0.3, P,x =−2.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.7 Velocity Profile w(y) for different values of P,x, keeping β = 0.3, P,z =

−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.8 Velocity profile w(y) for different values of P,z, keeping β = 0.3, P,x =−2.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.9 Velocity profile s(y) for different values of P,x, keeping β = 0.3, P,z =−2.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.10 Velocity profile s(y) for different values of P,z, keeping β = 0.3, P,x =−2.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

xviii

2.11 Velocity profile s(y) for different values of φ, keeping β = 0.3, P,x =−2.0

and P,z =−2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.12 Variation of shear stress Sxy for different values of β, keeping P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 54

2.13 Variation of shear stress Syz for different values of β, keeping P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1 Velocity profile u(y) for different values of Wi2, keeping P,x =−1.5, P,z =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2 Velocity profile w(y) for different values of Wi2, keeping P,x =−1.5, P,z =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Velocity profile s(y) for different values of Wi2, keeping P,x =−1.5, P,z =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4 Velocity profile u(y) for different values of P,x, keeping Wi2 = 0.25, P,z =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 Velocity profile u(y) for different values of P,z, keeping Wi2 = 0.25, P,x =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.6 Velocity profile w(y) for different values of P,x, keeping Wi2 = 0.25, P,z =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.7 Velocity profile w(y) for different values of P,z, keeping Wi2 = 0.25, P,x =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.8 Velocity profile s(y) for different values of P,x, keeping Wi2 = 0.25, P,z =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.9 Velocity profile s(y) for different values of P,z, keeping Wi2 = 0.25, P,x =

−1.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.10 Velocity profile s(y) for different values of φ, keeping Wi2 = 0.25, P,x =

−1.5 and P,z =−1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.11 Variation of shear stress Sxy for different values of Wi2, keeping P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 75

xix

3.12 Variation of shear stress Syz for different values of Wi2, keeping P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 76

4.1 Velocity profile u(y) for different values of ˜α, keeping P,x =−0.5, ˜β = 0.5

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2 Velocity profile w(y) for different values of ˜α, keeping P,z =−0.5, ˜β = 0.5

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 Velocity profile s(y) for different values of ˜α keeping ˜β = 0.5 P,x =

−0.5, P,z =−0.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Velocity profile u(y) for different values of ˜β keeping ˜α = 5.0 P,x =−0.5

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.5 Velocity profile w(y) for different values of ˜β, keeping ˜α = 5.0, P,z =−0.5

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.6 Velocity profile s(y) for different values of ˜β, keeping ˜α = 5.0, P,x =

−0.5, P,z =−0.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 87

4.7 Velocity profile u(y) for different values of P,x keeping ˜α = 5.0, ˜β = 0.5

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.8 Velocity profile w(y) for different values of P,z, keeping ˜α = 5.0, ˜β = 0.5

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.9 Velocity profile s(y) for different values of P,x, keeping ˜α = 5.0, ˜β =

0.5 P,z =−0.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.10 Velocity profile s(y) for different values of P,z keeping ˜α = 5.0, ˜β =

0.5 P,x =−0.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.11 Velocity profile s(y) for different values of φ, keeping ˜α = 5.0, ˜β =

0.5, P,x =−0.5 and P,z =−0,5. . . . . . . . . . . . . . . . . . . . . . . . 90

5.1 Velocity profile u(y) for different values of ˜α, keeping P,x =−0.5,˜β = 1.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 Velocity profile w(y) for different values of ˜α, keeping P,z =−0.5,˜β = 1.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

xx

5.3 Velocity profile u(y) for different values of˜β, keeping P,x =−0.5, ˜α = 1.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4 Velocity profile w(y) for different values of˜β, keeping P,z =−0.5, ˜α = 1.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5 Velocity profile u(y) for different values of P,x, keeping ˜α = 1.0,˜β = 1.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.6 Velocity profile w(y) for different values of P,z, keeping ˜α = 1.0,˜β = 1.0

and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.7 Velocity profile s(y) for different values of ˜α, keeping˜β = 1.0, P,x =

−0.5, P,z =−0.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 109

5.8 Velocity profile s(y) for different values of˜β, keeping ˜α = 1.0, P,x =

−0.5, P,z =−0.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 109

5.9 Velocity profile s(y) for different values of P,x, keeping ˜α = 1.0,˜β =

1.0, P,z =−0.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.10 Velocity profile s(y) for different values of P,z, keeping ˜α = 1.0,˜β =

1.0, P,x =−0.5 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.11 Velocity profile s(y) for different values of φ, keeping ˜α = 1.0,˜β =

1.0, P,x =−0.5 and P,z =−0,5. . . . . . . . . . . . . . . . . . . . . . . . 111

5.12 Variation of shear stress Sxy for different values of ˜α, keeping P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 111

5.13 Variation of shear stress Sxy for different values of˜β, keeping P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 112

5.14 Variation of shear stress Syz for different values of ˜α, keeping P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 112

5.15 Variation of shear stress Syz for different values of˜β, keeping P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 113

6.1 Velocity profile u(y) for different values of α, keeping β = 0, 0.4, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 131

xxi

6.2 Velocity profile u(y) for different values of β, keeping α = 0, 0.4, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 132

6.3 Velocity profile w(y) for different values of α, keeping β = 0, 0.4, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4 Velocity profile w(y) for different values of β, keeping α = 0, 0.4, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 133

6.5 Velocity profile s(y) for different values of α, keeping β = 0, 0.4, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 133

6.6 Velocity profile s(y) for different values of β, keeping α = 0, 0.4, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 134

6.7 Velocity profile u(y) for different values of P,x, keeping α = 0.4, β =

0.2, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.8 Velocity profile u(y) for different values of P,z, keeping α = 0.4, β =

0.2, P,x =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.9 Velocity profile w(y) for different values of P,x, keeping α = 0.4, β =

0.2, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.10 Velocity profile w(y) for different values of P,z, keeping α = 0.4, β =

0.2, P,x =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.11 Velocity profile s(y) for different values of P,x, keeping α = 0.4, β =

0.2, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.12 Velocity profile s(y) for different values of P,z, keeping α = 0.4, β =

0.2, P,x =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.13 Velocity profile s(y) for different values of φ, keeping α = 0.4, β =

0.2, P,x =−2.0 and P,z =−2.0. . . . . . . . . . . . . . . . . . . . . . . . 137

6.14 Variation of shear stress Sxy for different values of α, keeping β = 0.2, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 138

6.15 Variation of shear stress Sxy for different values of β, keeping α = 0.4, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 138

xxii

6.16 Variation of shear stress Syz for different values of α, keeping β = 0.2, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 139

6.17 Variation of shear stress Syz for different values of β, keeping α = 0.4, P,x =

−2.0, P,z =−2.0 and φ = 45. . . . . . . . . . . . . . . . . . . . . . . . . 139

7.1 Vertical concentric annulus. . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.2 Velocity profile v(r) for different values of β, keeping δ = 2.0 and λ = 0.4. 158

7.3 Velocity profile w(r) for different values of β, keeping keeping δ = 2.0 and

λ = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.4 Velocity profile v(r) for different values of β, keeping δ = 2.0 and λ = 0.4. 159

7.5 Velocity profile w(r) for different values of β, keeping keeping δ = 2.0 and

λ = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.1 Geometry of problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.2 Fluid element bounded by helical surface, root and barrel diameter and

planes θ and z constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.3 Velocity profile v(r) for different values of β, keeping P,θ = −4.0 , P,z =

−4.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8.4 Velocity profile w(r) for different values of β, keeping P,θ = −4.0, P,z =

−4.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8.5 Velocity profile v(r) for different values of P,θ, keeping β = 0.4, P,z =−4.0

and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.6 Velocity profile v(r) for different values of P,z, keeping β = 0.4, P,θ =−4.0

and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.7 Velocity profile w(r) for different values of P,θ, keeping β = 0.4, P,z =

−4.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.8 Velocity profile w(r) for different values of P,z, keeping β = 0.4, P,θ =

−4.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.9 Variation of shear stress Srθ for different values of β, keeping P,θ =

−2.0, P,z =−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 181

xxiii

8.10 Variation of shear stress Srθ for different values of β, keeping P,θ =

−2.0, P,z =−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.11 Variation of shear stress Srz for different values of β, keeping P,θ =

−2.0, P,z =−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.1 Velocity profile v(r) for different values of α, keeping P,θ = −2.0 , P,z =

−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

9.2 Velocity profile w(r) for different values of α, keeping P,θ = −2.0, P,z =

−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

9.3 Velocity profile v(r) for different values of P,θ, keeping α = 0.3, P,z =−2.0

and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9.4 Velocity profile v(r) for different values of P,z, keeping α = 0.3, P,θ =−2.0

and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9.5 Velocity profile w(r) for different values of P,θ, keeping α = 0.3, P,z =

−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.6 Velocity profile w(r) for different values of P,z, keeping α = 0.3, P,θ =

−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.7 Variation of shear stress Srθ for different values of α, keeping P,θ =

−2.0, P,z =−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.8 Variation of shear stress Srz for different values of α, keeping P,θ =

−2.0, P,z =−2.0 and δ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 203

xxiv

LIST OF ABBREVIATIONS

V Velocity fieldT Cauchy stress tensorS Extra stress tensorρ Fluid densityf Body force per unit massI Unit tensorP Dynamic pressureµ Fluid viscosityDDt

Material time derivative

h Channel depthB Width of the channelV Velocity of the barrelU Component of barrel velocity along x-axisW Component of barrel velocity along z-axisV0 Non-dimensional velocity of barrelφ Flight anglex Direction perpendicular to the wall of channelz Direction along down channely Direction along depth of channelu Component of fluid velocity in x-directionw Component of fluid velocity in z-directionv Azimuthal velocity componentr1 Radius of inner cylinderr2 Radius of outer cylinders Velocity in the direction of the axis of screwβ Dimensionless non-Newtonian parameter for third grade fluidα Dimensionless non-Newtonian parameter for co-rotational Maxwell fluid˜β Dimensionless non-Newtonian parameter for Eyring fluid˜α Dimensionless non-Newtonian parameter for Eyring fluid˜β Dimensionless non-Newtonian parameter for Eyring-Powell fluid

xxv

˜α Dimensionless non-Newtonian parameter for Eyring-Powell fluidC material constantB material constantα Dilatant constantβ Pseudoplastic constantWi Weissenberg numberδ Non-dimensional ratioP,x Pressure gradient in x−directionP,z Pressure gradient in z−directionP,θ Pressure gradient in θ−directionQx Volume flow rate in x−directionQθ Volume flow rate in θ−directionQz Volume flow rate in z−directionQ Resultant volume flow rate forward in the screw channelL Differential operatorL−1 Inverse of differential operatorN Number of parallel flights in a multiflight screw∇

(∗) Upper contravariant convected derivativeη0 Zero shear viscosityλ1 Relaxation timeλ2 Retardation time

xxvi

Preface

Helical Screw Rheometer (HSR) is used for rheological measurements of fluid food

suspensions. It contains a helical screw enclosed in a tight fitting cylinder called barrel, with

inlet and outlet ports. Rotation of screw creates a pressure gradient which increases linearly

along the axis of the screw, and is proportional to the viscosity of the fluid. Geometry

of HSR is very similar to a single screw extruder. Single Screw extruders are mostly

used for food processing such as breakfast cereals, pasta, cookie dough, sevai, snacks,

candy, confectionery products, french fries, baby food, and pet foods etc. During extrusion

processing screw extruder performs several functions, from the moment the raw material

enters the hopper to when melted material exits through the die with a specific cross

sectional profile. These functions include melting process, pressing and pumping, mixing

the melt and finally pushing the melt through the die. Carley et al.[1], Mohr and Mallouk

[2], Booy [3], Squires [4], Tadmor and Klein [5], Tadmor and Gogos [6], Rauwendaal [7],

Bird et al.[8] and of many others have studied such operations using mostly, Newtonian and

power law fluid models. Tamura et al.[9], had tried successfully, the preceding analysis in

the geometry of HSR (see Figure 1.1), for Newtonian and power law fluids.

In recent years, the study of non-Newtonian fluids have attracted many researchers. This

is mostly due to their wide use in food industry, chemical process industry, construction

engineering, power engineering, petroleum production, commercial and technological

applications. The rheological knowledge of such fluids is of special importance owing

to its application to many industrial problems. Rheological properties of fluids are

based, in general, on their so-called constitutive equations. The analysis of the behavior

of the fluid motion of the non-Newtonian fluids is more complicated than Newtonian

1

fluids due to the nonlinear relationship between stress and rate of strain. The governing

equation that describes the flow of an incompressible Newtonian fluid is the Navier-

Stokes equation. For the flow of non-Newtonian fluids, there is not a single governing

equation which shows their properties in entirety. Many models or constitutive equations

have been proposed, most of them are empirical or semi-empirical. Amongst fluids of

existing viscoelastic behavior, the fluids of differential type and those of rate type have

gained a great deal of attention. The constitutive equations of non-Newtonian fluids make

the governing equations more complicated involving a number of parameters and the

exact solutions are even rare in literature for these equations. Most of the natural and

industrial occurring problems when modeled, show nonlinearity. Nonlinearity increases

the mathematical complexity of the problems which reduces the chance of getting exact

solutions. In view of such difficulties, in past three decades, researchers and scientists

developed numerous analytical and numerical techniques to overcome nonlinearity and get

approximate solutions. Various analytical techniques such as perturbation method (PM),

Adomian decomposition method (ADM), homotopy analysis method (HAM), optimal

homotopy analysis method (OHAM), homotopy perturbation method (HPM), optimal

homotopy perturbation method (OHPM), variational iteration method (VIM) and some

others methods have been proven to be valuable tools to solve these types of complex

problems. These techniques have found profuse application in industry and technology.

This study develops the theoretical flow analyses for non-Newtonian fluids in HSR. The

obtained nonlinear ordinary differential equations are solved using PM [10, 11], HPM

[12–14] and ADM [15–17], with the help of symbolic computation software Wolfram

Mathematica 7. Many fluids used in processing, manufacturing and chemical industry are

considerably non-Newtonian in nature. Industrial applications demand concentrated study

of this kind of fluids. Therefore it is essential to extend the theoretical analysis in a more

concise way to study the flow behavior of the non-Newtonian fluid. Thus, in this study, the

focus is on non-Newtonian fluids. In this perspective, we present chapters 2 - 12.

In this thesis, we have considered the flow of non-Newtonian fluid models, namely

2

• Third-grade fluid

• Co-rotational Maxwell’s fluid

• Eyring fluid

• Eyring-Powell fluid

• Oldroyd 8−constant fluid

in HSR, by calculating the velocity profiles, shear stresses and volume flow rates, which

are of great importance during the production process. Expressions for the shear stresses,

shear at barrel surface, forces exerted on fluid, and average velocity are also calculated. In

this work, two types of geometries have been considered.

1. Using Cartesian co-ordinates flow of third-grade, co-rotational Maxwell, Eyring-

Powell and Oldroyd 8−constant fluids are studied in HSR by unwrapping or

flattening the channel, lands and the outside rotating barrel. The geometry is

approximated as a shallow infinite channel, by assuming the width of the channel

large as compared to the depth. Using analytical techniques, PM and ADM, the

analytical expressions for the velocity components in x and z−directions, also of the

resultant velocity in direction of the screw axis are obtained. Volume flow rates, shear

stresses, shear at barrel surface, forces exerted on fluid, and average velocity are also

calculated. Exact solution is obtained for velocity profiles and volume flow rates,

shear stresses, shear at barrel surface, forces exerted on fluid, and average velocity in

the case of Eyring fluid. The results have been discussed with the help of graphs as

well. The effects of the rheological parameters, pressure gradients and flight angle

on the velocity distribution are investigated. Problem (i) covers chapter 2, chapter 3,

chapter 4, chapter 5, and chapter 6 of this thesis.

2. Using cylindrical co-ordinates the flow of third grade fluid is considered in HSR with

and without flight angle. Also, the flow of co-rotational Maxwell fluid with the effect

of flight angle is considered, assuming that the barrel of radius r2 is stationary and

3

the screw of radius r1 rotates with angular velocity Ω. Using analytical techniques,

HPM and ADM, the analytical expressions for the velocity components in θ and

z−directions are obtained. Volume flow rates, average velocity, shear and normal

stresses and the shear stresses exerted by the fluid on the screw are also calculated.

The result for velocities have been discussed with the help of graphs. The effects

of the rheological parameters and pressure gradients on the velocity distribution are

investigated. The velocity profiles strongly depend on these factors. Problems with

this geometry are discussed in chapters 7, 8, and 9 of this thesis.

Chapter wise summary of this thesis is as follows:

Chapter 2 covers the study of steady flow of an incompressible, third grade fluid in

HSR. The developed second order nonlinear coupled differential equations are reduced

to single differential equation using a transformation. Using ADM we obtained analytical

expressions for the velocity components, also the resultant velocity along the screw axis.

Expressions for the shear stresses, shear at barrel surface, forces exerted on fluid, and

average velocity are also calculated. The behavior of velocities have been discussed with

the help of graphs. We observe that the velocity profiles are strongly dependant on non-

Newtonian parameter, pressure gradient, and flight angle. The behavior of the shear stresses

is also discussed with the help of graphs for different values of non-Newtonian parameters.

The contents of this chapter are published in the Journal of Applied Mathematics, Vol

2013 Article ID 620238, 11 pages.

Chapter 3 concerns with the study of steady flow of an incompressible, co-rotational

Maxwell fluid. The developed second order nonlinear coupled differential equations are

reduced to single differential equation by using a transformation. Using PM we obtained

analytical expressions for the velocities. Volume flow rates, shear stresses, shear at barrel

surface, forces exerted on fluid, and average velocity are also derived. We observed that the

velocity profiles are strongly depend on non-dimensional parameter , pressure gradients,

and flight angle. Thus extrusion process can be increased by increasing the involved

non-dimensional parameters. Graphical representation is also given to note the variation

4

of shear stresses with respect to non-Newtonian parameter. The work of this chapter is

accepted in Analysis and Applications.

Chapter 4 is devoted to the flow of an incompressible, isothermal Eyring fluid in HSR.

Exact solutions are obtained for the velocities in x, z−directions and also in the direction

of the axis of the screw. Shear stresses, shear at barrel surface, forces exerted on fluid,

volume flow rates and average velocity by solving the second order nonlinear differential

equations are also obtained. The flow profiles are discussed with the help of graphs. We

observed that the velocities are increasing with the increase in Eyring fluid parameters. It

is also noticed that the flow increases as the flight angle increases. This work conclude

that involved parameters play a vital role in the extrusion process. The contents of the this

chapter are submitted to the Proceedings of the Romanian Academy-series A.

Chapter 5 aims to study the flow of an incompressible, isothermal Eyring-Powell fluid

in HSR. The developed second order nonlinear differential equations are solved by using

ADM. Analytical expressions are obtained for the velocity profiles, shear stresses, shear

at barrel surface, force exerted on fluid, volume flow rates and average velocity. The flow

profiles and the changing behavior of shear stresses are discussed with the help of graphs

as well. We observed that the velocity profiles are strongly depend on dimensionless non-

Newtonian parameters with the increase in the value of flow parameters velocity profiles

increase progressively, which conclude that extrusion process increases with the increase in

the values of involved parameters. It is also noticed that the flow increases as the flight angle

increase. The material of the this chapter is accepted in The Scientific World Journal.

Chapter 6 provides the study of steady flow of an incompressible, Oldroyd 8-constant

fluid in HSR. The developed second order nonlinear coupled differential equations are

transformed to a single differential equation and then are solved using ADM. We obtained

analytical expressions for the velocity components, also the resultant velocity in direction

of the screw axis. Volume flow rates, shear stresses, shear at barrel surface, force exerted

on fluid and average velocity are calculated from the velocity components in x and

z−directions. The results have been discussed with the help of graphical representations.

We observe that the velocity profiles are strongly depend on non-dimensional parameter,

5

pressure gradients and flight angle.

Chapter 7 considers a theoretical study on steady incompressible flow of third grade

fluid with zero flight angles (a vertical concentric annulus). The developed second order

non linear coupled differential equations are solved by HPM. Expressions for velocity

components, shear and normal stresses, the shear stresses exerted by the fluid on the screw,

volume flow rate and average velocity are derived. The results are also shown graphically.

The work of this chapter is published in International Journal of Nonlinear Science and

Numerical Simulation, 13 (2012) 281 - 187.

Chapter 8 gives a theoretical study of steady incompressible flow of a third grade fluid. The

model developed in cylindrical co-ordinates pertains to second order non linear coupled

differential equations that are solved by HPM. Expressions for velocity components in θ

and z−direction are obtained. The volume flow rates are calculated for the azimuthal and

axial components of velocity profiles by introducing the effect of flights. Expressions for

shear and normal stresses, the shear stresses exerted by the fluid to the screw and average

velocity are also calculated. The contents of this chapter are submitted to the Journal of

University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics

and Physics for possible publication.

Chapter 9 investigates a theoretical study of steady incompressible flow of co-rotational

Maxwell fluid. The rheological constitutive equation for co-rotational Maxwell fluid

model gives the second order nonlinear coupled differential equations which could not

be solved explicitly. An iterative procedure, ADM is used to obtain the analytical solution.

Expressions for velocity components in θ and z−direction are obtained. Expressions for

volume flow rates, shear and normal stresses, the shear stresses exerted by the fluid on

the screw and average velocity are also calculated. The effect of involved flow parameters

investigated on the flow profiles through graphs. The work of this chapter is submitted to

the Journal of Proceedings of the Romanian Academy - Series A for possible publication.

Chapter 10 concludes this thesis. Appendices and references are given at the end of this

thesis.

6

Chapter 1

Introduction

7

Introduction

Extrusion process plays an important role in our daily life. Processing of food materials

has become an increasingly important manufacturing method with substantially broadened

applications. Today, a variety of products such as breakfast cereals, pasta, cookie dough,

sevai, snacks, candy, confectionery products, french fries, baby food, and pet foods are

made through extrusion processes. Extrusion process also includes fluids like multi-

grade oils, liquid detergents, paints, polymer solutions, ceramics, concrete and polymer

melts [18]. The injection molding process for polymeric materials, the production of

pharmaceutical products and processing of plastics materials [19] comes in the extrusion

processing. These products determine a high standard of living that we nowadays take for

granted. During extrusion process a material is pushed or drawn through a die of the desired

cross-section to create objects of a fixed cross-sectional profile. Extrusion processing is a

combination of several processes, including fluid flow, heat and mass transfer, mixing,

shearing, particle size reduction, melting, texturizing, caramelizing, plasticizing, shaping,

and forming. Depending upon the product, one or many of these processes will take place

in an extruder.

During processing, noticeable physical and chemical changes can occur. Study of

rheological characteristics of fluids is essential in the process of processing, to obtain

the desired quality and shape of the products. Viscosity is one of the many rheological

parameters, significant in the physical and chemical composition of the fluid.

1.1 Types of Viscometers

For measuring rheological properties of fluids in industries, mostly in food industry, the

available instruments are different types of viscometers.

1.1.1 Standard Viscometers

These viscometers are mostly used in laboratories and are mainly of two types

8

• Flow type Viscometers

• Rotational Viscometers

The flow type viscometers are Bostwick consistometer and capillary viscometers. The three

most common rotational viscometers are concentric cylinder viscometers, cone and plate

viscometers and parallel plate viscometers.

1.1.2 Process Viscometers

At industrial level, the process viscometers used for processing have complicated flow

fields. They are mostly devised for Newtonian fluids and usually adaptations of laboratory

viscometers. These viscometers are:

• Vibrational Viscometer

• Automatic Efflux Cup Viscometer

• Capillary Viscometer

• Falling Cylinder Viscometer

• Falling Ball Viscometer

• Rotational Viscometers

All these viscometers have their own advantages and limitations, they do not measure the

fundamental physical parameters absolutely, and are empirical in nature [20, 21].

An alternate instrument which was first developed by Kraynik, et al., for use in coal lique-

faction processing and could characterize fluid suspensions accurately and consistently is

called Helical Screw Rheometer [21].

9

1.2 Helical Screw Rheometer

Helical Screw Rheometer (HSR) is used for rheological measurements of fluid food

suspensions. It contains a helical screw enclosed in a tight fitting cylinder called barrel,

with inlet and outlet ports. Rotation of screw creates a pressure gradient which increases

linearly along the axis of the screw, and is proportional to the viscosity of the fluid [21].

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r

pre

ssure

tra

nsd

uce

r

pressure differential

z

r

B

r

r

1

2

Chan

nel

wid

th

Chan

nel

dep

th

h

xx

xxxxxxxx

xxxxxxxxx

xxxxxxxxx

Screw Flight

Ω

Inlet port (hopper)

Outlet port (die)

Figure 1.1: Geometry of Helical Screw Rheometer.

1.3 Fluid

Our universe is made up of different types of materials. The material which has no

resistance to shearing force is called fluid (liquids and gases). In other words, a substance

that continuously deforms (flow) under an applied shear stress is called fluid. The fluids

are divided in two groups on the basis of their response to normal stresses (or pressure)

acting on fluid elements. When a fluid element adjusts its volume, consequently density, in

the reaction of applied pressure is called compressible fluid. When no volume or density

change occurs with applied force (pressure or temperature) in the fluid element is called

incompressible fluid [22].

10

Fluids are further classified on the basis of their behavior in shear (viscosity), inviscid fluids

(ideal fluids) and viscous fluids.

1.4 Inviscid Fluids

The fluids of vanishingly small viscosity µ≈ 0 are called inviscid (ideal) fluids.

1.5 Viscous Fluids

Viscous fluids are of two types

• Newtonian fluids (Linearly viscous fluids)

• Non-Newtonian fluids.

1.5.1 Newtonian Fluids

The fluids have small, stiff molecules and exhibit no memory, also having constant viscosity

independent of shear rate and type of flow, shear or extensional. Such types of fluids follow

a linear relationship between shear stress, τ and rate of shear strain,dγdt

. Mathematically it

can be written as:

τ = µdγdt

= µdudy

, (1.1)

where µ is the dynamic viscosity of the fluid, γ is infinitesimal strain tensor anddudy

is the

velocity gradient. The linear relationship (1.1) implies that these fluids are linearly viscous

or Newtonian fluids.

1.5.2 Non-Newtonian Fluids

The fluids of complex molecular structures, particularly with long chain molecules, exhibit

viscosity which changes with the rate of shear strain and with type of deformation. Such

types of fluids do not follow a linear relationship between shear stress and rate of shear

11

strain. Example of non-Newtonian fluids are industrial materials, such as polymer melts,

drilling muds, clay coatings, paints, gels, oils, soaps, rubbers, inks, concrete, ketchup,

pastes, suspensions, slurries, biological liquids such as blood and foodstuffs [8, 23]. Non-

Newtonian fluids are divided into three main groups;

• Time Independent (Visco-Inelastic) Fluids

• Time Dependent Fluids

• Viscoelastic Fluids

Time Independent (Visco-Inelastic) Fluids

Fluids of this type are isotropic and homogeneous at rest. In simple unidirectional shear,

this sub-set of fluids is characterized by the fact that the current value of the rate of shear

strain at a point in the fluid is determined only by the corresponding current value of the

shear stress and vice versa [24]. This means that rate of shear strain for these fluids at any

point of the fluid is only the function of the instantaneous shear stress at that point, which

can be mathematically written asdγdt

= f (τ), (1.2)

These nonlinearly viscous fluids are also named as generalized Newtonian fluids because

their constitutive equations are similar to Newton’s law of viscosity; however, the viscosity

itself is a function of shear rate. Moreover, fluids of this type exhibit shear thinning or shear

thickening viscosity, never showing normal stresses in viscometric flow and no elasticity

or memory. These fluids are further classified into three types depending on the nature of

equation (1.2): which are: pseudoplastic, dilatant and bingham plastic fluids.

Pseudoplastic Fluids These fluids show no yield stress. A flow curve for these materials

indicate that the ratio of shear stress to the rate of shear strain gradually decreases with

increase in the shear rate. At a very low and very high shear rates, these fluids behave like

Newtonian fluids as the slopes are almost linear [25].

Dilatant Fluids These types of fluids are similar to pseudoplastic fluids showing no yield

12

stress, but their apparent viscosity increases with increasing the shear rate and hence they

are named as shear-thickening. Dilatancy of the material is the isothermal reversible

increase of viscosity with increasing shear rate with no measurable time dependence. This

process may or may not be accompanied by notable volume change.

Bingham plastics Bingham plastic fluids exhibit a solid like configuration and flow when

sheared by an external stress bigger than a characteristic stress, the yield stress τ0. This

can be shown by a flow curve that is a straight line having an intercept τ0 on the shear

stress axis. Beyond the point where the yield stress occurs these materials behave like

Newtonian, shear thinning, or shear thickening fluids and below this point, they behave

like elastic solids. paints, ketchup, mayonnaise, are examples of bingham plastic.

Time Dependent Fluids

In some fluids the change in viscosity does not only depend on the applied shear stress or

on the shear rate alone, but also on the duration for which the fluid has been subjected to

shearing as well as their previous kinematic history [24]. Such changes are reversible or

irreversible. The fluids that show an increase in viscosity with passage of time are called

rheopectic [25], which means shear thickening with time. On the other hand, the fluids

that show a decrease in viscosity with time are called thixotropic, i.e., shear thinning with

time. Rheopexy and thixotropy are time dependent effects; not shear dependent effects.

Rheopexy is essentially the reverse of thixotropy. In this case, gradual formation of a

structure is accompanied by shear. Polymeric melts and polymeric solutions are example

of shear thinning while suspensions and emulsions are examples of shear thickening.

Viscoelastic Fluids

The nonlinearly viscous fluids which accommodate a certain level of elasticity and memory

in addition to the shear thinning or shear thickening viscosity are called the viscoelastic

fluids. Due to this characteristic of material a certain amount of energy is stored in the

fluid as a strain energy in addition to viscous dissipation in the form of heat during flow. In

13

viscoelastic fluids, the strain no matter how small it may be is responsible for the fluids’s

partial recovery to its original state and the reverse flow that ensues, on the removal of the

stress. The natural state of a viscoelastic fluid changes continuously in flow. It tries to

attain the instantaneous state, but it never completely regain. This lag helps in measuring

the elasticity of the fluid or the so-called memory of the fluid. For linearly viscous fluids

the deformation and the stress are in phase, while for viscoelastic fluids, both are out of

phase, and therefore viscoelastic fluids may continue to be under stress even under zero

deformation or shear rate. Their viscous character is controlled by their ability to orient

themselves differently under different flow conditions, which gives rise to shear thinning.

The flexibility and ability of elastic macromolecules to respond to shear and extensional

deformation controls elastic character of the fluid. All materials of polymeric origin (melts,

solutions, suspensions, emulsions) are viscoelastic. Maxwell model is a simple viscoelastic

model, which is based on the assumption that a viscoelastic fluid exhibits both viscous

resistance to flow and the elastic resistance to continuous deformation, measured by its

viscosity, and by relaxation time, respectively.

1.6 Basic Equations

The basic equations governing the motion of an isothermal, homogeneous and incompress-

ible fluid are conservation of mass and balance of linear momentum.

1.6.1 Conservation of Mass

The law of conservation of mass states that “The rate of increase of the mass of the fluid

within the control volume V is equal to the net influx of fluid across the bounding surface S”

[26]. Mathematical equation for the conservation of mass is called equation of continuity.

For unsteady flow of compressible fluids this equation can be written as

DρDt

+ρdivV = 0. (1.3)

14

In equation (1.3), ρ denotes fluid density and V is the velocity vector andDDt

is the material

time derivative defined asD(∗)Dt

=∂∂t

(∗)+(V ·∇)(∗), (1.4)

which is the combination of local contribution and convective contribution respectively.

Equation (1.3) can also be written as

∂ρ∂t

+div(ρV) = 0. (1.5)

If the fluids have constant density then (1.5) takes the form

divV = 0, (1.6)

where equation (1.6) is the continuity equation for incompressible fluids.

Component form of equation (1.6) in Cartesian co-ordinates (x,y,z) is,

∂ux

∂x+

∂uy

∂y+

∂uz

∂z= 0, (1.7)

where ux, uy and uz are the velocity components in x, y and z directions, respectively. In

cylindrical co-ordinates this equation can be written as,

1r

∂(r ur)∂r

+1r

∂uθ∂θ

+∂uz

∂z= 0. (1.8)

where ur, uθ and uz are the velocity components in cylindrical coordinates.

1.6.2 Balance of Linear Momentum

The balance of momentum leaving and entering a control volume, has to be in equilibrium

with the stresses T and the body forces ρf gives a typical equation in vector form is

ρDVDt

= divT+ρf, (1.9)

15

where f is the body force per unit mass and T is the Cauchy stress tensor, T is given as:

T =−PI+S, (1.10)

where P denotes the dynamic pressure, I the unit tensor and S denotes the extra stress

tensor.

Equation (1.9) with equation (1.10) in Cartesian co-ordinate (x,y,z) results in the following

three components,

x−Component of Momentum Equation

ρ[

∂ux

∂t+ux

∂ux

∂x+uy

∂ux

∂y+uz

∂ux

∂z

]=−∂P

∂x+

∂Sxx

∂x+

∂Sxy

∂y+

∂Sxz

∂z+ρ fx. (1.11)

y−Component of Momentum Equation

ρ[

∂uy

∂t+ux

∂uy

∂x+uy

∂uy

∂y+uz

∂uy

∂z

]=−∂P

∂y+

∂Syx

∂x+

∂Syy

∂y+

∂Syz

∂z+ρ fy. (1.12)

z−Component of Momentum Equation

ρ[

∂uz

∂t+ux

∂uz

∂x+uy

∂uz

∂y+uz

∂uz

∂z

]=−∂P

∂z+

∂Szx

∂x+

∂Szy

∂y+

∂Szz

∂z+ρ fz, (1.13)

where V = (ux,uy,uz), f = ( fx, fy, fz) and

S =

Sxx Sxy Sxz

Syx Syy Syz

Szx Szy Szz

. (1.14)

In cylindrical co-ordinates (r,θ,z), if velocity vector V is defined as V = (ur,uθ,uz), then

16

r−Component of Momentum Equation

ρ[

∂ur

∂t+ur

∂ur

∂r+

uθr

∂ur

∂θ+uz

∂ur

∂z− u2

θr

]= −∂P

∂r+

1r

∂(rSrr)∂r

+1r

∂Sθr

∂θ+

∂Szr

∂z

− Sθθr

+ρ fr. (1.15)

θ−Component of Momentum Equation

ρ[

∂uθ∂t

+ur∂uθ∂r

+uθr

∂uθ∂θ

+uz∂uθ∂z

+uruθ

r

]= −1

r∂P∂θ

+1r2

∂(r2Srθ)∂r

+1r

∂Sθθ∂θ

+∂Szθ∂z

− (Sθr−Srθ)r

+ρ fθ. (1.16)

z−Component of Momentum Equation

ρ[

∂uz

∂t+ur

∂uz

∂r+

uθr

∂uz

∂θ+uz

∂uz

∂z

]= −∂P

∂z+

1r

∂(rSrz)∂r

+1r

∂Sθz

∂θ

+∂Szz

∂z+ρ fz. (1.17)

where f = ( fx, fθ, fz) and

S =

Srr Srθ Srz

Sθr Sθθ Sθz

Szr Szθ Szz

. (1.18)

1.7 Constitutive Equation

The fundamental relation between stress and deformation during flow is called the consti-

tutive equation. It is an equation of state under flow and deformation that differentiates

the behavior of the rheologically different fluids, even when subjected under the same

flow conditions. Some fluids, for example, water can flow easily even under infinitesimal

pressure and stress gradient. While other fluids, such as ketchup requires comparatively

large pressure and stress gradients. The constitutive equation relates the stress components

to the velocity field and therefore provides solution to the equation of motion. The

17

constitutive equation is not a conservation equation. It is only a characteristic of the fluid,

which keeps a relationship among the stress, velocity and velocity derivatives. In this thesis,

we have considered the homogeneous, incompressible and isothermal non-Newtonian fluid

models particularly for third-grade fluid, co-rotational Maxwell fluid, Eyring fluid, Eyring-

Powell fluid and Oldroyd 8−constant fluid. Their constitutive equations are as follows:-

1.7.1 Third Grade Fluid Model

The constitutive equation for a third-grade fluid can be expressed as [27],

S = µA1 +α1A2 +α2A21 +β1A3 +β2(A1A2 +A2A1)+β3(trA2

1)A1, (1.19)

where µ is the viscosity of the fluid, α1, α2, β1, β2 and β3 are the material constants,

A1, A2 and A3 are the first three Rivlin-Ericksen tensors defined as

A1 = (∇V)+(∇V)T , (1.20)

An+1 =DAn

Dt+[An(∇V)+(∇V)T An], (n = 1,2), (1.21)

where superscript T stands for the transpose of the tensor and ∇V is the velocity gradient

in Cartesian co-ordinate (x,y,z) can be expressed as

∇V =

∂ux

∂x∂uy

∂x∂uz

∂x∂ux

∂y∂uy

∂y∂uz

∂y∂ux

∂z∂uy

∂z∂uz

∂z

. (1.22)

and in cylindrical co-ordinates (r,θ,z),

∇V =

∂ur

∂r1r

∂ur

∂θ− uθ

r∂ur

∂z∂uθ∂r

1r

∂uθ∂θ

+ur

r∂uθ∂z

∂uz

∂r1r

∂uz

∂θ∂uz

∂z

. (1.23)

18

On substituting α1 = α2 = β1 = β2 = β3 = 0 and β1 = β2 = β3 = 0 respectively, in equation

(1.19), we get the constitutive equations for Newtonian and second-grade fluids

S = µA1, (1.24)

S = µA1 +α1A2 +α2A21. (1.25)

1.7.2 Co-rotational Maxwell Fluid Model

The constitutive equation for co-rotational Maxwell fluid in terms of extra stress tensor and

first Rivlin-Ericksen tensor is defined as:

S+λ1∇S+

12

λ1(A1S+SA1) = η0A1, (1.26)

where η0 and λ1 are zero shear viscosity and relaxation time, respectively. The upper

contravariant convected derivative designated by ∇ over S is defined as

∇(∗) =

D(∗)Dt

−(∇V)T (∗)+(∗)(∇V)

. (1.27)

The constitutive equation for Newtonian fluid can be obtained by setting λ1 = 0 and η0 = µ

in equation (1.26).

1.7.3 Eyring Fluid Model

The constitutive equation for Eyring fluid [28] is

S =

B sinh−1(− 1

C|A1|

)

|A1|

A1, (1.28)

19

where |A1| =√

12

tr(A21) and B , C are material constants. This model predicts pseudo-

plastic behaviour at finite values of stress components. Furthermore this model reduces to

Newtonian fluid model for µ =BC

, µ is the viscosity of the fluid.

1.7.4 Eyring-Powell Fluid Model

The constitutive equation for Eyring-Powell fluid [29] is given by

S = µA1 +

1B

sinh−1(

1C|A1|

)

|A1|

A1, (1.29)

where µ is viscosity, C and B are material constants. As β→∞, model reduce to Newtonian

fluid.

1.7.5 Oldroyd 8−Constant Fluid Model

The constitutive equation for Oldroyd 8−constant fluid can be defined as:

S+λ1∇S+

12(λ1−µ1)(A1S+SA1)+

12

µ0(tr(S))A1 +12

ν1(tr(SA1))I =

= η0

A1 +λ2

∇A1 +(λ2−µ2)A2

1 +12

ν2(tr(A21))I

, (1.30)

where η0, λ1 and λ2 are zero shear viscosity, relaxation time and retardation time,

respectively. The other five constants µ0, µ1, µ2, ν1 and ν2 are associated with nonlinear

terms. The upper contravariant convected derivative designated by ∇ over S and A1 are

given in equation (1.27). Different fluid models can be retrieved from constitutive equation

(1.30) as:

20

1. Newtonian Fluid Model

If λ1 = λ2 = µ0 = µ1 = µ2 = ν1 = ν2 = 0, the equation (1.30) reduce to

S = η0A1. (1.31)

2. Second-grade Fluid Model

Letting λ1 = µ0 = µ2 = ν1 = ν2 = 0 and µ1 = λ1, the equation (1.30) gives

S = η0

(A1 +λ2

∇A1 +λ2A2

1

). (1.32)

3. Upper Convected Maxwell Fluid Model

If λ2 = µ0 = µ2 = ν1 = ν2 = 0 and µ1 = λ1, the equation (1.30) provides

S+λ1∇S = η0A1. (1.33)

4. Co-rotational Maxwell Fluid Model

Put λ2 = µ0 = µ1 = µ2 = ν1 = ν2 = 0 in equation (1.30) we obtained the constitutive

equation for co-rotational Maxwell fluid model:

S+λ1∇S+

12

λ1(A1S+SA1) = η0A1. (1.34)

5. Upper Convected Jeffrey’s Fluid Model (Oldroyd B Model)

If µ0 = ν1 = ν2 = 0 and µ1 = λ1, µ2 = λ2, the equation (1.30) reduce to

S+λ1∇S = η0

(A1 +λ2

∇A1

). (1.35)

21

6. Co-rotational Jeffrey’s Fluid Model

Setting µ0 = µ1 = µ2 = ν1 = ν2 = 0 the equation (1.30) results in

S+λ1∇S+

12

λ1(A1S+SA1) = η0

(A1 +λ2

∇A1 +λ2A2

1

). (1.36)

7. Oldroyd 4−constant Fluid Model

Letting µ1 = λ1, µ2 = λ2 and ν1 = ν2 = 0, the equation (1.30) reduce to

S+λ1∇S+

12

µ0(tr(S))A1 = η0

(A1 +λ2

∇A1

). (1.37)

Likewise we can obtain model for Oldroyd 6−constant fluid, more similar in behavior to

Oldroyd 8−constant fluid.

1.8 Methods of Solutions

The fluid models which we are going to use in the coming chapters with the geometry under

consideration along with assumptions, developed highly nonlinear ordinary differential

equations. For these equations exact solutions seem to be very cumbersome. Therefore, we

use the following approximate methods:

• Perturbation Method,

• Homotopy Perturbation Method,

• Adomian Decomposition Method,

1.8.1 Perturbation Method (PM)

Exact solutions are very rare when solving the equation of motion for constitutive equations

of non-Newtonian fluids. Asymptotic methods have then proved to be powerful tools to

obtain approximate solutions of these equations. The PM [10, 11, 30] is extensively applied

for getting approximate solutions for the problems arising in engineering and science. The

22

PM needs the existence of a small parameter in the given problem. By taking that parameter

as a perturbation parameter, say ξ in the given problem, we expand the dependent variable

say u(y,ξ) with respect to ξ as,

u = u0 +ξu1 +ξ2u2 +ξ3u3 + · · · . (1.38)

After substituting equation (1.38) in the differential equation, and equating coefficients of

like powers of ξ , we get linear problems of various orders. These problems are then solved

in conjunction to the initial/boundary conditions, which provide solutions of the nonlinear

differential equations.

1.8.2 Homotopy Perturbation Method (HPM)

A kind of analytical technique, proposed by He [12–14], which is coupling of the traditional

perturbation method and homotopy. The main advantage of this method is that it does not

require a small parameter in the equation modeling the phenomena. This method has been

successfully applied to get the solution of nonlinear boundary value problem, nonlinear

problems on bifurcation, asymptotology, wave equation and oscillator with discontinuities

[31–33].

The equations modeling non-Newtonian fluids often result in highly nonlinear differential

equations [32]. The literature survey revealed that the HPM is an effective and reliable

technique to get the solution of these problems.

To illustrate the basic idea of this method, we consider the following nonlinear differential

equation:

A(u)− f (r) = 0, r ∈Ω, (1.39)

with the boundary condition

℘(u,∂u∂n

) = 0, r ∈ Γ, (1.40)

23

where A is a general differential operator,℘a boundary operator, f (r) is a known analytical

function and Γ is the boundary of the domain Ω,∂

∂ndenotes the differentiation along the

normal drawn outwards from Ω. The operator A can be divided into two parts: G the linear

operator and N a nonlinear operator. Equation (1.39) can, therefore, be rewritten as:

G(u)+N(u)− f (r) = 0. (1.41)

By the homotopy technique, we construct a homotopy as v(r, p) : Ω× [0,1] −→ ℜ which

satisfies:

H(v, p) = (1− p)[G(v)−G(u0)]+ p[G(v)+N(v)− f (r)] = 0, p ∈ [0,1], r ∈Ω,

(1.42)

or

H(v, p) = G(v)−G(u0)+ p[G(u0)+N(v)− f (r)] = 0, (1.43)

where p ∈ [0,1], with the conditions for p = 0, v(r, p) = u0 and limp−→1 v(r, p) = u is

an embedding parameter and u0 is an initial approximation which satisfies the boundary

conditions. In topology, this is called deformation, and G(v)−G(u0) and G(v)+ N(v)−f (r) are called homotopic. Here, the embedding parameter is introduced much more

naturally, unaffected by artificial factors. So the solution of (1.42) can be written as a

power series in p [31, 34]:

v =∞

∑i=0

pivi = v0 + pv1 + p2v2 + · · · (1.44)

As p→ 1, approximate solution of (1.39) becomes

u = limp−→1

v = v0 + v1 + v2 + · · · (1.45)

24

1.8.3 Adomian Decomposition Method (ADM)

The iterative technique ADM [35, 36] was introduced and developed by George Adomian

and well addressed in the literature. ADM has recently received ample attention in the

area of series solutions. A considerable amount of research work has been invested in the

application of this method to a wide class of linear, nonlinear, partial differential equations

and integral equations [37]. A number of interesting problems in applied sciences and

engineering have been successfully solved using ADM to their higher degree of accuracy

[38, 39]. A useful quality of the ADM is that it has proved to be a competitive substitute

to the Taylor series method and other series techniques [40]. This method has been used

in obtaining analytic and approximate solutions to a wide class of linear and nonlinear,

differential and integral equations, homogeneous or inhomogeneous, with constant as

well as variable coefficients. The ADM is comparatively easy to program in engineering

problems than other series methods and provides immediate and visible solution terms

without linearization, perturbation or discretization of the problem, while the physical

behavior of the solution remains unchanged. It provides analytical solution in the form

of infinite series in which each term can be easily determined [37, 41]. If an exact solution

exists for the problem, the obtained series converges rapidly to the exact solution. For

concrete problems, where a closed form solution is not obtainable, a truncated number of

terms is usually used for numerical purposes [37, 42].

To explain the basic idea of ADM, reconsider equation (1.41). Decompose linear operator

G into L + R, where L is invertible, and taken as the highest order derivative to avoid

difficult integrations and R is the remainder of the linear operator. Thus, the equation

(1.41) becomes

L(u)+R(u)+N(u) = f (r), (1.46)

where N(u) indicates the nonlinear terms and f (r) is forcing function. since L is invertible,

so L−1 exist. Applying L−1 on both sides of equation(1.46),we get

L−1L(u) = L−1 f (r)−L−1R(u)−L−1N(u). (1.47)

25

Equation (1.47) becomes

u = g(r)+L−1 ( f (r))−L−1 (R(u))−L−1 (N(u)) , (1.48)

where g(r) is a function such that Lg(r) = 0 and can be determined by using boundary or

initial conditions. ADM assumes that the solution u can be expanded into infinite series

as u =∞∑

n=0un, also the nonlinear term N(u) will be written as N(u) =

∞∑

n=0An, where An are

special Adomian polynomials [43] which can be defined as

An =1n!

[dn

dλn

N

(∞

∑i=0

λiui

)]

λ=0

n = 0,1,2,3, ..., (1.49)

finally, the solution can be written as

∞

∑n=0

un = u0−L−1R

(∞

∑n=0

un

)−L−1

(∞

∑n=0

An

), (1.50)

where

u0 = f (y)+L−1g(y), (1.51)

is initial solution and

un+1 =−L−1R(un)−L−1An, n≥ 0, (1.52)

is nth-order solution. Then the n-term approximation will be

φn =n+1

∑i=0

ui, (1.53)

which gives solution as [36, 38],

u = limn−→∞

φn+1 =n

∑i=0

ui. (1.54)

26

Chapter 2

Analysis of Third-Grade Fluid in Helical Screw

Rheometer: Adomian Decomposition Method

27

This chapter deals with the steady flow of an incompressible, third-grade fluid in HSR, by

unwrapping or flattening the channel, lands and the outside rotating barrel. The geometry

is approximated as a shallow infinite channel, by assuming that the width of the channel is

large as compared to the depth. The developed second order nonlinear coupled differential

equations are reduced to single differential equation by using a transformation. Using

ADM, analytical expressions are calculated for the velocity profiles, shear and normal

stresses, shear at barrel surface, forces exerted on fluid, average velocity and volume flow

rates. The results have been discussed with the help of graphs. We observed that the

velocity profiles are strongly dependant on non-Newtonian parameter (β) and with the

increase in β the velocity profile increases progressively, which concludes that extrusion

process increases with increase in β. We also observed that the increase in pressure

gradients in x and z− directions increases the net flow inside the helical screw rheometer,

results in an increase in the extrusion process. We also noticed that the flow increases as

the flight angle increases.

2.1 Problem Formulation

Consider the steady flow of an isothermal, incompressible and homogeneous third-grade

fluid in HSR. The curvature of the screw channel is ignored; unrolled and laid out on a flat

surface. The barrel surface is also flattened. Assume that the screw surface, the lower plate,

is stationary and the barrel surface, the upper plate, is moving across the top of the channel

with velocity V at an angle φ to the direction of the channel Fig.2.1. The phenomena is

same as, the barrel held stationary and the screw rotates. The geometry is approximated as

a shallow infinite channel, by assuming the width B of the channel large compared with the

depth h; edge effects in the fluid at the land are ignored. The coordinate axes are positioned

in such a way that the x-axis is perpendicular to the flight walls, y-axis is normal to the

barrel surface and z-axis is in down channel direction. The liquid wets all the surfaces and

moves by the shear stresses produced by the relative movement of the barrel and channel.

No leakage of the fluid occurs across the flights. For simplicity, the velocity of the barrel

28

relative to the channel is decomposed into two components (see fig. 2.1b): U along x-axis

and W along z-axis [2]. Under these assumptions the velocity profile and extra stress tensor

can be considered as

V = (ux(y),0,uz(y)), S = S(y). (2.1)

z- axis

x-axis

y-axis

X'

X

Flight angleφ

Flow Channel

B

V

Moving flattenedbarrel surface

Stationary unrolled flatscrew surface

Channel width

Axis of srew

h

Small gap between barrel and flight landChannel depth

Fig. 2.1a

Flight

Cross sectional view

B

sin φ

φ

V

U

Decomposition of the barrel velocity

W

Fig.2.1bV

axisz-x-axis

X

φ90 φ

φ Flight angle90 φ

X'

φ

Fig.2.1c

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Flight angle

φ

Unrolled flat screw surface

Flattened barrel surface

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Figure 2.1: The geometry of the “unwrapped” screw channel and barrel surface.

29

The associated boundary conditions can be taken as (see Fig. 2.1)

ux = 0, uz = 0, at y = 0,

ux = U, uz = W, at y = h.(2.2)

where

U =−V sinφ, W = V cosφ.

Using velocity profile (2.1), continuity equation (1.7) is identically satisfied and momentum

equations (1.11 - 1.13) in the absence of body forces result in

0 = −∂P∂x

+∂Sxy

∂y, (2.3)

0 = −∂P∂y

+∂Syy

∂y, (2.4)

0 = −∂P∂z

+∂Syz

∂y. (2.5)

On defining the modified pressure P = P− Syy in equation (2.4), which implies that P =

P(x,z) only, thus equations (2.3 - 2.5) reduce to

∂P∂x

=∂Sxy

∂y, (2.6)

∂P∂z

=∂Syz

∂y. (2.7)

To calculate the components Sxy and Syz in equations (2.6) and (2.7) we use the constitutive

equation for third-grade fluid (1.19). For this we proceed as follow: using velocity profile

(2.1) in equation (1.22) we get

∇V =

0 0 0dux

dy0

duz

dy0 0 0

, (2.8)

30

then

(∇V)T =

0dux

dy0

0 0 0

0duz

dy0

, (2.9)

using (2.8) and (2.9) in equation (1.20) we obtain

A1 =

0dux

dy0

dux

dy0

duz

dy

0duz

dy0

, (2.10)

A21 = A1AT

1 =

(dux

dy

)2

0dux

dyduz

dy

0(

dux

dy

)2

+(

duz

dy

)2

0

dux

dyduz

dy0

(duz

dy

)2

, (2.11)

and

tr(A21) = 2

[(dux

dy

)2

+(

duz

dy

)2]

= P1, (2.12)

then

tr(A21)A1 =

0 P1dux

dy0

P1dux

dy0 P1

duz

dy

0 P1duz

dy0

. (2.13)

For n = 1 equation (1.21) takes the form

A2 =∂A1

∂t+(V ·∇)A1 +[A1(∇V)+(∇V)T A1]. (2.14)

31

Steady flow assumption gives∂A1

∂t= 0, and velocity profile (2.1) suggest that

(V ·∇)A1 =(

uxddx

+0ddy

+uzddz

)A1 = 0.

Using tensors (2.8) and (2.10), we get

A1(∇V) =

(dux

dy

)2

0dux

dyduz

dy

0 0 0

dux

dyduz

dy0

(duz

dy

)2

, (2.15)

and tensors (2.9) and (2.10), give

(∇V)T A1 = (A1∇V)T =

(dux

dy

)2

0dux

dyduz

dy

0 0 0

dux

dyduz

dy0

(duz

dy

)2

, (2.16)

then

A2 =

2(

dux

dy

)2

0 2dux

dyduz

dy

0 0 0

2dux

dyduz

dy0 2

(duz

dy

)2

, (2.17)

tr(A2) = 2

[(dux

dy

)2

+(

duz

dy

)2]

= tr(A21).

Equation (1.21) for n = 2 gives

A3 =∂A2

∂t+(V ·∇)A2 +[A2(∇V)+(∇V)T A2]. (2.18)

32

Steady flow assumption results in∂A2

∂t= 0, and velocity profile (2.1) gives that

(V ·∇)A2 =(

uxddx

+0ddy

+uzddz

)A2 = 0.

Tensors (2.8), (2.9) and (2.17) give

A2(∇V) =

0 0 0

0 0 0

0 0 0

, (2.19)

(∇V)T A2 = (A2∇V)T =

0 0 0

0 0 0

0 0 0

, (2.20)

then

A3 =

0 0 0

0 0 0

0 0 0

. (2.21)

Tensors (2.10) and (2.17) give

A1A2 =

0 0 0

P2 0 P3

0 0 0

, (2.22)

A2A1 =

0 P2 0

0 0 0

0 P3 0

, (2.23)

33

then

A1A2 +A2A1 =

0 P2 0

P2 0 P3

0 P3 0

, (2.24)

where

P2 = 2(

dux

dy

)3

+2dux

dy

(duz

dy

)2

,

P3 = 2(

dux

dy

)2 duz

dy+2

(duz

dy

)3

.

On substituting equations (2.10 - 2.24) in constitutive equation (1.19), we obtain non-zero

components of extra stress tensor as

Sxx = (2α1 +α2)(

dux

dy

)2

, (2.25)

Sxy = Syx = µdux

dy+2(β2 +β3)

(dux

dy

)2

+(

duz

dy

)2

dux

dy, (2.26)

Sxz = Szx = (2α1 +α2)dux

dyduz

dy, (2.27)

Syy = α2

[(dux

dy

)2

+(

duz

dy

)2]

, (2.28)

Syz = Szy = µduz

dy+2(β2 +β3)

(dux

dy

)2

+(

duz

dy

)2

duz

dy, (2.29)

Szz = (2α1 +α2)(

duz

dy

)2

, (2.30)

where S = [Si j], i, j = x, y, z.

Therefore, equations (2.6) and (2.7) result in

0 = −∂P∂x

+∂∂y

[µ

dux

dy+2(β2 +β3)

(dux

dy

)2

+(

duz

dy

)2

dux

dy

], (2.31)

34

0 = −∂P∂z

+∂∂y

[µ

duz

dy+2(β2 +β3)

(dux

dy

)2

+(

duz

dy

)2

duz

dy

], (2.32)

or

d2ux

dy2 +2(β2 +β3)

µddy

[(dux

dy

)2

+(

duz

dy

)2

dux

dy

]=

1µ

∂P∂x

, (2.33)

d2uz

dy2 +2(β2 +β3)

µddy

[(dux

dy

)2

+(

duz

dy

)2

duz

dy

]=

1µ

∂P∂z

. (2.34)

Introducing dimensionless parameters

x∗ =xh, y∗ =

yh, z∗ =

zh, u∗ =

ux

W, w∗ =

uz

W, P∗ =

P

µ(

Wh

) , (2.35)

equations (2.33), (2.34) and (2.2) take the form

d2u∗

dy∗2 + β∗d

dy∗

[(du∗

dy∗

)2

+(

dw∗

dy∗

)2

du∗

dy∗

]=

∂P∗

∂x∗, (2.36)

d2w∗

dy∗2 + β∗d∗

dy∗

[(du∗

dy∗

)2

+(

dw∗

dy∗

)2

dw∗

dy∗

]=

∂P∗

∂z∗, (2.37)

u∗ = 0, w∗ = 0, at y∗ = 0,

u∗ =UW

, w∗ = 1, at y∗ = 1,(2.38)

where β∗ =(β2 +β3)W 2

µ h2 is dimensionless non-Newtonian parameter. Dropping “*” from

equations (2.36 - 2.38) and defining

F = u+ ιw, V0 =UW

+ ι, G = P,x + ιP,z, (2.39)

35

where ι =√−1,

∂P∂x

= P,x and∂P∂z

= P,z , equations (2.36 - 2.38) reduce to

d2Fdy2 = G− β

(dFdy

)2 d2Fdy2 +2

dFdy

d2Fdy2

dFdy

, (2.40)

where F is the complex conjugate of F . The boundary conditions become

F = 0 at y = 0,

F = V0 at y = 1.(2.41)

Equation (2.40) is second order nonlinear inhomogeneous ordinary differential equation,

the exact solution seems to be difficult. In the following section we use ADM to obtain the

approximate solution.

2.2 Solution of the problem

ADM (see sec:1.8.3) suggests to write equation (2.40) in the form

Lyy(F) = G− β

(dFdy

)2 d2Fdy2 +2

dFdy

d2Fdy2

dFdy

, (2.42)

where Lyy =d2

dy2 is assumed to be invertible differential operator, defined by L−1yy =

∫ ∫(∗)dydy.

Applying L−1yy to both sides of equation (2.42), we get

F = C1 +C2y+L−1yy (G)− βL−1

yy

(dFdy

)2 d2Fdy2 +2

dFdy

d2Fdy2

dFdy

, (2.43)

where C1 and C2 are constants of integration and can be determined, using boundary

conditions. According to procedure of ADM, F and F can be written in component form

36

as:F =

∞∑

n=0Fn,

F =∞∑

n=0Fn.

(2.44)

Thus equation (2.43) takes the form

∞

∑n=0

Fn = C1 +C2y+L−1yy (G)− βL−1

yy

(ddy

(∞

∑n=0

Fn

))2 (d2

dy2

(∞

∑n=0

Fn

))

+ 2

(ddy

(∞

∑n=0

Fn

))(d2

dy2

(∞

∑n=0

Fn

))(ddy

(∞

∑n=0

Fn

)). (2.45)

Adomian also suggested that the nonlinear terms can be explored in the form of Adomian

polynomials, say An and Bn as

∞∑

n=0An =

(ddy

(∞

∑n=0

Fn

))2 (d2

dy2

(∞

∑n=0

Fn

)),

∞∑

n=0Bn = 2

(ddy

(∞

∑n=0

Fn

))(d2

dy2

(∞

∑n=0

Fn

))(ddy

(∞

∑n=0

Fn

)),

(2.46)

equation (2.45) is transformed as

∞

∑n=0

Fn = C1 +C2y+L−1yy (G)− βL−1

yy

(∞

∑n=0

An +∞

∑n=0

Bn

), (2.47)

and the boundary conditions (2.41) become

∞∑

n=0Fn = 0, y = 0,

∞∑

n=0Fn = V0, y = 1.

(2.48)

From the recursive relations (2.47) and (2.48), we can identify the zeroth order problem as,

F0 = C1 +C2y+L−1yy (G) , (2.49)

37

with boundary conditions

F0 = 0, y = 0,

F0 = V0, y = 1.(2.50)

The remaining order problems are in the following form

Fn = −βL−1yy (An−1 +Bn−1) , n≥ 1 (2.51)

with the boundary conditions

∞∑

n=1Fn = 0, y = 0,

∞∑

n=1Fn = 0, y = 1.

(2.52)

The ADM solution to equation (2.47) along with the boundary conditions (2.48) will be

then

F =∞

∑n=0

Fn. (2.53)

2.2.1 Zeroth Component Solution

The equation (2.49) along with the boundary condition (2.50) give the zeroth component

solution as

F0 = V0y+12

G(y2− y

), (2.54)

using equation (2.39) real and imaginary parts become

u0 =UW

y+12

P,x(y2− y

), (2.55)

w0 = y+12

P,z(y2− y

), (2.56)

which are also the solution for Newtonian case.

38

2.2.2 First Component Solution

For n = 1 equations (2.51 - 2.52) yield

F1 =−βL−1yy (A0 +B0) , (2.57)

F1 = 0 at y = 0,

F1 = 0 at y = 1.(2.58)

Equation (2.57) depends on zeroth order Adomian polynomials which can be obtained from

(2.46) as

A0 =(

dF0

dy

)2 d2F0

dy2 , (2.59)

B0 = 2dF0

dyd2F0

dy2dF0

dy. (2.60)

Using equation (2.54) in equations (2.59 - 2.60) then equation (2.57) with the help of

boundary conditions (2.58) gives

F1 = −β(A1 + ιB1)

(y2− y

)+(A2 + ιB2)

(y3− y

)

+ (A3 + ιB3)(y4− y

), (2.61)

real and imaginary parts are then

u1 = −βA1

(y2− y

)+A2

(y3− y

)+A3

(y4− y

), (2.62)

w1 = −βB1

(y2− y

)+B2

(y3− y

)+B3

(y4− y

), (2.63)

where A1, A2, A3, B1, B2 and B3 are constants coefficients given in Appendix I.

39

2.2.3 Second Component Solution

Equations (2.51 - 2.52) give the second order problem for n = 2 as

F2 =−βL−1yy (A1 +B1) , (2.64)


F2 = 0 at y = 0,

F2 = 0 at y = h.(2.65)

Equation (2.64) can be calculated using equations (2.54) and (2.61) as

A1 =(

dF0

dy

)2 d2F1

dy2 +2dF0

dydF1

dyd2F0

dy2 , (2.66)

B1 = 2(

dF0

dyd2F0

dy2dF1

dy+

dF0

dyd2F1

dy2dF0

dy+

dF1

dyd2F0

dy2dF0

dy

), (2.67)

then solution will be

F2 = β2 (A4 + ιB4)

(y2− y

)+(A5 + ιB5)

(y3− y

)+(A6 + ιB6)

(y4− y

)

+ (A7 + ιB7)(

y5− y)

+(A8 + ιB8)(

y6− y)

. (2.68)

Equating real and imaginary parts we get

u2 = β2 A4

(y2− y

)+A5

(y3− y

)+A6

(y4− y

)

+ A7

(y5− y

)+A8

(y6− y

), (2.69)

w2 = β2 B4

(y2− y

)+B5

(y3− y

)+B6

(y4− y

)

+ B7

(y5− y

)+B8

(y6− y

), (2.70)

where A4, A5, A6, A7, A8, B4, B5, B6, B7 and B8 are constants coefficients given in

Appendix I.

40

2.2.4 Velocity fields

Velocity profile in x-direction

Combining equations (2.55), (2.62) and (2.69), give the solution for the velocity profile in

the transverse plane

u =UW

y+(

12

P,x + βA1 + β2A4

)(y2− y

)+

(βA2 + β2A5

)(y3− y

)

+(

βA3 + β2A6

)(y4− y

)+ β2A7

(y5− y

)+ β2A8

(y6− y

). (2.71)

Velocity profile in z-direction

Combining equations (2.56), (2.63) and (2.70), the solution for the velocity profile in the

down channel direction becomes

w = y+(

12

P,z + βB1 + β2B4

)(y2− y

)+

(βB2 + β2B5

)(y3− y

)

+(

βB3 + β2B6

)(y4− y

)+ β2B7

(y5− y

)+ β2B8

(y6− y

). (2.72)

Velocity in the direction of the axis of screw

The resultant velocity, s in the direction of the axis of the screw at any depth in the channel

can be computed from equations (2.71) and (2.72) as (see fig. 2.1c)

s = wsinφ+ucosφ, (2.73)

= y+(

12

P,z + βB1 + β2B4

)(y2− y

)+

(βB2 + β2B5

)(y3− y

)

+(

βB3 + β2B6

)(y4− y

)+ β2B7

(y5− y

))

+ β2B8

(y6− y

)sinφ

41

+

UW

y+(

12

P,x + βA1 + β2A4

)(y2− y

)+

(βA2 + β2A5

)(y3− y

)

+(

βA3 + β2A6

)(y4− y

)+ β2A7

(y5− y

))

+ β2A8

(y6− y

)cosφ. (2.74)

The expression (2.74) shows that forward velocity at any point in the channel depends on

pressure gradients P,x and P,z only.

2.2.5 Stresses

Using equations (2.71) and (2.72) in equations (2.26), (2.27) and (2.29) we obtained the

shear stresses as

S∗xy = S∗yx =

UW

+(

βA1 + β2A4 +P,x

2

)(−1+2y)+

(βA2 + β2A5

)(−1+3y2)

+(

βA3 + β2A6

)(−1+4y3)+ β2A7(−1+5y4)+ β2A8

(−1+6y5

)

+ β[

UW

+(

βA1 + β2A4 +P,x

2

)(−1+2y)+

(βA2 + β2A5

)(−1+3y2)

+(

βA3 + β2A6

)(−1+4y3)+ β2A7(−1+5y4)+ β2A8

(−1+6y5

)2

+

1+(

βB1 + β2B4 +P,z

2

)(−1+2y)+

(βB2 + β2B5

)(−1+3y2)

+(

βB3 + β2B6

)(−1+4y3)+ β2B7(−1+5y4)+ β2B8

(−1+6y5

)2]

UW

+(

βA1 + β2A4 +P,x

2

)(−1+2y)+

(βA2 + β2A5

)(−1+3y2)

+(

βA3 + β2A6

)(−1+4y3)+ β2A7(−1+5y4)+ β2A8

(−1+6y5

), (2.75)

S∗yz = S∗zy =

1+(

βB1 + β2B4 +P,z

2

)(−1+2y)+

(βB2 + β2B5

)(−1+3y2)

+(

βB3 + β2B6

)(−1+4y3)+ β2B7(−1+5y4)+ β2B8

(−1+6y5

)

+ β[

UW

+(

βA1 + β2A4 +P,x

2

)(−1+2y)+

(βA2 + β2A5

)(−1+3y2)

42

+(

βA3 + β2A6

)(−1+4y3)+ β2A7(−1+5y4)+ β2A8

(−1+6y5

)2

+

1+(

βB1 + β2B4 +P,z

2

)(−1+2y)+

(βB2 + β2B5

)(−1+3y2)

+(

βB3 + β2B6

)(−1+4y3)+ β2B7(−1+5y4)+ β2B8

(−1+6y5

)2]

1+

(βB1 + β2B4 +

P,z

2

)(−1+2y)+

(βB2 + β2B5

)(−1+3y2)

+(

βB3 + β2B6

)(−1+4y3)+ β2B7(−1+5y4)+ β2B8

(−1+6y5

), (2.76)

S∗xz = S∗zx =(2α1 +α2)

µWh

UW

+(

βA1 + β2A4 +P,x

2

)(−1+2y)+

(βA2 + β2A5

)

(−1+3y2)+(

βA3 + β2A6

)(−1+4y3)+ β2A7(−1+5y4)

+ β2A8

(−1+6y5

)1+

(βB1 + β2B4 +

P,z

2

)(−1+2y)

+(

βB2 + β2B5

)(−1+3y2)+(

βB3 + β2B6

)(−1+4y3)

+ β2B7(−1+5y4)+ β2B8

(−1+6y5

), (2.77)

where S∗i j =Si jµW

h

, i, j = x,y,z and i 6= j are the non-dimensional shear stresses.

The shears exerted by the fluid on the barrel surface at y = 1 are

S∗wxy= S∗wxy

=

UW

+(

βA1 + β2A4 +P,x

2

)+2

(βA2 + β2A5

)+3

(βA3 + β2A6

)

+ 4β2A7 +5β2A8

+ β

[UW

+(

βA1 + β2A4 +P,x

2

)+2

(βA2 + β2A5

)

+ 3(

βA3 + β2A6

)+4β2A7 +5β2A8

2+

1+

(βB1 + β2B4 +

P,z

2

)

+ 2(

βB2 + β2B5

)+3

(βB3 + β2B6

)+4β2B7 +5β2B8

2]

UW

+(

βA1 + β2A4 +P,x

2

)+2

(βA2 + β2A5

)

+ 3(

βA3 + β2A6

)+4β2A7 +5β2A8

, (2.78)

43

S∗wyz= S∗wzy

=

1+(

βB1 + β2B4 +P,z

2

)+2

(βB2 + β2B5

)+3

(βB3 + β2B6

)

+ 4β2B7 +5β2B8

+ β

[UW

+(

βA1 + β2A4 +P,x

2

)+2

(βA2 + β2A5

)

+ 3(

βA3 + β2A6

)+4β2A7 +5β2A8

2+

1+

(βB1 + β2B4 +

P,z

2

)

+ 2(

βB2 + β2B5

)+3

(βB3 + β2B6

)+4β2B7 +5β2B8

2]

1+

(βB1 + β2B4 +

P,z

2

)+2

(βB2 + β2B5

)

+ 3(

βB3 + β2B6

)+4β2B7 +5β2B8

, (2.79)

S∗wxz= S∗wzx

=(2α1 +α2)

µWh

UW

+(

βA1 + β2A4 +P,x

2

)+2

(βA2 + β2A5

)

+ 3(

βA3 + β2A6

)+4β2A7 +5β2A8

1+

(βB1 + β2B4 +

P,z

2

)

+ 2(

βB2 + β2B5

)+3

(βB3 + β2B6

)+4β2B7 +5β2B8

, (2.80)

Using equations (2.71) and (2.72) in equations (2.25), (2.28) and (2.30) we calculate the

normal stresses as

S∗xx =(2α1 +α2)

µWh

UW

+(

βA1 + β2A4 +P,x

2

)(−1+2y)+

(βA2 + β2A5

)(−1+3y2)

+(

βA3 + β2A6

)(−1+4y3)+ β2A7(−1+5y4)+ β2A8

(−1+6y5

)2, (2.81)

S∗yy =α2

µWh

[UW

+(

βA1 + β2A4 +P,x

2

)(−1+2y)+

(βA2 + β2A5

)(−1+3y2)

+(

βA3 + β2A6

)(−1+4y3)+ β2A7(−1+5y4)+ β2A8

(−1+6y5

)2

+

1+(

βB1 + β2B4 +P,z

2

)(−1+2y)+

(βB2 + β2B5

)(−1+3y2)

+(

βB3 + β2B6

)(−1+4y3)+ β2B7(−1+5y4)+ β2B8

(−1+6y5

)2], (2.82)

S∗zz =(2α1 +α2)

µWh

1+

(βB1 + β2B4 +

P,z

2

)(−1+2y)+

(βB2 + β2B5

)(−1+3y2)

+(

βB3 + β2B6

)(−1+4y3)+ β2B7(−1+5y4)+ β2B8

(−1+6y5

)2. (2.83)

44

The shear forces per unit width required to move the barrel in x and z−directions, are

Fx

B=−

∫ q1

0Swxydx, (2.84)

or

F∗x =−S∗wxyδ1, (2.85)

Fz

B=−

∫ q2

0Swyzdz, (2.86)

or

F∗z =−S∗wyzδ2, (2.87)

where F∗i =Fi

µWB, i = x, y are dimensionless shear forces, δ1 =

q1

hand δ2 =

q2

hare

dimensionless lengths of the channel in x and z−directions, respectively, and q1 and q2

are dimensional lengths of the channel in x and z−directions, respectively. Therefore

F∗ = F∗z sinφ+F∗x cosφ, (2.88)

is the net shear force per unit width in the direction of the axis of the screw.

2.2.6 Volume flow rates

Volume flow rate in x-direction per unit width is

Q∗x

B=

∫ y=h

y=0udy, (2.89)

using dimensionless parameters (2.35) we get

Qx =∫ 1

0udy, (2.90)

=U

2W− 1

6

(12

P,x + βA1 + β2A4

)− 1

4

(βA2 + β2A5

)

− 310

(βA3 + β2A6

)− 1

3β2A7− 5

14β2A8, (2.91)

45

where Qx =Q∗

xWhB

.

Volume flow rate in z-direction per unit width is

Q∗z

B=

∫ y=h

y=0wdy, (2.92)

non-dimensionalization gives

Qz =∫ 1

0wdy, (2.93)

Qz =12− 1

6

(12

P,z + βB1 + β2B4

)− 1

4

(βB2 + β2B5

)

− 310

(βB3 + β2B6

)− 1

3β2B7− 5

14β2B8, (2.94)

where Qz =Q∗

z

WhB.

The resultant volume flow rate forward in the screw channel is the product of the resultant

velocity and cross-sectional area integrated from the root of the screw to the barrel surface

therefore,

Q∗ =N

sinφ

∫ y=h

y=0sdy, (2.95)

non-dimensionalization implies

Q =N

sinφ

∫ 1

0sdy, (2.96)

where Q =Q∗

WhBand N is the number of parallel flights in a multiflight screw. Using

equation (2.74) we get

Q =N

sinφ

[12− 1

6

(12

P,z + βB1 + β2B4

)− 1

4

(βB2 + β2B5

)

− 310

(βB3 + β2B6

)− 1

3β2B7− 5

14β2B8

sinφ

46

+

U2W

− 16

(12

P,x + βA1 + β2A4

)− 1

4

(βA2 + β2A5

)

− 310

(βA3 + β2A6

)− 1

3β2A7− 5

14β2A8

cosφ

], (2.97)

=N

sinφQz sinφ+Qx cosφ . (2.98)

2.2.7 Average velocity

The average velocity in the direction of the axis of the screw is

s∗ = N∫ 1

0sdy, (2.99)

where s∗ =s

Wis non-dimensional average velocity. Using equation (2.74) gives

s∗ = N[

12− 1

6

(12

P,z + βB1 + β2B4

)− 1

4

(βB2 + β2B5

)

− 310

(βB3 + β2B6

)− 1

3β2B7− 5

14β2B8

sinφ

+

U2W

− 16

(12

P,x + βA1 + β2A4

)− 1

4

(βA2 + β2A5

)

− 310

(βA3 + β2A6

)− 1

3β2A7− 5

14β2A8

cosφ

]. (2.100)

The study reveals that if we set β =αW 2

h2 in the above equations, we can calculate the

solutions for dilatant fluid, where α is dilatant constant.

2.3 Results and Discussion

In this chapter steady flow of an incompressible, third-grade fluid in an unwrapped HSR

is considered. The geometry is discussed using Fig 2.1. The developed second order

nonlinear coupled differential equations are reduced to a single differential equation by

using a transformation. Using ADM, analytical expressions are calculated for the velocities

u and w in x and z− directions, respectively and also in the direction of the axis of the

47

screw, s. Expressions for the shear stresses in the region, shear stresses at barrel surface,

forces exerted on fluid, volume flow rates and average velocity are also calculated. Here

we discussed the effect of material constant β, flight angle φ and pressure gradients P,x and

P,z, on the velocity profiles given by equations (2.71 - 2.72) and (2.74) with the help of

graphical representation. From the figures (2.2 - 2.4), it is seen that the velocity profiles are

strongly dependent on the non-Newtonian parameter β. Figure (2.2) is sketched for u, back

flow is seen toward the barrel surface after some points in the channel height which suggest

that the fluid circulates inside the confined channel, thus the velocity in x−direction helps

in the process of mixing during processing. In figure (2.3) we observe that with the increase

in value of β the velocity w increases, helps to move the fluid in the forward direction in the

channel. The resultant velocity s is shown in figure (2.4), which resembles to the Poiseuille

flow in the channel. Due to s the fluid moves toward the die. It is worthwhile to note that

the shear thinning occurs with the increase in value of β, which increases extrusion process.

The velocity profiles for Newtonian case are retrieved for β = 0 [2].

Figure 2.2: Velocity profile u(y) for different values of β, keeping P,x =−2.0, P,z =−2.0and φ = 45.

48

Figure 2.3: Velocity profile w(y) for different values of β, keeping P,x =−2.0, P,z =−2.0and φ = 45.

Moreover Figures (2.5), (2.7) and (2.9) are sketched for the velocity profiles u, w and s

for different values of P,x . Figures (2.5) and (2.9) show that the rise in pressure gradient

increases speed of flow. However figure (2.7) depicts very small effect of P,x on the w and

there are also damping effects of P,x on w at some points in the channel, which show that

P,x resist the flow in z−direction. Figures (2.6), (2.8) and (2.10) are plotted for the velocity

profiles u, w and s for different values of P,z. Figures (2.8) and (2.10) show that the increase

in the value of P,z increases speed of flow. However figure (2.6) depicts very small effect

of P,z on the u and there are also damping effects of P,z on u at some points in the channel,

which show that P,z resist the flow in x−direction, and responsible for forward flow.

The figure (2.11) is plotted for different values of φ. It is observed that, the resultant

velocity attains its maximum value at φ = 45, which conforms the results given in [6].

From the resultant velocity (2.74) we conclude that for φ = 0 velocity has only component

in x−direction and φ = 90 gives the velocity in z−direction only. The figures (2.12) and

(2.13) are sketched to observe the effect of β on shear stresses Sxy and Syz. Shear stresses

49

Sxy and Syz are zero at some points in the channel, these are the points where the velocities

in x and z−directions attain their maximum values.

Figure 2.4: Velocity profile s(y) for different values of β, keeping P,x =−2.0, P,z =−2.0and φ = 45.

Figure 2.5: Velocity profile u(y) for different values of P,x, keeping β = 0.3, P,z = −2.0and φ = 45.

50

Figure 2.6: Velocity profile u(y) for different values of P,z, keeping β = 0.3, P,x = −2.0and φ = 45.

Figure 2.7: Velocity Profile w(y) for different values of P,x, keeping β = 0.3, P,z = −2.0and φ = 45.

51

Figure 2.8: Velocity profile w(y) for different values of P,z, keeping β = 0.3, P,x = −2.0and φ = 45.

Figure 2.9: Velocity profile s(y) for different values of P,x, keeping β = 0.3, P,z = −2.0and φ = 45.

52

Figure 2.10: Velocity profile s(y) for different values of P,z, keeping β = 0.3, P,x = −2.0and φ = 45.

Figure 2.11: Velocity profile s(y) for different values of φ, keeping β = 0.3, P,x = −2.0and P,z =−2.0.

53

Figure 2.12: Variation of shear stress Sxy for different values of β, keeping P,x =−2.0, P,z =−2.0 and φ = 45.

Figure 2.13: Variation of shear stress Syz for different values of β, keeping P,x =−2.0, P,z =−2.0 and φ = 45.

54

2.4 Conclusion

The steady, homogeneous flow of an isothermal and incompressible third-grade fluid is

investigated in HSR. The geometry of the problem under consideration gives second

order nonlinear coupled differential equations which are reduced to single differential

equation using a transformation. Adomian decomposition method is used to obtain

analytical expressions for the velocity profiles. Shear stresses, shear stresses at barrel

surface, shear forces exerted on the fluid, volume flow rates and average velocity of the

fluid are also calculated. It is noticed that the zeroth component solution matches with

solution of the Newtonian fluid in HSR and also found that the net velocity of the fluid

is due to the pressure gradient only. Graphical representation shows that the velocity

profiles are strongly dependent on non-Newtonian parameter (β), β also controls the shear

thinning/thicking of the fluid. The speed of the fluid can be controlled with the proper

values of pressure gradients in x and z− direction. The zero points of the shear stresses are

also noted for different values of β.

55

Chapter 3

Study of Co-rotational Maxwell Fluid in Helical Screw

Rheometer: Perturbation Method

56

This chapter discuss the steady flow of an incompressible, co-rotational Maxwell fluid

in HSR. The geometry of HSR is same as given in previous chapter (see fig.2.1). The

developed second order nonlinear coupled differential equations are transformed to single

differential equation. Using PM, analytical expressions are obtained for the velocity

components in x and z−directions. Expression for the resultant velocity in direction of

the screw axis is also derived. Volume flow rates, shear and normal stresses, shear at barrel

surface, forces exerted on fluid and average velocity are also calculated. The results have

been discussed with the help of graphs as well.


Consider the steady flow of an isothermal, incompressible and homogeneous co-rotational

Maxwell fluid in HSR using the simplified geometry given in section 2.1. Using velocity

profile (2.1) equation (1.7) is identically satisfied and momentum equations give (2.6)

and (2.7). To calculate the velocity profiles of co-rotational Maxwell fluid in HSR, we

solve these equations using constitutive equation (1.26). For this we proceed as follows:

Equations (1.14) and (2.8) give

S(∇V) =

Sxydux

dy0 Sxy

duz

dy

Syydux

dy0 Syy

duz

dy

Szydux

dy0 Szy

duz

dy

, (3.1)

and equation (1.14) with equation (2.9) results in

(∇V)T S =

Sxydux

dySyy

dux

dySyz

dux

dy0 0 0

Syxduz

dySyy

duz

dySyz

duz

dy

. (3.2)

57

Equation (1.27) implies

∇S =

∂S∂t

+(V ·∇)S−(∇V)T S+S(∇V)

, (3.3)

steady flow assumption results in∂S∂t

= 0, and velocity profile (2.1) gives

(V ·∇)S =(

uxddx

+0ddy

+uzddz

)S = 0.

Using equations (3.1) and (3.2) in equation (3.3) we get

∇S =−

2Sxydux

dySyy

dux

dySyz

dux

dy+Sxy

duz

dy

Syydux

dy0 Syy

duz

dy

Syzdudy

+Sxyduz

dySyy

duz

dy2Syz

duz

dy

. (3.4)


A1S+SA1 =

2Sxydux

dyP4 P5

P4 2(

Sxydux

dy+Szy

duz

dy

)P6

P5 P6 2Syzduz

dy

, (3.5)

P4 = (Syy +Sxx)dux

dy+Sxz

duz

dy,

P5 = Syzdux

dy+Sxy

duz

dy,

P6 = (Syy +Szz)duz

dy+Sxz

dux

dy.

58

On substituting equations (1.14), (2.10), (3.4) and (3.5) in equation (1.26), we obtained

the following components of extra stress tensor S,

Sxx = λ1Sxydux

dy, (3.6)

Sxy = Syx =η0

dux

dy

1+λ21

(dux

dy

)2

+(

duz

dy

)2 , (3.7)

Sxz = Szx =12

λ1

Syz

dux

dy+Sxy

duz

dy

, (3.8)

Syy = −λ1

Syz

duz

dy+Sxy

dux

dy

, (3.9)

Syz = Szy =η0

duz

dy

1+λ21

(dux

dy

)2

+(

duz

dy

)2 , (3.10)

Szz = λ1Syzduz

dy. (3.11)

Using equations (3.7) and (3.10), equations (2.6) and (2.7) become

∂P∂x

=ddy

η0dux

dy

1+λ21

(dux

dy

)2

+(

duz

dy

)2

, (3.12)

∂P∂z

=ddy

η0duz

dy

1+λ21

(dux

dy

)2

+(

duz

dy

)2

. (3.13)

59

Introducing dimensionless parameters (2.35) in equations (3.12) and (3.13) we obtained

∂P∗

∂x∗=

ddy∗

du∗

dy∗

1+(Wi)2

(du∗

dy∗

)2

+(

dw∗

dy∗

)2

, (3.14)

∂P∗

∂z∗=

ddy∗

dw∗

dy∗

1+(Wi)2

(du∗

dy∗

)2

+(

dw∗

dy∗

)2

, (3.15)

along with the boundary conditions (2.2)

u∗ = 0, w∗ = 0, at y∗ = 0,

u∗ =UW

, w∗ = 1, at y∗ = 1,(3.16)

where Wi =λ1W

his the Weissenberg number. Dropping “*” from equations (3.14 - 3.16)

and then integrating equations (3.14) and (3.15) with respect to y, we get

dudy

= (P,xy+C1)

[1+(Wi)2

(dudy

)2

+(

dwdy

)2]

, (3.17)

dwdy

= (P,zy+C2)

[1+(Wi)2

(dudy

)2

+(

dwdy

)2]

, (3.18)

where P,x =∂P∂x

, P,z =∂P∂z

and C1 and C2 are arbitrary constants of integration, can be

determined using the associated boundary conditions.

Using equation (2.39) in equations (3.17) and (3.18) yield

dFdy

= (Gy+K)+(Wi)2(Gy+K)dFdy

dFdy

, (3.19)

60

the boundary conditions (3.16) become

F = 0 at y = 0,

F = V0 at y = 1,(3.20)

where F is the complex conjugate of F and K = C1 + ιC2.

Equation (3.19) is second order nonlinear inhomogeneous ordinary differential equation,

and its exact solution seems to be difficult. In the following section we use PM (discussed

in sec: 1.8.1) to obtain the approximate solution.


Assume ξ = (Wi)2 to be a small parameter in equation (3.19) and expand F(y) and K in a

series of the form

F(y) = F0(y)+ξF1(y)+ξ2F2(y)+ · · · , (3.21)

K = K0 +ξK1 +ξ2K2 + · · · , (3.22)

where K0, K1, K2, · · · , are arbitrary constants to be determined using boundary conditions.

Substituting series (3.21) and (3.22) into equations (3.19 - 3.20) and equating the

coefficients of like powers of ξ, we get the following problems of different orders.

3.2.1 Zeroth order problem

ξ0 :dF0

dy= (Gy+K0), (3.23)

where K0 is an arbitrary constant. The boundary conditions associated with the equation

(3.23) are,

F0 = 0 at y = 0,

F0 = V0 at y = 1,(3.24)

61

has the solution

F0 = V0y+12

G(y2− y

), (3.25)

separating real and imaginary parts we get

u0 =UW

y+12

P,x(y2− y

), (3.26)

w0 = y+12

P,z(y2− y

), (3.27)

equations (3.26) and (3.27), are same as (2.55) and (2.56), solution for Newtonian case.

3.2.2 First order problem

ξ1 :dF1

dy= K1 +(Gy+K0)

dF0

dydF0

dy, (3.28)

with corresponding boundary conditions

F1 = 0 at y = 0,

F1 = 0 at y = 1,(3.29)

where K1 is a constant to be determined, gives solution of the form

F1 = (C0 + ιD0)(y2− y

)+(C1 + ιD1)

(y3− y

)+(C2 + ιD2)

(y4− y

), (3.30)

has real and imaginary parts as

u1 = C0(y2− y

)+C1

(y3− y

)+C2

(y4− y

), (3.31)

w1 = D0(y2− y

)+D1

(y3− y

)+D2

(y4− y

), (3.32)

where Ci, D j, i = 0, · · · ,2, j = 0, · · · ,2 are constant coefficients given in Appendix II.

62

3.2.3 Second order problem

ξ2 :dF2

dy= K2 +(Gy+K0)

dF0

dydF1

dy+

dF0

dydF1

dy

+K1

dF0

dydF0

dy, (3.33)

using the boundary conditions

F2 = 0 at y = 0,

F2 = 0 at y = h,(3.34)

where K2 is a constant, results in

F2 = (C3 + ιD3)(y2− y

)+(C4 + ιD4)

(y3− y

)+(C5 + ιD5)

(y4− y

)

+ (C6 + ιD6)(

y5− y)

+(C7 + ιD7)(

y6− y)

. (3.35)

Separation of real and imaginary parts give

u2 = C3(y2− y

)+C4

(y3− y

)+C5

(y4− y

)

+ C6

(y5− y

)+C7

(y6− y

), (3.36)

w2 = D3(y2− y

)+D4

(y3− y

)+D5

(y4− y

)

+ D6

(y5− y

)+D7

(y6− y

), (3.37)

where Ci, D j, i = 3, · · · ,7, j = 3, · · · ,7 are constant coefficients given in Appendix II.



Combining equations (3.26), (3.31) and (3.36), give the approximate solution for the

velocity profile in the x-direction as

63

u =UW

y+(

12

P,x +ξC0 +ξ2C3

)(y2− y

)+

(ξC1 +ξ2C4

)(y3− y

)

+(ξC2 +ξ2C5

)(y4− y

)+ξ2C6

(y5− y

)+ξ2C7

(y6− y

). (3.38)

In terms of Weissenberg number Wi,

u =UW

y+(

12

P,x +(Wi)2C0 +(Wi)4C3

)(y2− y

)+

((Wi)2C1 +(Wi)4C4

)(y3− y

)

+((Wi)2C2 +(Wi)4C5

)(y4− y

)+(Wi)4C6

(y5− y

)+(Wi)4C7

(y6− y

). (3.39)


Combining equations (3.27), (3.32) and (3.37) we obtain the solution for the velocity profile

in the z-direction as

w = y+(

12

P,z +ξD0 +ξ2D3

)(y2− y

)+

(ξD1 +ξ2D4

)(y3− y

)

+(ξD2 +ξ2D5

)(y4− y

)+ξ2D6

(y5− y

)+ξ2D7

(y6− y

), (3.40)

= y+(

12

P,z +(Wi)2D0 +(Wi)4D3

)(y2− y

)+

((Wi)2D1 +(Wi)4D4

)(y3− y

)

+((Wi)2D2 +(Wi)4D5

)(y4− y

)+(Wi)4D6

(y5− y

)

+ (Wi)4D7

(y6− y

). (3.41)

Velocity along the axis of screw

The velocity along the axis of the screw at any depth in the channel can be calculated using

equations (3.39) and (3.41) as


=(

12

P,z +(Wi)2D0 +(Wi)4D3

)(y2− y

)+

((Wi)2D1 +(Wi)4D4

)(y3− y

)

64

+((Wi)2D2 +(Wi)4D5

)(y4− y

)+(Wi)4D6

(y5− y

)

+ (Wi)4D7

(y6− y

)sinφ

+(

12

P,x +(Wi)2C0 +(Wi)4C3

)(y2− y

)+

((Wi)2C1 +(Wi)4C4

)(y3− y

)

+((Wi)2C2 +(Wi)4C5

)(y4− y

)+(Wi)4C6

(y5− y

)

+ (Wi)4C7

(y6− y

)cosφ, (3.43)

expression (3.43) has no drag term which shows that the net velocity at any point in the

channel depends on pressure gradients P,x and P,z.

3.2.5 Stresses

Substituting the derivatives of velocity components (3.39) and (3.41) in equations (3.7), (3.8)

and (3.10) we obtained the shear stresses as

S∗xy = S∗yx =1

1+Wi2Π1

UW

+(

WiC0 +Wi2C3 +P,x

2

)(−1+2y)

+(WiC1 +Wi2C4

)(−1+3y2)+(WiC2 +Wi2C5

)(−1+4y3)+Wi2C6(−1+5y4)

+ Wi2C7

(−1+6y5

), (3.44)

S∗yz = S∗zy =1

1+Wi2Π1

1+

(WiD0 +Wi2D3 +

P,z

2

)(−1+2y)

+(WiD1 +Wi2D4

)(−1+3y2)+(WiD2 +Wi2D5

)(−1+4y3)+Wi2D6(−1+5y4)

+ Wi2D7

(−1+6y5

), (3.45)

S∗xz = S∗zx =Wi2

[S∗yz

UW

+(

WiC0 +Wi2C3 +P,x

2

)(−1+2y)

+(WiC1 +Wi2C4

)(−1+3y2)+(WiC2 +Wi2C5

)(−1+4y3)+Wi2C6(−1+5y4)

+ Wi2C7

(−1+6y5

)+S∗xy

1+

(WiD0 +Wi2D3 +

P,z

2

)(−1+2y)

+(WiD1 +Wi2D4

)(−1+3y2)+(WiD2 +Wi2D5

)(−1+4y3)

+ Wi2D6(−1+5y4)+Wi2D7

(−1+6y5

)], (3.46)

65

where

Π1 =[

UW

+(

WiC0 +Wi2C3 +P,x

2

)(−1+2y)+

(WiC1 +Wi2C4

)(−1+3y2)

+(WiC2 +Wi2C5

)(−1+4y3)+Wi2C6(−1+5y4)+Wi2C7

(−1+6y5

)2

+

1+(

WiD0 +Wi2D3 +P,z

2

)(−1+2y)+

(WiD1 +Wi2D4

)(−1+3y2)

+(WiD2 +Wi2D5

)(−1+4y3)+Wi2D6(−1+5y4)

+ Wi2D7

(−1+6y5

)2]. (3.47)

Then the shears exerted by the fluid on the barrel surface at y = 1 are

S∗wxy= S∗wxy

=1

1+Wi2Π2

UW

+(

WiC0 +Wi2C3 +P,x

2

)

+ 2(WiC1 +Wi2C4

)+3

(WiC2 +Wi2C5

)+4Wi2C6

+ 5Wi2C7

, (3.48)

S∗wyz= S∗wzy

=1

1+Wi2Π2

1+

(WiD0 +Wi2D3 +

P,z

2

)

+ 2(WiD1 +Wi2D4

)+3

(WiD2 +Wi2D5

)+4Wi2D6

+ 5Wi2D7

, (3.49)

S∗wxz= S∗wzx

=Wi2

[S∗yz

UW

+(

WiC0 +Wi2C3 +P,x

2

)

+ 2(WiC1 +Wi2C4

)+3

(WiC2 +Wi2C5

)+4Wi2C6

+ 5Wi2C7 +S∗xy

1+

(WiD0 +Wi2D3 +

P,z

2

)

+ 2(WiD1 +Wi2D4

)+3

(WiD2 +Wi2D5

)

+ 4Wi2D6 +5Wi2D7]

, (3.50)

where

Π2 =[

UW

+(

WiC0 +Wi2C3 +P,x

2

)+2

(WiC1 +Wi2C4

)

+ 3(WiC2 +Wi2C5

)+4Wi2C6 +5Wi2C7

2

66

+

1+(

WiD0 +Wi2D3 +P,z

2

)+2

(WiD1 +Wi2D4

)

+ 3(WiD2 +Wi2D5

)+4Wi2D6

+ 5Wi2D72

]. (3.51)

Similarly, we can calculate normal stresses (3.6), (3.9) and (3.11) as

S∗xx = Wi[

S∗xy

UW

+(

WiC0 +Wi2C3 +P,x

2

)(−1+2y)

+(WiC1 +Wi2C4

)(−1+3y2)+(WiC2 +Wi2C5

)(−1+4y3)+Wi2C6(−1+5y4)

+ Wi2C7

(−1+6y5

)], (3.52)

S∗yy = −Wi[

S∗xy

UW

+(

WiC0 +Wi2C3 +P,x

2

)(−1+2y)

+(WiC1 +Wi2C4

)(−1+3y2)+(WiC2 +Wi2C5

)(−1+4y3)+Wi2C6(−1+5y4)

+ Wi2C7

(−1+6y5

)+S∗yz

1+

(WiD0 +Wi2D3 +

P,z

2

)(−1+2y)

+(WiD1 +Wi2D4

)(−1+3y2)+(WiD2 +Wi2D5

)(−1+4y3)

+ Wi2D6(−1+5y4)+Wi2D7

(−1+6y5

)], (3.53)

S∗zz = Wi[

S∗yz

1+

(WiD0 +Wi2D3 +

P,z

2

)(−1+2y)

+(WiD1 +Wi2D4

)(−1+3y2)+(WiD2 +Wi2D5

)(−1+4y3)

+ Wi2D6(−1+5y4)+Wi2D7

(−1+6y5

)], (3.54)


h

, i, j = x,y,z and i 6= j are the non-dimensional stresses.

Using equations (2.84) and (2.86) we can calculate the shear forces per unit width required

to move the barrel in x and z−directions as

F∗x =−S∗wxyδ1, (3.55)

F∗z =−S∗wyzδ2. (3.56)

67

The net shear force per unit width in the direction of the axis of the screw then can be

calculated as

F∗ = F∗z sinφ+F∗x cosφ. (3.57)

3.2.6 Volume flow rate

Volume flow rate (2.90) in x-direction when u is given by (3.39), becomes

Qx =U

2W− 1

6

(12

P,x +(Wi)2C0 +(Wi)4C3

)− 1

4((Wi)2C1 +(Wi)4C4

)

− 310

((Wi)2C2 +(Wi)4C5

)− 13(Wi)4C6− 5

14(Wi)4C7, (3.58)

and volume flow rate in z-direction using (3.41) takes the form

Qz =12− 1

6

(12

P,z +(Wi)2D0 +(Wi)4D3

)− 1

4((Wi)2D1 +(Wi)4D4

)

− 310

((Wi)2D2 +(Wi)4D5

)− 13(Wi)4D6− 5

14(Wi)4D7. (3.59)

Resultant volume flow rate forward in the screw channel (2.96) with the help of equation

(3.43) is

Q =N

sinφ

[12− 1

6

(12

P,z +(Wi)2D0 +(Wi)4D3

)− 1

4((Wi)2D1 +(Wi)4D4

)

− 310

((Wi)2D2 +(Wi)4D5

)− 13(Wi)4D6− 5

14(Wi)4D7

sinφ

+

U2W

− 16

(12

P,x +(Wi)2C0 +(Wi)4C3

)− 1

4((Wi)2C1 +(Wi)4C4

)

− 310

((Wi)2C2 +(Wi)4C5

)− 13(Wi)4C6− 5

14(Wi)4C7

cosφ

], (3.60)

=N



The average velocity in the direction of the axis of the screw (2.99) is

68

s∗ = N[

12− 1

6

(12

P,z +(Wi)2D0 +(Wi)4D3

)− 1

4((Wi)2D1 +(Wi)4D4

)

− 310

((Wi)2D2 +(Wi)4D5

)− 13(Wi)4D6− 5

14(Wi)4D7

sinφ

+

U2W

− 16

(12

P,x +(Wi)2C0 +(Wi)4C3

)− 1

4((Wi)2C1 +(Wi)4C4

)

− 310

((Wi)2C2 +(Wi)4C5

)− 13(Wi)4C6− 5

14(Wi)4C7

cosφ

]. (3.62)

During this study we found that, if we put Wi2 =βW 2

h2 in the above equations we can obtain

the solutions for Pseudoplastic fluid, where β is Pseudoplastic constant.


In the present work we have considered the steady flow of an incompressible, isothermal

and homogeneous co-rotational Maxwell fluid in HSR. The geometry is same as discussed

in the previous chapter. Second order nonlinear coupled differential equations are

transformed to a single differential equation. Using perturbation method analytical

expressions are obtained for velocities u and w in x and z− directions, respectively, and

also in the direction of the axis of the screw, s. Expressions for the shear stresses in the

flow field and at barrel surface, forces exerted on fluid, volume flow rates and average

velocity are also derived. Here we discussed the behavior of co-rotational Maxwell fluid

in HSR in terms of non-Newtonian parameter Wi2, flight angle φ and pressure gradients P,x

and P,z on the velocities given by equations (3.39), (3.41) and (3.43). The figures (3.1 -

3.3) are plotted for u, w and s respectively, for different values of Wi2. It is noticed that the

behavior of the velocity profiles is same as discussed in chapter 2 for β , the only difference

is that now the shear thinning effects due to Wi2 are larger. For Wi2 = 0.0, solution for

Newtonian case are retrieved.

Figures (3.4 -3.9) are sketched for the velocity profiles u, w and s for different values of

69

P,x and P,z, it is observed that here the patterns of velocity profiles are more obvious than

third-grade fluid case.

The figure (3.10) shows the effect of φ on the velocity profile s, more clear picture due

to larger shear thinning behavior of co-rotational Maxwell fluid than third-grade fluid is

observed. The figures (3.11) and (3.12) are plotted to observe effect of Wi2 on shear stresses

Sxy and Syz. The regions where velocities in x and z−directions attain their maximum values

are also observed.

Figure 3.1: Velocity profile u(y) for different values of Wi2, keeping P,x =−1.5, P,z =−1.5and φ = 45.

70

Figure 3.2: Velocity profile w(y) for different values of Wi2, keeping P,x = −1.5, P,z =−1.5 and φ = 45.

Figure 3.3: Velocity profile s(y) for different values of Wi2, keeping P,x =−1.5, P,z =−1.5and φ = 45.

71

Figure 3.4: Velocity profile u(y) for different values of P,x, keeping Wi2 = 0.25, P,z =−1.5and φ = 45.

Figure 3.5: Velocity profile u(y) for different values of P,z, keeping Wi2 = 0.25, P,x =−1.5and φ = 45.

72

Figure 3.6: Velocity profile w(y) for different values of P,x, keeping Wi2 = 0.25, P,z =−1.5and φ = 45.

Figure 3.7: Velocity profile w(y) for different values of P,z, keeping Wi2 = 0.25, P,x =−1.5and φ = 45.

73

Figure 3.8: Velocity profile s(y) for different values of P,x, keeping Wi2 = 0.25, P,z =−1.5and φ = 45.

Figure 3.9: Velocity profile s(y) for different values of P,z, keeping Wi2 = 0.25, P,x =−1.5and φ = 45.

74

Figure 3.10: Velocity profile s(y) for different values of φ, keeping Wi2 = 0.25, P,x =−1.5and P,z =−1.5.

Figure 3.11: Variation of shear stress Sxy for different values of Wi2, keeping P,x =−2.0, P,z =−2.0 and φ = 45.

75

Figure 3.12: Variation of shear stress Syz for different values of Wi2, keeping P,x =−2.0, P,z =−2.0 and φ = 45.

3.4 Conclusion

The steady, homogeneous flow of an isothermal and incompressible Co-rotational Maxwell

fluid is investigated in HSR. The geometry of the problem under consideration gives second


equation using a transformation. Perturbation method is used to obtained analytical

expressions for the velocity profiles. Expressions for volume flow rate shear and normal

stresses, shear at barrel surface, forces exerted on fluid and average velocity are also

calculated. It is noticed that the zeroth component solution matches with solution of the

Newtonian fluid in HSR and also found that the net velocity of the fluid is due to the

pressure gradient only. Graphical representation shows that the velocity profiles strongly

depend on non-Newtonian parameter and pressure gradients in x and z− direction. It is

also observed that the shear thinning effects in co-rotational Maxwell fluid are larger than

third-grade fluid for non-Newtonian parameter. The points where the velocities in x and

z−directions attain their maximum values are also noted.

76

Chapter 4

Analysis of Eyring Fluid in Helical Screw Rheometer:

Exact Solution

77

In this part of the thesis, the flow of an incompressible, isothermal Eyring fluid in a helical

screw rheometer is analyzed. The geometry of the HSR is same as discussed in the previous

chapters (see fig.2.1). Exact solutions are obtained for the velocities in x and z−directions

and also in the direction of the axis of the screw. Expressions for shear stresses, shear

at barrel surface, forces exerted on fluid, volume flow rates and average velocity are also

obtained. The flow profiles are discussed with the help of graphs. We observed that the

velocity profiles are strongly dependent on parameters involved. It is also noticed that

the flow increases as the flight angle increases. This work concludes that the involved

parameters play a vital role in the extrusion process.


Consider the steady flow of an isothermal, incompressible and homogeneous Eyring fluid

in HSR (fig 2.1). Using velocity profile (2.1) equation (1.7) is identically satisfied and

momentum equations result in (2.3 - 2.5). Using equation(2.10) in equation (1.28), we get

the following components of the extra stress tensor S = S(y) for Eyring fluid:

Sxx = 0, (4.1)

Sxy = Syx = B sinh−1(− 1

Cdux

dy

), (4.2)

Sxz = Szx = 0, (4.3)

Syy = 0, (4.4)

Syz = Szy = B sinh−1(− 1

Cduz

dy

), (4.5)

Szz = 0. (4.6)

Using equations (4.2), (4.4) and (4.5) in equations (2.3 - 2.5) we get

∂P∂x

= Bddy

sinh−1

(− 1

Cdux

dy

), (4.7)

∂P∂y

= 0 (4.8)

78

∂P∂z

= Bddy

sinh−1

(− 1

Cduz

dy

). (4.9)

The equation (4.8) suggest that P = P(x,z). Since the right sides of equations (4.7) and

(4.9) are functions of y alone and P 6= P(y), this implies∂P∂x

and∂P∂z

are constant pressure

gradients. Introducing the dimensionless parameters (2.35) in equations (4.7), (4.9) and

(2.2) result in

∂P∗

∂x∗= ˜β∗

ddy∗

sinh−1

(− 1

˜α∗du∗

dy∗

), (4.10)

∂P∗

∂z∗= ˜β∗

ddy∗

sinh−1

(− 1

˜α∗dw∗

dy∗

), (4.11)

u∗ = 0, at y∗ = 0, and u∗ =−tanφ, at y∗ = 1, (4.12)

w∗ = 0, at y∗ = 0, and w∗ = 1, at y∗ = 1, (4.13)

where ˜α∗ =ChW

and ˜β∗ =B

µWh

are dimensionless Eyring fluid parameters. Dropping “*”

equations (4.10 - 4.13), we get

ddy

sinh−1

(− 1

˜αdudy

)=

1˜β

∂P∂x

, (4.14)

ddy

sinh−1

(− 1

˜αdwdy

)=

1˜β

∂P∂z

, (4.15)

u = 0, at y = 0, and u =−tanφ, at y = 1, (4.16)

w = 0, at y = 0, and w = 1, at y = 1. (4.17)

Equations (4.14) and (4.15) are second order nonlinear inhomogeneous ordinary differen-

tial equations together with the inhomogeneous boundary conditions (4.16) and (4.17). To

get their exact solution we proceed as follows:

79


Integrating equations (4.14) and (4.15) with respect to y, we get

sinh−1(− 1

˜αdudy

)=

1˜β

P,xy+C1, (4.18)

sinh−1(− 1

˜αdwdy

)=

1˜β

P,zy+C2, (4.19)

where P,x =∂P∂x

and P,z =∂P∂z

. Simplification implies

dudy

= − ˜αsinh

[1˜β

P,xy+C1

], (4.20)

dwdy

= − ˜αsinh

[1˜β

P,zy+C2

]. (4.21)

Integration with respect to y, results in

u = −˜α ˜βP,x

cosh

[1˜β

P,xy+C1

]+K1, (4.22)

w = −˜α ˜βP,z

cosh

[1˜β

P,zy+C2

]+K2, (4.23)

where C1, K1, C2 and K2 are arbitrary constants of integration. Using boundary conditions

(4.16) and (4.17) in these equations, we get four different values for each C1, K1, C2 and

K2 in which more suitable values of C1, K1, C2 and K2 satisfying the boundary conditions

of the problem for the flow profiles in forward direction are

C1 = cosh−1

(K1

˜α ˜βX1

), (4.24)

K1 =K1

X1P,x, (4.25)

80

C2 = cosh−1

(K2

˜α ˜βX2

), (4.26)

K2 =K2

X2P,z, (4.27)

where

K1 = −2 ˜α ˜βsinh2

(P,x

2 ˜β

)P,x tanφ

−√√√√ ˜α2 ˜β2 sinh2

(P,x˜β

)[2 ˜α2 ˜β2

−1+ cosh

(P,x˜β

)+P2

,x tan2 φ

],

X1 = 2 ˜α ˜β

−1+ cosh

(P,x˜β

),

K2 = 2 ˜α ˜βsinh2

(P,z

2 ˜β

)P,z−

√√√√ ˜α2 ˜β2 sinh2

(P,z˜β

)[2 ˜α2 ˜β2

−1+ cosh

(P,z˜β

)+P2

,z

],

X2 = 2 ˜α ˜β

−1+ cosh

(P,z˜β

).

Thus equations (4.22) and (4.23) take the form

u = −˜α ˜βP,x

cosh

[1˜β

P,xy+ cosh−1

(K1

˜α ˜βX1

)]+

K1

X1P,x, (4.28)

w = −˜α ˜βP,z

cosh

[1˜β

P,zy+ cosh−1

(K2

˜α ˜βX2

)]+

K2

X2P,z, (4.29)

which are the velocity components in x and z−directions.


The velocity profile in the direction of the axis of the screw, s, at any depth in the channel

can be computed from equations (4.28) and (4.29) as


81

s =

[−

˜α ˜βP,z

cosh

P,z˜β

y+ cosh−1

(K2

˜α ˜βX2

)+

K2

X2P,z

]sinφ

+

[−

˜α ˜βP,x

cosh

P,x˜β

y+ cosh−1

(K1

˜α ˜βX1

)+

K1

X1P,x

]cosφ (4.31)

4.2.2 Shear Stresses

Using (4.22) and (4.23) in equations (4.2) and (4.5) we obtain as

S∗xy = S∗yx = P,xy+ ˜βcosh−1

(K1

˜α ˜βX1

), (4.32)

S∗yz = S∗zy = P,zy+ ˜βcosh−1

(K2

˜α ˜βX2

). (4.33)


S∗wx= P,x + ˜βcosh−1

(K1

˜α ˜βX1

), (4.34)

S∗wz= P,z + ˜βcosh−1

(K2

˜α ˜βX2

), (4.35)


h


Using equations (2.84) and (2.86) we can calculate the shear forces per unit width required

to move the barrel in x and z−directions as

F∗x =−S∗wxδ1, (4.36)

F∗z =−S∗wzδ2. (4.37)

The net shear force per unit width in the direction of the axis of the screw can be computed

using equation (2.88) as

F∗ = F∗z sinφ+F∗x cosφ. (4.38)

82


Using equations (4.28) and (4.29), equations (2.90) and (2.93), respectively, give the

volume flow rate in x and z−directions, respectively, as

Qx =˜α ˜β2

P2,x

sinh

(cosh−1

(K1

˜α ˜βX1

))− sinh

(cosh−1

(K1

˜α ˜βX1

)+

P,x˜β

)

+K1

X1P,x, (4.39)

Qz =˜α ˜β2

P2,z

sinh

(cosh−1

(K2

˜α ˜βX2

))− sinh

(cosh−1

(K2

˜α ˜βX2

)+

P,z˜β

)

+K2

X2P,z, (4.40)

and the resultant volume flow rate forward in the screw channel (2.96), using equation

(4.31) becomes

Q =N

sinφ

[˜α ˜β2

P2,x

(sinh

(cosh−1

(K1

˜α ˜βX1

))− sinh

(cosh−1

(K1

˜α ˜βX1

)+

P,x˜β

))

+K1

X1P,x

cosφ+

˜α ˜β2

P2,z

(sinh

(cosh−1

(K2

˜α ˜βX2

))

− sinh

(cosh−1

(K2

˜α ˜βX2

)+

P,z˜β

))+

K2

X2P,z

sinφ

], (4.41)

=N

sinφQx cosφ+Qz sinφ . (4.42)


The average velocity in the direction of the axis of the screw is obtained, using equation

(4.31) in (2.99) as

s∗ = N

[˜α ˜β2

P2,x

(sinh

(cosh−1

(K1

˜α ˜βX1

))− sinh

(cosh−1

(K1

˜α ˜βX1

)+

P,x˜β

))

83

+K1

X1P,x

cosφ+

˜α ˜β2

P2,z

(sinh

(cosh−1

(K2

˜α ˜βX2

))

− sinh

(cosh−1

(K2

˜α ˜βX2

)+

P,z˜β

))+

K2

X2P,z

sinφ

]. (4.43)

It may be mentioned here that the solution for the Johnson-Tevaarwerk fluid can be obtained

from equations (4.20) and (4.21) by relating ˜α and ˜β with material constants of Johnson-

Tevaarwerk fluid.



and homogeneous Eyring fluid in the same geometry as discussed in chapter 2 (see fig.2.1).

Exact solutions are obtained for velocities u, w in x, z− directions respectively and also in

the direction of the axis of the screw s. Expressions for the shear stresses, shear stresses

at barrel surface, forces exerted on fluid, volume flow rates and average velocity are also

calculated. Here we discussed the effect of dimensionless parameters ˜α, ˜β, flight angle

φ and pressure gradients P,x and P,z, on the velocity profiles with the help of graphical

representation. From figures (4.1 - 4.2) and (4.3), we can observe the behavior of ˜α. For

˜α the pattern of velocities are same as discussed in chapter 2. In this case the velocity

profiles steadily increase from the screw surface toward the barrel up to some points and

then steadily decrease toward the barrel surface. This may be due to the fluid property or

due to zero normal stresses. In the previous case steady increase observed in the velocity

profiles up to some points from screw to barrel surface and then a sharp decrease observed

in the graphs near the barrel. Figures (4.4 - 4.5) and (4.6) show that an increase in the

magnitude of the ˜β reduce the velocities u, w and s monotonically, which suggest the

happening of shear thickening in the fluid for increasing values of ˜β. From figures (4.7 -

4.10), it can be noticed that an increase in the value of pressure gradient increases the flow

profile. Figure (4.11) shows that the effect of φ on s is same as previously discussed.

84

Figure 4.1: Velocity profile u(y) for different values of ˜α, keeping P,x =−0.5, ˜β = 0.5 andφ = 45.

Figure 4.2: Velocity profile w(y) for different values of ˜α, keeping P,z = −0.5, ˜β = 0.5and φ = 45.

85

Figure 4.3: Velocity profile s(y) for different values of ˜α keeping ˜β = 0.5 P,x =−0.5, P,z =−0.5 and φ = 45.

Figure 4.4: Velocity profile u(y) for different values of ˜β keeping ˜α = 5.0 P,x =−0.5 andφ = 45.

86

Figure 4.5: Velocity profile w(y) for different values of ˜β, keeping ˜α = 5.0, P,z = −0.5and φ = 45.

Figure 4.6: Velocity profile s(y) for different values of ˜β, keeping ˜α = 5.0, P,x =−0.5, P,z =−0.5 and φ = 45.

87

Figure 4.7: Velocity profile u(y) for different values of P,x keeping ˜α = 5.0, ˜β = 0.5 andφ = 45.

Figure 4.8: Velocity profile w(y) for different values of P,z, keeping ˜α = 5.0, ˜β = 0.5 andφ = 45.

88

Figure 4.9: Velocity profile s(y) for different values of P,x, keeping ˜α = 5.0, ˜β = 0.5 P,z =−0.5 and φ = 45.

Figure 4.10: Velocity profile s(y) for different values of P,z keeping ˜α = 5.0, ˜β = 0.5 P,x =−0.5 and φ = 45.

89

Figure 4.11: Velocity profile s(y) for different values of φ, keeping ˜α = 5.0, ˜β = 0.5, P,x =−0.5 and P,z =−0,5.

4.4 conclusion

The steady, homogeneous flow of an isothermal and incompressible Eyring fluid is

investigated in HSR. The velocity profiles, shear stresses, shear stresses at barrel surface,

shear forces exerted on the fluid, volume flow rates and average velocity of the fluid are

calculated. We observed that the velocity field depends on the involved parameters. The

increase in the values of non-dimensional parameters ˜α and pressure gradient increases the

flow of the fluid. However graphical representation shows monotonically decrease in the

flow profiles with the increase in the value of ˜β, which suggest shear thickening behavior

of fluid for larger values of ˜β. It is also observed that the net velocity of the fluid attain its

maximum value at φ = 45.

90

Chapter 5

Analysis of Eyring-Powell Fluid in Helical Screw

Rheometer: Adomian Decomposition Method

91

This chapter aims to study the flow of an incompressible, isothermal Eyring-Powell fluid

in HSR (see fig.2.1). The developed second order nonlinear inhomogeneous differential

equations are solved using ADM. Analytical expressions are obtained for the velocity

profiles, shear stresses, shear at barrel surface, force exerted on fluid, volume flow rates

and average velocity. The flow profiles are discussed with the help of graphs. We observed

that the velocity profiles strongly depend on dimensionless non-Newtonian parameters.

With the increase in the value of flow parameters velocity increases, which conclude that

extrusion process increases with increasing values of involved parameters. It is also noticed

that the flow increases as the flight angle increases.


Consider the steady flow of an isothermal, incompressible and homogeneous Eyring-

Powell fluid in HSR, in the same geometry as given in section 2.1. Using velocity profile

(2.1), equation (1.7) is identically satisfied. To calculate the components of extra stress

tensor S=S(y), for the constitutive equation (1.29) of Eyring-Powell fluid involved in

equations (2.3 - 2.5) we proceed as: Using equation (2.10) in equation (1.29), we obtained

the nonzero components of the extra stress tensor as

Sxy = Syx = µdux

dy+

1B

sinh−1(

1C

dux

dy

), (5.1)

Syz = Szy = µduz

dy+

1B

sinh−1(

1C

duz

dy

). (5.2)

Using equations (5.1) and (5.2) in equations (2.3 - 2.5), we get

∂P∂x

= µd2ux

dy2 +1B

ddy

[sinh−1

(1C

dux

dy

)], (5.3)

∂P∂y

= 0, (5.4)

∂P∂z

= µd2uz

dy2 +1B

ddy

[sinh−1

(1C

duz

dy

)]. (5.5)

92

The equation (5.4) shows that P = P(x,z). Since the right sides of equations (5.3) and

(5.5) are functions of y alone and P 6= P(y), this implies∂P∂x

= constant and∂P∂z

= constant.

Maclaurin series expansion of the inverse sine hyperbolic function in equations (5.3) and

(5.5), give

∂P∂x

= µd2ux

dy2 +1B

ddy

[1C

dux

dy− 1

6

(1C

dux

dy

)3

+O(

1C

)5]

, (5.6)

∂P∂z

= µd2uz

dy2 +1B

ddy

[1C

duz

dy− 1

6

(1C

duz

dy

)3

+O(

1C

)5]

, (5.7)

neglecting higher powers of non-Newtonian parameter C , and simplifying equations (5.6)

and (5.7), we obtained

d2ux

dy2 =(

CBµCB +1

)∂P∂x

+(

12C 2(µCB +1)

)(dux

dy

)2 d2ux

dy2 , (5.8)

d2uz

dy2 =(

CBµCB +1

)∂P∂z

+(

12C 2(µCB +1)

)(duz

dy

)2 d2uz

dy2 . (5.9)

Introducing the dimensionless parameters (2.35) in equations (5.8), (5.9) and (2.2) we get

d2u∗

dy∗2 = ˜α∗∂P∗

∂x∗+

˜β∗

(du∗

dy∗

)2 d2u∗

dy∗2 , (5.10)

d2w∗

dy∗2 = ˜α∗∂P∗

∂z∗+

˜β∗

(dw∗

dy∗

)2 d2w∗

dy∗2 , (5.11)

u∗ = 0, at y∗ = 0, and u∗ =−tanφ, at y∗ = 1,

w∗ = 0, at y∗ = 0, and w∗ = 1, at y∗ = 1,(5.12)

where ˜α∗=(

µCBµCB +1

)and

˜β∗=

(W 2

2C 2h2(µCB +1)

)are dimensionless non-Newtonian

parameters. Dropping “*” equations (5.10) and (5.11), give

d2udy2 = ˜αP,x +

˜β(

dudy

)2 d2udy2 , (5.13)

93

d2wdy2 = ˜αP,z +

˜β(

dwdy

)2 d2wdy2 , (5.14)

where∂P∂x

= P,x and∂P∂z

= P,z and boundary conditions (5.12) become

u(0) = 0, and u(1) =−tanφ,

w(0) = 0, and w(1) = 1.(5.15)

Equations (5.13) and (5.14) are second order nonlinear inhomogeneous ordinary differ-

ential equations, with boundary conditions (5.15). It is difficult to find exact solution,

therefore, we are using ADM to solve the problem in hand.


ADM (discussed in sec: 1.8.3) describes that in operator form equations (5.13) and (5.14),

can be written as

Lyy(u) = ˜αP,x +˜β(

dudy

)2 d2udy2 , (5.16)

Lyy(w) = ˜αP,z +˜β(

dwdy

)2 d2wdy2 , (5.17)

where Lyy =d2

dy2 is the invertible differential operator. Applying L−1yy (=

∫ ∫(∗)dydy) to

both sides of equations (5.16) and (5.17), we obtained

u = C1 +C2y+L−1yy

(˜αP,x

)+

˜βL−1

yy

[(dudy

)2 d2udy2

], (5.18)

w = C3 +C4y+L−1yy

(˜αP,z

)+

˜βL−1

yy

[(dwdy

)2 d2wdy2

], (5.19)

where C1, C2, C3 and C4 are arbitrary constants of integration, can be determined using

boundary conditions. According to procedure of ADM, u and w can be written in

94

component form as:

u =∞

∑n=0

un, (5.20)

w =∞

∑n=0

wn. (5.21)

Using equations (5.20) and (5.21) in equations (5.18) and (5.19), result in

u = C1 +C2y+L−1yy

(˜αP,x

)+

˜βL−1

yy

(ddy

∞

∑n=0

un

)2 (d2

dy2

∞

∑n=0

un

) , (5.22)

w = C3 +C4y+L−1yy

(˜αP,z

)+

˜βL−1

yy

(ddy

∞

∑n=0

wn

)2 (d2

dy2

∞

∑n=0

wn

) . (5.23)

In terms of Adomian polynomial, these equations can be written as

u = C1 +C2y+L−1yy

(˜αP,x

)+

˜βL−1

yy

(∞

∑n=0

Λn

), (5.24)

w = C3 +C4y+L−1yy

(˜αP,z

)+

˜βL−1

yy

(∞

∑n=0

Γn

), (5.25)

where

∞

∑n=0

Λn =

(ddy

∞

∑n=0

un

)2 (d2

dy2

∞

∑n=0

un

), (5.26)

∞

∑n=0

Γn =

(ddy

∞

∑n=0

wn

)2 (d2

dy2

∞

∑n=0

wn

), (5.27)

are Adomian polynomials, and the boundary conditions (5.15) will take the form

∞

∑n=0

un(0) = 0 and∞

∑n=0

un(1) =−tanφ, (5.28)

∞

∑n=0

wn(0) = 0 and∞

∑n=0

wn(1) = 1. (5.29)

95

From the recursive relations (5.24 - 5.29), we can identify the zeroth order problems as,

u0 = C1 +C2y+L−1yy

(˜αP,x

), (5.30)

w0 = C3 +C4y+L−1yy

(˜αP,z

), (5.31)

u0(0) = 0 and u0(1) =−tanφ, (5.32)

w0(0) = 0 and w0(1) = 1. (5.33)

The remaining order problems are in the following form

un+1 =˜βL−1

yy (Λn) , n≥ 0, (5.34)

wn+1 =˜βL−1

yy (Γn) , n≥ 0, (5.35)


∞

∑n=1

un(0) = 0 and∞

∑n=1

un(1) = 0, (5.36)

∞

∑n=1

wn(0) = 0 and∞

∑n=1

wn(1) = 0. (5.37)

From equations (5.26) and (5.27) we can calculate Adomian polynomials as

Λ0 =(

du0

dy

)2 d2u0

dy2 , (5.38)

Λ1 =(

du0

dy

)2 d2u1

dy2 +2du0

dydu1

dyd2u0

dy2 , (5.39)

Λ2 =(

du1

dy

)2 d2u0

dy2 +2du0

dydu1

dyd2u1

dy2 +2du0

dydu2

dyd2u0

dy2 +(

du0

dy

)2 d2u2

dy2 , (5.40)

Γ0 =(

dw0

dy

)2 d2w0

dy2 , (5.41)

Γ1 =(

dw0

dy

)2 d2w1

dy2 +2dw0

dydw1

dyd2w0

dy2 , (5.42)

96

Γ2 =(

dw1

dy

)2 d2w0

dy2 +2dw0

dydw1

dyd2w1

dy2 +2dw0

dydw2

dyd2w0

dy2

+(

dw0

dy

)2 d2w2

dy2 , (5.43)

the remaining components of Adomian polynomials can be generated easily.

The ADM solutions to equations (5.24) and (5.25) with the boundary conditions (5.28) and

(5.29) will be the sum of all order solutions, that is

u =∞

∑n=1

un, (5.44)

w =∞

∑n=1

wn. (5.45)

5.2.1 Zeroth order Solution

Zeroth order solutions of equations (5.13 - 5.15) can be calculated from the relations given

by equations (5.30 - 5.33), which are

u0 = −y tanφ+˜α2

P,x(y2− y), (5.46)

w0 = y+˜α2

P,z(y2− y). (5.47)

These equations give the solution for Newtonian fluid if we put ˜α =1µ

.

5.2.2 First order Solution

Equations (5.34 - 5.37) give the first order problems as

u1 =˜βL−1

yy (Λ0) , (5.48)

w1 =˜βL−1

yy (Γ0) , (5.49)

97


u1(0) = = u1(1) = 0, (5.50)

w1(0) = = w1(1) = 0, (5.51)

which have the solution

u1 =˜β(ε0

2(y2− y

)+

ε1

6(y3− y

)+

ε2

12(y4− y

)), (5.52)

w1 =˜β(σ0

2(y2− y

)+

σ1

6(y3− y

)+

σ2

12(y4− y

)), (5.53)

where constant coefficients ε0, ε1, ε2, σ0, σ1 and σ2 are given in Appendix III.

5.2.3 Second order Solution

The second order problems have the form

u2 =˜βL−1

yy (Λ1) , (5.54)

w2 =˜βL−1

yy (Γ1) , (5.55)

along with their boundary conditions

u2(0) = = u2(1) = 0, (5.56)

w2(0) = = w2(1) = 0. (5.57)

Using equations (5.39) and (5.42), we have the solutions

u2 =˜β2

(ε3

2(y2− y

)+

ε4

6(y3− y

)+

ε5

12(y4− y

)+

ε6

20

(y5− y

)

+ε7

30

(y6− y

)), (5.58)

98

w2 =˜β2

(σ3

2(y2− y

)+

σ4

6(y3− y

)+

σ5

12(y4− y

)+

σ6

20

(y5− y

)

+σ7

30

(y6− y

)), (5.59)

where εi, σ j, i = 3, · · · ,7, j = 3, · · · ,7 are constant coefficients given in Appendix III.

5.2.4 Third order Solution

Equations (5.34 - 5.37) give the third order problems for n = 2 as

u3 =˜βL−1

yy (Λ2) , (5.60)

w3 =˜βL−1

yy (Γ2) , (5.61)

with

u3(0) = = u3(1) = 0, (5.62)

w3(0) = = w3(1) = 0. (5.63)

Using equations (5.40) and (5.43) in equations (5.60 - 5.63), we get the solutions

u3 =˜β3

(ε8

2(y2− y

)+

ε9

6(y3− y

)+

ε10

12(y4− y

)+

ε11

20

(y5− y

)+

ε12

30

(y6− y

)

+ε13

42(y7− y

)+

ε14

56(y8− y

)), (5.64)

w3 =˜β3

(σ8

2(y2− y

)+

σ9

6(y3− y

)+

σ10

12(y4− y

)+

σ11

20

(y5− y

)+

σ12

30

(y6− y

)

+σ13

42(y7− y

)+

σ14

56(y8− y

)), (5.65)

where εi, σ j, i = 8, · · · ,14, j = 8, · · · ,14 are constant coefficients given in Appendix III.

99



Combining equations (5.46), (5.52), (5.58) and (5.64), the solution for the velocity profile

upto order three in the transverse plane becomes

u = −y tanφ+12

(˜αP,x +

˜βε0 +

˜β2ε3 +

˜β3ε8

)(y2− y

)+

16

(˜βε1 +

˜β2ε4 +

˜β3ε9

)(y3− y

)

+1

12

(˜βε2 +

˜β2ε5 +

˜β3ε10

)(y4− y

)+

120

(˜β2ε6 +

˜β3ε11

)(y5− y

)

+1

30

(˜β2ε7 +

˜β3ε12

)(y6− y

)+

˜β3ε13

42(y7− y

)+

˜β3ε14

56(y8− y

). (5.66)


Sum of the equations (5.47), (5.53), (5.59) and (5.65) give the solution for the velocity

profile upto order three in the down channel direction

w = y+12

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)(y2− y

)+

16

(˜βσ1 +

˜β2σ4 +

˜β3σ9

)(y3− y

)

+1

12

(˜βσ2 +

˜β2σ5 +

˜β3σ10

)(y4− y

)+

120

(˜β2σ6 +

˜β3σ11

)(y5− y

)

+1

30

(˜β2σ7 +

˜β3σ12

)(y6− y

)+

˜β3σ13

42(y7− y

)+

˜β3σ14

56(y8− y

). (5.67)


The velocity in the direction of the axis of the screw at any depth in the channel can be

computed from equations (5.66) and (5.67) as


=12

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)sinφ+

(˜αP,z +

˜βε0 +

˜β2ε3 +

˜β3ε8

)cosφ

(y2− y

)

+16

(˜βσ1 +

˜β2σ4 +

˜β3σ9

)sinφ+

(˜βε1 +

˜β2ε4 +

˜β3ε9

)cosφ

(y3− y

)

100

+112

(˜βσ2 +

˜β2σ5 +

˜β3σ10

)sinφ+

(˜βε2 +

˜β2ε5 +

˜β3ε10

)cosφ

(y4− y

)

+120

(˜β2σ6 +

˜β3σ11

)sinφ+

(˜β2ε6 +

˜β3ε11

)cosφ

(y5− y

)

+130

(˜β2σ7 +

˜β3σ12

)sinφ+

(˜β2ε7 +

˜β3ε12

)cosφ

(y6− y

)

+˜β3

42(σ13 sinφ+ ε13 cosφ)

(y7− y

)+

˜β3


(y8− y

), (5.69)

which is the resultant velocity of the flow. It can be noticed that the forward velocity at any

point in the channel depends only on pressure gradient.

5.2.6 Shear Stresses

Shear stresses can now be calculated with the help of (5.66 - 5.67) as follows:

S∗xy = S∗yx =[− tanφ+

12(−1+2y)

(˜αP,x +

˜βε0 +

˜β2ε3 +

˜β3ε8

)+

16

(−1+3y2)(

˜βε1

+˜β2ε4 +

˜β3ε9

)+

112

(−1+4y3)(

˜βε2 +

˜β2ε5 +

˜β3ε10

)+

120

(−1+5y4)(

˜β2ε6

+˜β3ε11

)+

130

(−1+6y5

)(˜β2ε7 +

˜β3ε12

)+

˜β3ε13

42

(−1+7y6

)+

˜β3ε14

56(−1+8y7)

+h

µBWsinh−1

[WhC

− tanφ+

12(−1+2y)

(˜αP,x +

˜βε0 +

˜β2ε3 +

˜β3ε8

)

+16

(−1+3y2)(

˜βε1 +

˜β2ε4 +

˜β3ε9

)+

112

(−1+4y3)(

˜βε2 +

˜β2ε5 +

˜β3ε10

)

+1

20(−1+5y4)

(˜β2ε6 +

˜β3ε11

)+

130

(−1+6y5

)(˜β2ε7 +

˜β3ε12

)

+˜β3ε13

42

(−1+7y6

)+

˜β3ε14

56(−1+8y7)

, (5.70)

S∗yz = S∗zy =[

1+12(−1+2y)

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)+

16

(−1+3y2)(

˜βσ1

+˜β2σ4 +

˜β3σ9

)+

112

(−1+4y3)(

˜βσ2 +

˜β2σ5 +

˜β3σ10

)+

120

(−1+5y4)(

˜β2σ6

101

+˜β3σ11

)+

130

(−1+6y5

)(˜β2σ7 +

˜β3σ12

)+

˜β3σ13

42

(−1+7y6

)+

˜β3σ14

56(−1+8y7)

+h

µBWsinh−1

[WCh

1+

12(−1+2y)

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)

+16

(−1+3y2)(

˜βσ1 +

˜β2σ4 +

˜β3σ9

)+

112

(−1+4y3)(

˜βσ2 +

˜β2σ5 +

˜β3σ10

)

+1

20(−1+5y4)

(˜β2σ6 +

˜β3σ11

)+

130

(−1+6y5

)(˜β2σ7 +

˜β3σ12

)

+˜β3σ13

42

(−1+7y6

)+

˜β3σ14

56(−1+8y7)

, (5.71)


h


The shear stresses exerted by the fluid on the barrel surface at y = 1 are

S∗xy|y=1 = S∗wx= −

[− tanφ+

12

(˜αP,x +

˜βε0 +

˜β2ε3 +

˜β3ε8

)+

13

(˜βε1 +

˜β2ε4 +

˜β3ε9

)

+14

(˜βε2 +

˜β2ε5 +

˜β3ε10

)+

15

(˜β2ε6 +

˜β3ε11

)+

16

(˜β2ε7 +

˜β3ε12

)

+˜β3ε13

7+

˜β3ε14

8

− h

µBWsinh−1

[WhC

− tanφ+

12

(˜αP,x +

˜βε0

+˜β2ε3 +

˜β3ε8

)+

13

(˜βε1 +

˜β2ε4 +

˜β3ε9

)+

14

(˜βε2 +

˜β2ε5 +

˜β3ε10

)

+15

(˜β2ε6 +

˜β3ε11

)+

16

(˜β2ε7 +

˜β3ε12

)+

˜β3ε13

7+

˜β3ε14

8

, (5.72)

S∗yz|y=1 = S∗wz= −

[1+

12

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)+

13

(˜βσ1 +

˜β2σ4 +

˜β3σ9

)

+14

(˜βσ2 +

˜β2σ5 +

˜β3σ10

)+

15

(˜β2σ6 +

˜β3σ11

)+

16

(˜β2σ7 +

˜β3σ12

)

+˜β3σ13

7+

˜β3σ14

8

− h

µBWsinh−1

[WhC

1+

12

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)

102

+13

(˜βσ1 +

˜β2σ4 +

˜β3σ9

)+

14

(˜βσ2 +

˜β2σ5 +

˜β3σ10

)+

15

(˜β2σ6 +

˜β3σ11

)

+16

(˜β2σ7 +

˜β3σ12

)+

˜β3σ13

7+

˜β3σ14

8

. (5.73)

Equation (2.84) gives the shear force per unit width required to move the barrel in

x−direction as

F∗x =−S∗wxδ1. (5.74)

Similarly equation (2.86) gives the shear force per unit width required to move the barrel

in z−direction as

F∗z =−S∗wzδ2, (5.75)

therefore




Volume flow rates (2.90) and (2.93) in x and z−directions respectively, are obtained, using

equations (5.66) and (5.67) as

Qx = −12

tanφ− 112

(˜αP,x +

˜βε0 +

˜β2ε3 +

˜β3ε8

)− 1

24

(˜βε1 +

˜β2ε4 +

˜β3ε9

)

− 140

(˜βε2 +

˜β2ε5 +

˜β3ε10

)− 1

60

(˜β2ε6 +

˜β3ε11

)

− 184

(˜β2ε7 +

˜β3ε12

)−

˜β3ε13

112−

˜β3ε14

144, (5.77)

Qz =12− 1

12

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)− 1

24

(˜βσ1 +

˜β2σ4 +

˜β3σ9

)

− 140

(˜βσ2 +

˜β2σ5 +

˜β3σ10

)− 1

60

(˜β2σ6 +

˜β3σ11

)

103

− 184

(˜β2σ7 +

˜β3σ12

)−

˜β3σ13

112−

˜β3σ14

144. (5.78)

Resultant volume flow rate (2.96) forward in the screw channel with the help of equation

(5.69) is

Q =N

sinφ

[− 1

12

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)sinφ+

(˜αP,x +

˜βε0 +

˜β2ε3 +

˜β3ε8

)cosφ

− 124

(˜βσ1 +

˜β2σ4 +

˜β3σ9

)sinφ+

(˜βε1 +

˜β2ε4 +

˜β3ε9

)cosφ

− 140

(˜βσ2 +

˜β2σ5 +

˜β3σ10

)sinφ+

(˜βε2 +

˜β2ε5 +

˜β3ε10

)cosφ

− 160

(˜β2σ6 +

˜β3σ11

)sinφ+

(˜β2ε6 +

˜β3ε11

)cosφ

− 184

(˜β2σ7 +

˜β3σ12

)sinφ+

(˜β2ε7 +

˜β3ε12

)cosφ

−˜β3

112(σ13 sinφ+ ε13 cosφ)−

˜β3


, (5.79)

or

Q =N



Average velocity in the direction of the axis of the screw can easily be calculated by

substituting equation (5.69) in equation (2.99) results in

s∗ = N[− 1

12

(˜αP,z +

˜βσ0 +

˜β2σ3 +

˜β3σ8

)sinφ+

(˜αP,x +

˜βε0 +

˜β2ε3 +

˜β3ε8

)cosφ

− 124

(˜βσ1 +

˜β2σ4 +

˜β3σ9

)sinφ+

(˜βε1 +

˜β2ε4 +

˜β3ε9

)cosφ

− 140

(˜βσ2 +

˜β2σ5 +

˜β3σ10

)sinφ+

(˜βε2 +

˜β2ε5 +

˜β3ε10

)cosφ

− 160

(˜β2σ6 +

˜β3σ11

)sinφ+

(˜β2ε6 +

˜β3ε11

)cosφ

104

− 184

(˜β2σ7 +

˜β3σ12

)sinφ+

(˜β2ε7 +

˜β3ε12

)cosφ

−˜β3

112(σ13 sinφ+ ε13 cosφ)−

˜β3


. (5.81)

Setting˜β = 0 and ˜α =

1µ

, solution for the Newtonian fluid in HSR can be obtained and

setting ˜α and˜β both equal to zero, flow only due to the drag of the plate (barrel) can be

calculated.



and homogeneous Eyring-Powell fluid in HSR. The geometry of the HSR is same as

discussed in the previous chapters (see fig.2.1). Using ADM solutions are obtained for

velocity profiles in x and z− directions and also in the direction of the axis of the screw

s. Expressions for the shear stresses (Sxy and Syz), shear stresses at barrel surface, forces

exerted on fluid, volume flow rates and average velocity are also calculated. Here we

discussed the effect of non-Newtonian parameters ˜α,˜β, flight angle φ and pressure gradients

P,x and P,z, on the velocity profiles with the help of graphical representation. From figures

(5.1 - 5.4) and (5.7 - 5.8) we can observed that the velocities u, w and s are in the same

pattern for ˜α and˜β as discussed in chapter 2. It is also noticed that both non-Newtonian

parameters depict the shear thinning effects in the fluid. However graphical representation

shows that shear thinning effects of˜β are larger than ˜α, as the increase in velocity profiles

is observed larger for˜β.

From figures (5.5 - 5.6) and (5.9 - 5.10), it can be noticed that an increase in the value of

pressure gradient increases the flow of the fluid. Figure (5.11) shows the same effects of φ

at s as discussed in the previous work. The figures (5.12 - 5.15) are sketched for Sxy and

Syz using different values of ˜α and˜β. Shear stresses show the same behavior for both the

parameters and findings are same as discussed in the previous chapters.

105

Figure 5.1: Velocity profile u(y) for different values of ˜α, keeping P,x =−0.5,˜β = 1.0 and

φ = 45.

Figure 5.2: Velocity profile w(y) for different values of ˜α, keeping P,z = −0.5,˜β = 1.0

and φ = 45.

106

Figure 5.3: Velocity profile u(y) for different values of˜β, keeping P,x =−0.5, ˜α = 1.0 and

φ = 45.

Figure 5.4: Velocity profile w(y) for different values of˜β, keeping P,z = −0.5, ˜α = 1.0

and φ = 45.

107

Figure 5.5: Velocity profile u(y) for different values of P,x, keeping ˜α = 1.0,˜β = 1.0 and

φ = 45.

Figure 5.6: Velocity profile w(y) for different values of P,z, keeping ˜α = 1.0,˜β = 1.0 and

φ = 45.

108

Figure 5.7: Velocity profile s(y) for different values of ˜α, keeping˜β = 1.0, P,x =

−0.5, P,z =−0.5 and φ = 45.

Figure 5.8: Velocity profile s(y) for different values of˜β, keeping ˜α = 1.0, P,x =

−0.5, P,z =−0.5 and φ = 45.

109

Figure 5.9: Velocity profile s(y) for different values of P,x, keeping ˜α = 1.0,˜β = 1.0, P,z =

−0.5 and φ = 45.

Figure 5.10: Velocity profile s(y) for different values of P,z, keeping ˜α = 1.0,˜β =

1.0, P,x =−0.5 and φ = 45.

110

Figure 5.11: Velocity profile s(y) for different values of φ, keeping ˜α = 1.0,˜β = 1.0, P,x =

−0.5 and P,z =−0,5.

Figure 5.12: Variation of shear stress Sxy for different values of ˜α, keeping P,x =−2.0, P,z =−2.0 and φ = 45.

111

Figure 5.13: Variation of shear stress Sxy for different values of˜β, keeping P,x =

−2.0, P,z =−2.0 and φ = 45.

Figure 5.14: Variation of shear stress Syz for different values of ˜α, keeping P,x =−2.0, P,z =−2.0 and φ = 45.

112

Figure 5.15: Variation of shear stress Syz for different values of˜β, keeping P,x =−2.0, P,z =

−2.0 and φ = 45.

5.4 Conclusion

The steady flow of an isothermal, homogeneous and incompressible Eyring-Powell fluid is

investigated in HSR. Using ADM the expressions for the velocity profiles are calculated.

Expressions for the shear stresses, shear stresses at barrel surface, shear forces exerted on

the fluid, volume flow rates and average velocity of the fluid are also calculated. Graphical

representation is given for the velocity profiles. It is observed that the velocity field depends

on the involved parameters. The increase in the value of non-Newtonian parameters and

pressure gradients increase the flow of the fluid. It can be seen that the shear thinning effect

of˜β is larger than ˜α in the fluid. It is also observed that the net velocity of the fluid is due

to the pressure gradient. Shear stresses show the same pattern for both the ˜α and˜β.

113

Chapter 6

Analytical Solution For the Flow of Oldroyd 8-Constant

Fluid in Helical Screw Rheometer

114

Steady flow of an incompressible, Oldroyd 8-constant fluid in HSR (fig.2.1) is considered

here. The developed second order nonlinear coupled differential equations are transformed

to single nonlinear differential equation, then solved by ADM. We obtained analytical

expressions for the velocity components, also the resultant velocity in direction of the screw

axis. Volume flow rates are calculated for all three velocities. The shear stresses, shear at

barrel surface, force exerted on fluid and average velocity are also calculated here. The

results have been discussed with the help of graphs as well.


Consider the steady flow of an incompressible Oldroyd 8-constant fluid in HSR. The

geometry of the problem under consideration is same as given in section 2.1. Using velocity

profile (2.1), equation (1.7) is identically satisfied. To calculate the components of extra

stress tensor S = S(y) for equations (2.6) and (2.7), we take the following steps:

Equation (1.14) implies

tr(S) = Sxx +Syy +Szz, (6.1)

equations (6.1) and (2.10) give

tr(S)A1 =

0 (Sxx +Syy +Szz)dux

dy0

(Sxx +Syy +Szz)dux

dy0 (Sxx +Syy +Szz)

duz

dy

0 (Sxx +Syy +Szz)duz

dy0

. (6.2)

Using equations (2.10) and (1.14), we obtain

SA1 =

Sxydux

dySxx

dux

dy+Sxz

duz

dySxy

duz

dy

Syydux

dySyx

dux

dy+Syz

duz

dySyy

duz

dy

Szydux

dySzx

dux

dy+Szz

duz

dySzy

duz

dy

, (6.3)

115

then

tr(SA1) = 2(

Syxdux

dy+Syz

duz

dy

)= P7, (6.4)

and

tr(SA1)I =

P7 0 0

0 P7 0

0 0 P7

, (6.5)

using velocity profile (2.1) and equations (2.8 - 2.10) in equation (1.27) we get

∇A1 =−

2(

dux

dy

)2

0 2dux

dyduz

dy

0 0 0

2dux

dyduz

dy0 2

(duz

dy

)2

, (6.6)

as steady flow assumption gives∂A1

∂t= 0, and velocity profile (2.1) suggest that

(V ·∇)A1 =(

uxddx

+0ddy

+uzddz

)A1 = 0.

Multiplication of equation (2.21) with a unit tensor gives

tr(A21)I =

P3 0 0

0 P3 0

0 0 P3

. (6.7)

Using equations (1.14), (2.10), (2.12), (3.4), (3.5) and (6.1 - 6.7) in the constitutive

equation (1.30) for Oldroyd 8-constant fluid, then simplifying we obtained

Sxx = η0(ν2−λ2−µ2)(

dux

dy

)2

+η0ν2

(duz

dy

)2

− (ν1−λ1−µ1)Sxydux

dy−ν1Syz

duz

dy, (6.8)

116

Sxy = Syx =

η0

[1+α

(dux

dy

)2

+(

duz

dy

)2]

dux

dy

1+β

(dux

dy

)2

+(

duz

dy

)2 , (6.9)

Sxz = Szx =12(λ1 +µ1)

Syz

dux

dy+Sxy

duz

dy

−η0(λ2 +µ2)

dux

dyduz

dy, (6.10)

Syy = η0(ν2 +λ2−µ2)

(dux

dy

)2

+(

duz

dy

)2

− (ν1 +λ1−µ1)

Sxydux

dy+Syz

duz

dy

, (6.11)

Syz = Szy =

η0

[1+α

(dux

dy

)2

+(

duz

dy

)2]

duz

dy

1+β

(dux

dy

)2

+(

duz

dy

)2 , (6.12)

Szz = η0(ν2−λ2−µ2)(

duz

dy

)2

+η0ν2

(dux

dy

)2

− (ν1−λ1−µ1)Syzduz

dy−ν1Sxy

dux

dy, (6.13)

α = λ1λ2 +µ1(ν2−µ2)+µ0(µ2− 32

ν2),

β = λ21 +µ1(ν1−µ1)+µ0(µ1− 3

2ν1),

(6.14)

where α is dilatant constant and β is pseudoplastic constant. Using equations (6.9) and

(6.12) in equations (2.6) and (2.7) we obtained momentum equations of the form

∂P∂x

=ddy

η0

[1+α

(dux

dy

)2

+(

duz

dy

)2]

dux

dy

1+β

(dux

dy

)2

+(

duz

dy

)2

, (6.15)

∂P∂z

=ddy

η0

[1+α

(dux

dy

)2

+(

duz

dy

)2]

duz

dy

1+β

(dux

dy

)2

+(

duz

dy

)2

. (6.16)

117

Eliminating hat from P and introducing the dimensionless parameters (2.35) in equations

(2.2), (6.15) and (6.16), we obtained

∂P∗

∂x∗=

ddy∗

[1+α∗

(du∗

dy∗

)2

+(

dw∗

dy∗

)2]

du∗

dy∗

1+β∗(

du∗

dy∗

)2

+(

dw∗

dy∗

)2

, (6.17)

∂P∗

∂z∗=

ddy∗

[1+α∗

(du∗

dy∗

)2

+(

dw∗

dy∗

)2]

dw∗

dy∗

1+β∗(

du∗

dy∗

)2

+(

dw∗

dy∗

)2

, (6.18)

andu∗ = 0, w∗ = 0, at y∗ = 0,

u∗ =UW

, w∗ = 1, at y∗ = 1,(6.19)

where α∗ =αW 2

h2 and β∗ =βW 2

h2 dimensionless material constants. Dropping ” ∗ “ from

equations (6.17 - 6.19) and using equation (2.39) in equations (6.17 - 6.19), we get,

ddy

dFdy

+α(

dFdy

)2 dFdy

1+βdFdy

dFdy

= G, (6.20)

the associated boundary conditions will become

F = 0 at y = 0,

F = V0 at y = 1.(6.21)

Integrating equation (6.20) with respect to y which gives

dFdy

= (Gy+C1)(

1+βdFdy

dFdy

)−α

(dFdy

)2 dFdy

, (6.22)

118

where C1 is arbitrary constant of integration and can be determined with the help of

boundary conditions. Equation (6.22) is second order highly nonlinear inhomogeneous

ordinary differential equation with inhomogeneous boundary conditions. To get the

approximate solution ADM is applied.


The operator form of the ADM (discussed in sec: 1.8.3) suggests that equation (6.22) can

be written as

Ly(F) = (Gy+C1)(

1+βdFdy

dFdy

)−α

(dFdy

)2 dFdy

, (6.23)

where Ly =ddy

is invertible differential operator.

On applying L−1y (=

∫(∗)dy) to both sides of equation (6.23), we get

F = C2 +L−1y (Gy+C1)+βL−1

y

(Gy+C1)

dFdy

dFdy

−αL−1

y

(dFdy

)2 dFdy

, (6.24)

where C2 is also a constant of integration can be determined from boundary conditions. As

suggested by ADM F , F and C1 can be written in component form as,

F =∞

∑n=0

Fn, F =∞

∑n=0

Fn, C1 =∞

∑n=0

C1,n. (6.25)

Thus (6.24) takes the form

∞

∑n=0

Fn = C2 +L−1y

(Gy+

∞

∑n=0

C1,n

)

+ βL−1y

(Gy+

∞

∑n=0

C1,n

)ddy

(∞

∑n=0

Fn

)ddy

(∞

∑n=0

Fn

)

− αL−1y

(ddy

(∞

∑n=0

Fn

))2ddy

(∞

∑n=0

Fn

) . (6.26)

119

Nonlinear terms can be written in the form of Adomian polynomials say, An and Bn as

∞

∑n=0

An =

(Gy+

∞

∑n=0

C1,n

)ddy

(∞

∑n=0

Fn

)ddy

(∞

∑n=0

Fn

), (6.27)

∞

∑n=0

Bn =

(ddy

(∞

∑n=0

Fn

))2ddy

(∞

∑n=0

Fn

), (6.28)

thus equation (6.26), reduces to

∞

∑n=0

Fn = C2 +L−1y

(Gy+

∞

∑n=0

C1,n

)

+ βL−1y

(∞

∑n=0

An

)−αL−1

y

(∞

∑n=0

Bn

). (6.29)

The boundary conditions will take the form

∞∑

n=0Fn = 0 at y = 0,

∞∑

n=0Fn = V0 at y = 1.

(6.30)

Here we assumed that (Gy+C1,0) is forcing function and(

∞∑

n=1C1,n

)is remainder of the

linear part,[40], the recursive relation then becomes

F0 = C2 +L−1y (Gy+C1,0) , (6.31)

F0 = 0 at y = 0,

F0 = V0 at y = 1,(6.32)

Fn+1 = L−1y (C1,n+1)+βL−1

y (An)−αL−1y (Bn) , n≥ 0 (6.33)

∞∑

n=1Fn = 0 at y = 0,

∞∑

n=1Fn = 0 at y = 1.

(6.34)

120

Using the relation

F =∞

∑n=0

Fn, (6.35)

we can obtained the solutions for the velocity components in x and z−directions,

respectively.


The relations (6.31) and (6.32), give the zeroth component solution which is

F0 = V0y+12

G(y2− y

). (6.36)

With the help of (2.39) separating real and imaginary parts in equation (6.36) we get

u0 =UW

y+12

P,x(y2− y

), (6.37)

w0 = y+12

P,z(y2− y

), (6.38)

these equations describe the velocity profiles in x and z−directions, for Newtonian case.


The relations (6.33) and (6.34) give the first component problem as:

F1 = L−1y (C1,1)+βL−1

y (A0)−αL−1y (B0) , (6.39)

together with the boundary condition

F1 = 0 at y = 0,

F1 = 0 at y = 1.(6.40)

121

The Adomian polynomials for the first component problem are

A0 = (Gy+C1,0)dF0

dydF0

dy, (6.41)

B0 =(

dF0

dy

)2 dF0

dy. (6.42)

Calculating the Adomian polynomials (6.41) and (6.42), then equation (6.39) along with

boundary conditions gives

F1 = (β−α)(H0 + ιL0)

(y2− y

)+(H1 + ιL1)

(y3− y

)+(H2 + ιL2)

(y4− y

), (6.43)

Separating real and imaginary parts, we get

u1 = (β−α)

H0(y2− y

)+H1

(y3− y

)+H2

(y4− y

), (6.44)

w1 = (β−α)

L0(y2− y

)+L1

(y3− y

)+L2

(y4− y

), (6.45)

where constant coefficients H0, H1, H2, L0, L1 and L2 are given in Appendix IV.


The second component of recursive relations (6.33) along with (6.34) gives

F2 = L−1y (C1,2)+βL−1

y (A1)−αL−1y (B1) , (6.46)

F2 = 0 at y = 0,

F2 = 0 at y = 1,(6.47)

where A1 and B1 are given as

A1 = (Gy+C1,0)(

dF0

dydF1

dy+

dF1

dydF0

dy

)+C1,1

dF0

dydF0

dy, (6.48)

B1 =(

dF0

dy

)2 dF1

dy+2

dF0

dydF1

dydF0

dy. (6.49)

122

Therefore equation (6.46) with the boundary conditions (6.47) has the solution

F2 = (β−α)[

β(H3 + ιL3)−α(H4 + ιL4)(

y2− y)+

β(H5 + ιL5)

− α(H6 + ιL6)(

y3− y)+2(2β−3α)(H7 + ιL7)

(y4− y

)

+ (2β−3α)(H8 + ιL8)(

y5− y)

+(2β−3α)(H9 + ιL9)(

y6− y)]

. (6.50)

Equating real and imaginary parts, we get

u2 = (β−α)(βH3−αH4)

(y2− y

)+(βH5−αH6)

(y3− y

)

+ 2(2β−3α)H7(y4− y

)+(2β−3α)H8

(y5− y

)

+ (2β−3α)H9

(y6− y

), (6.51)

w2 = (β−α)(βL3−αL4)

(y2− y

)+(βL5−αL6)

(y3− y

)

+ 2 (2β−3α)L7(y4− y

)+(2β−3α)L8

(y5− y

)

+ (2β−3α)L9

(y6− y

), (6.52)

where Hi, L j, i = 3, · · · ,9, j = 3, · · · ,9 are constant coefficients given in Appendix IV.



Velocity profile in the transverse plane can be calculated by combining equations (6.37), (6.44)

and (6.51) as

u =UW

y+

P,x

2+(β−α)(H0 +(βH3−αH4))

(y2− y

)

+ (β−α)(H1 +(βH5−αH6))

(y3− y

)+(H2 +2(2β−3α)H7)

(y4− y

)

+ (2β−3α)H8

(y5− y

)+(2β−3α)H9

(y6− y

). (6.53)

123


Combining equations (6.38), (6.45) and (6.52), velocity profile in the down channel

direction can be obtained as

w = y+

P,z

2+(β−α)(L0 +(βL3−αL4))

(y2− y

)

+ (β−α)(L1 +(βL5−αL6))

(y3− y

)+(L2 +2(2β−3α)L7)

(y4− y

)

+ (2β−3α)L8

(y5− y

)+(2β−3α)L9

(y6− y

). (6.54)


The velocity in the direction of the axis of the screw at any depth in the channel can be

computed from equations (6.53) and (6.54) as


=[

y+

P,z

2+(β−α)(L0 +(βL3−αL4))

(y2− y

)

+ (β−α)(L1 +(βL5−αL6))

(y3− y

)+(L2 +2(2β−3α)L7)

(y4− y

)

+ (2β−3α)L8

(y5− y

)+(2β−3α)L9

(y6− y

)]sinφ

+[

UW

y+

P,x

2+(β−α)(H0 +(βH3−αH4))

(y2− y

)

+ (β−α)(H1 +(βH5−αH6))

(y3− y

)+(H2 +2(2β−3α)H7)

(y4− y

)

+ (2β−3α)H8

(y5− y

)+(2β−3α)H9

(y6− y

)]cosφ. (6.56)

6.2.5 Stresses

Differentiating equations (6.53) and (6.54) and substituting in equations (6.9), (6.10) and

(6.12) we get the shear stresses of the form

S∗xy = S∗yx =1+αΠ3

1+βΠ3

[UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

124

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)], (6.57)

S∗yz = S∗zy =1+αΠ3

1+βΠ3

[1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1

)

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

)], (6.58)

S∗xz = S∗zx =(λ1−µ1)W

2h

[S∗yz

UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)

+ S∗xy

1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1)

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

)]

− (λ2−µ2)Wh

UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)

1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1)

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

), (6.59)

where

Π3 =[

UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)]2

+[

1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1)

125

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

)]2. (6.60)


S∗wxy= S∗wxy

=1+αΠ4

1+βΠ4

[UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

+ (β−α)

2(H1 +(βH5−αH6))+3(H2 +2(2β−3α)H7)

+ 4(2β−3α)H8 +5(2β−3α)H9]

, (6.61)

S∗wyz= S∗wzy

=1+αΠ4

1+βΠ4

[1+

P,z

2+(β−α)(L0 +(βL3−αL4))

+ (β−α)2(L1 +(βL5−αL6))+3(L2 +2(2β−3α)L7)

+ 4(2β−3α)L8 +5(2β−3α)L9] , (6.62)

S∗xz = S∗zx =(λ1−µ1)W

2h

[S∗yz

UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

+ (β−α)

2(H1 +(βH5−αH6))+3(H2 +2(2β−3α)H7)+4(2β−3α)H8

+ 5(2β−3α)H9

+S∗xy

1+

P,z

2+(β−α)(L0 +(βL3−αL4))

+ 2(β−α)(L1 +(βL5−αL6))+3(L2 +2(2β−3α)L7)+4(2β−3α)L8

+ 5(2β−3α)L9]− (λ2−µ2)Wh

UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

+ (β−α)(H1 +2(βH5−αH6))+3(H2 +2(2β−3α)H7)+4(2β−3α)H8

+ 5(2β−3α)H9

1+

P,z

2+(β−α)(L0 +(βL3−αL4))

+ 2(β−α)(L1 +(βL5−αL6))+3(L2 +2(2β−3α)L7)

+ 4(2β−3α)L8 +5(2β−3α)L9 , (6.63)

where

Π4 =[

UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

+ (β−α)

2(H1 +(βH5−αH6))+3(H2 +2(2β−3α)H7)

+ 4(2β−3α)H8 +5(2β−3α)H9]2

+[

1+

P,z

2+(β−α)(L0 +(βL3−αL4))

126

+ (β−α)2(L1 +(βL5−αL6))+3(L2 +2(2β−3α)L7)

+ 4(2β−3α)L8 +5(2β−3α)L9]2 , (6.64)

and normal stresses (6.8), (6.11) and (6.13) are obtained as

S∗xx =(ν2−λ2−µ2)W

h

[UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)]2

+ν2W

h

[1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1)

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

)]2

− (ν1−λ1−µ1)Wh

S∗xy

[UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)]

− ν1Wh

S∗yz

[1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1)

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

)], (6.65)

S∗yy =(ν2 +λ2−µ2)W

hΠ3− (ν1 +λ1−µ1)W

h

[S∗xy

UW

+

P,x

2+ (β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)

+ S∗yz

1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1)

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

)], (6.66)

127

S∗zz =(ν2−λ2−µ2)W

h

[1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1)

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

)]2

+ν2W

h

[UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)]2

− (ν1−λ1−µ1)Wh

S∗yz

[1+

P,z

2+(β−α)(L0 +(βL3−αL4))

(2y−1)

+ (β−α)(L1 +(βL5−αL6))

(3y2−1

)+(L2 +2(2β−3α)L7)

(4y3−1

)

+ (2β−3α)L8(5y4−1

)+(2β−3α)L9

(6y5−1

)]

− ν1Wh

S∗xy

[UW

+

P,x

2+(β−α)(H0 +(βH3−αH4))

(2y−1)

+ (β−α)(H1 +(βH5−αH6))

(3y2−1

)+(H2 +2(2β−3α)H7)

(4y3−1

)

+ (2β−3α)H8(5y4−1

)+(2β−3α)H9

(6y5−1

)], (6.67)


h


Using equation (2.84) the shear force per unit width required to move the barrel in

x−direction is

F∗x =−S∗wxyδ1. (6.68)

Equation (2.86) gives the shear force per unit width required to move the barrel in

z−direction as

F∗z = −S∗wyzδ2, (6.69)



128


Volume flow rate in x-direction (2.90) for Oldroyd 8-constant fluid has the expression

Qx =U

2W− 1

6

P,x

2+(β−α)(H0 +(βH3−αH4))

− (β−α)

14

(H1 +(βH5−αH6))+3

10(H2 +2(2β−3α)H7)

+13(2β−3α)H8 +

514

(2β−3α)H9

. (6.71)

and in z-direction (2.93) becomes

Qz =12− 1

6

P,z

2+(β−α)(L0 +(βL3−αL4))

− (β−α)

14

(L1 +(βL5−αL6))+310

(L2 +2(2β−3α)L7)

+13(2β−3α)L8 +

514

(2β−3α)L9

. (6.72)

Therefore, the resultant volume flow rate forward in the screw channel (2.96) can be written

as

Q =N

sinφ

[12− 1

6

(P,z

2+(β−α)(L0 +(βL3−αL4))

)

− (β−α)(

14

(L1 +(βL5−αL6))+310

(L2 +2(2β−3α)L7)

+13(2β−3α)L8 +

514

(2β−3α)L9

)sinφ

+

U2W

− 16

(P,x

2+(β−α)(H0 +(βH3−αH4))

)

− (β−α)(

14

(H1 +(βH5−αH6))+3

10(H2 +2(2β−3α)H7)

+13(2β−3α)H8 +

514

(2β−3α)H9

)cosφ

], (6.73)

=N


129


Using equation (6.56) in (2.99) the average velocity in the direction of the axis of the screw

can be calculated as

s∗ = N[

12− 1

6

(P,z

2+(β−α)(L0 +(βL3−αL4))

)

− (β−α)(

14

(L1 +(βL5−αL6))+310

(L2 +2(2β−3α)L7)

+13(2β−3α)L8 +

514

(2β−3α)L9

)sinφ

+

U2W

− 16

(P,x

2+(β−α)(H0 +(βH3−αH4))

)

− (β−α)(

14

(H1 +(βH5−αH6))+3

10(H2 +2(2β−3α)H7)

+13(2β−3α)H8 +

514

(2β−3α)H9

)cosφ

]. (6.75)

When α = β velocity profile reduced to the profile of Newtonian fluid. The solutions for

dilatant fluids can be calculated by setting β = 0, similarly α = 0 gives the solutions for

pseudoplastic fluids. Setting α = µ(Wi)2(1−a2) and β = (Wi)2(1−a2), we can obtain the

solution for Johnson-Segalman fluid, where µ =µ

(µ+η)is dimensionless parameter and

Wi =mW

hdenotes the Weissenberg number and µ and η are viscosities, and m and a are

relaxation time and slip parameter, respectively in the constitutive equation of Johnson-

Segalman fluid. Furthermore, using different values of 8−constants, we get different fluid

models as discussed in section 1.7.5.


In this chapter, we deal with the steady flow of an incompressible, isothermal and

homogeneous Oldroyd 8-constant fluid in HSR (see fig.2.1). Using ADM solutions are

obtained for velocity profiles in x and z− directions and also in the direction of the axis

of the screw. Expressions for the shear stresses, shear stresses at barrel surface, forces

130

exerted on fluid, volume flow rates and average velocity are also calculated. The effect

of parameters α, β, flight angle φ, P,x and P,z, on the velocities given by equations (6.53

- 6.54) and (6.56) are discussed. In figures (6.1) - (6.6) we have plotted velocities for

different values of non-Newtonian parameters α and β. All three velocities u, w and s are

in the same pattern as discussed in chapter 2. It is also noticed that both non-Newtonian

parameters depict the shear thinning effects in the fluid. However, increase in α is more

effective in shear thinning of fluid. It is observed that for α = β velocities reduced to the

Newtonian case.

Figures (6.7 - 6.12) are plotted for different values of P,x and P,z same pattern of velocity

profiles observed as discussed in chapter 2. The figure (6.13) is plotted for the velocity s

for different values of φ, same result is obtained as discussed in chapter 2. The figures (6.14

- 6.17) are plotted to note the variation of shear stresses Sxy and Syz with respect to α and

β, same results are obtained as given in chapter 2.

Figure 6.1: Velocity profile u(y) for different values of α, keeping β = 0, 0.4, P,x =−2.0, P,z =−2.0 and φ = 45.

131

Figure 6.2: Velocity profile u(y) for different values of β, keeping α = 0, 0.4, P,x =−2.0, P,z =−2.0 and φ = 45.

Figure 6.3: Velocity profile w(y) for different values of α, keeping β = 0, 0.4, P,x =−2.0, P,z =−2.0 and φ = 45.

132

Figure 6.4: Velocity profile w(y) for different values of β, keeping α = 0, 0.4, P,x =−2.0, P,z =−2.0 and φ = 45.

Figure 6.5: Velocity profile s(y) for different values of α, keeping β = 0, 0.4, P,x =−2.0, P,z =−2.0 and φ = 45.

133

Figure 6.6: Velocity profile s(y) for different values of β, keeping α = 0, 0.4, P,x =−2.0, P,z =−2.0 and φ = 45.

Figure 6.7: Velocity profile u(y) for different values of P,x, keeping α = 0.4, β = 0.2, P,z =−2.0 and φ = 45.

134

Figure 6.8: Velocity profile u(y) for different values of P,z, keeping α = 0.4, β = 0.2, P,x =−2.0 and φ = 45.

Figure 6.9: Velocity profile w(y) for different values of P,x, keeping α = 0.4, β = 0.2, P,z =−2.0 and φ = 45.

135

Figure 6.10: Velocity profile w(y) for different values of P,z, keeping α = 0.4, β =0.2, P,x =−2.0 and φ = 45.

Figure 6.11: Velocity profile s(y) for different values of P,x, keeping α = 0.4, β =0.2, P,z =−2.0 and φ = 45.

136

Figure 6.12: Velocity profile s(y) for different values of P,z, keeping α = 0.4, β =0.2, P,x =−2.0 and φ = 45.

Figure 6.13: Velocity profile s(y) for different values of φ, keeping α = 0.4, β = 0.2, P,x =−2.0 and P,z =−2.0.

137

Figure 6.14: Variation of shear stress Sxy for different values of α, keeping β = 0.2, P,x =−2.0, P,z =−2.0 and φ = 45.

Figure 6.15: Variation of shear stress Sxy for different values of β, keeping α = 0.4, P,x =−2.0, P,z =−2.0 and φ = 45.

138

Figure 6.16: Variation of shear stress Syz for different values of α, keeping β = 0.2, P,x =−2.0, P,z =−2.0 and φ = 45.

Figure 6.17: Variation of shear stress Syz for different values of β, keeping α = 0.4, P,x =−2.0, P,z =−2.0 and φ = 45.

139

6.4 Conclusion

The steady flow of an isothermal, homogeneous and incompressible Oldroyd 8−constant

fluid is investigated in HSR. The geometry of the problem under consideration gives second


equation using a transformation. ADM is used to obtain analytical expressions for the

velocity profiles. Shear stresses, shear stresses at barrel surface, shear forces exerted on

the fluid, volume flow rates and average velocity of the fluid are also calculated. It is

noticed that the zeroth component solution matches with solution of the Newtonian fluid in

HSR and also found that the net velocity of the fluid is due to the pressure gradient as the

expression for the net velocity is free from the drag term. Graphical representation shows

that the velocity profiles strongly depend on non-Newtonian parameters, pressure gradients

and flight angle. The behavior of the shear stresses is also discussed with the help of graphs

for different values of non-Newtonian parameters. The graphical representation show that

pattern of flow profiles is similar to the third-grade fluid.

140

Chapter 7

Homotopy Perturbation Method for Flow of a

Third-Grade Fluid Through Helical Screw Rheometer

with Zero Flight Angle

141

In this chapter steady incompressible flow of third-grade fluid in HSR with zero flight angle

(a vertical concentric annulus) is discussed, using cylindrical coordinate system (r,θ,z)

which is more suitable choice for the flow analysis in HSR. The developed second order

nonlinear coupled differential equations are solved, to get azimuthal and axial velocity

components, using HPM. Expressions for shear and normal stresses, the shear stresses

exerted by the fluid on the screw, volume flow rate and average velocity are also calculated.

The effect of non-Newtonian parameter on the velocity profiles are presented through

graphs.


Steady, laminar flow of an incompressible, isothermal third-grade fluid through HSR with

zero flight angles is considered.

Ω

Flight angleφ

port

port

z

r

r

r

1

2

Channel depth

h

= 0

Figure 7.1: Vertical concentric annulus.

The barrel of radius r2 is assumed to be stationary and the screw of radius r1 rotates with

an angular velocity Ω. We also assume that pressure gradient in z−direction is negligibly

142

small and the effect of body force, say, gravity is only in z−direction. Let V = (ur,uθ,uz)

be the velocity field, with ur, uθ and uz velocity components in r, θ and z−directions,

respectively.

Geometry of the problem suggests the boundary conditions

uθ = Ωr1 uz = 0 at r = r1,

uθ = 0 uz = 0 at r = r2.(7.1)

For axisymmetric and fully developed flow the velocity profile and extra stress tensor can

be written as:

V = (0,uθ(r),uz(r)), S = S(r), (7.2)

where uθ and uz are azimuthal and axial velocity components, respectively. Using velocity

profile (7.2), equation (1.8) is identically satisfied and equations (1.15 - 1.17) result in

ρ(−u2

θr

)= −∂P

∂r+

1r

∂(rSrr)∂r

, (7.3)

0 =1r2

∂(r2Srθ)∂r

+(Sθr−Srθ)

r, (7.4)

0 =1r

∂(rSrz)∂r

−ρg. (7.5)

To calculate the components of extra stress tensor we proceed as follows: using velocity

profile (7.2) in the velocity gradient (1.23) we get

∇V =

0 −uθr

0duθdr

0 0duz

dr0 0

, (7.6)

143

then

(∇V)T =

0duθdr

duz

dr−uθ

r0 0

0 0 0

. (7.7)

Therefore,

A1 =

0 P8duz

drP8 0 0duz

dr0 0

, (7.8)

where P8 =duθdr

− uθr

. From the above tensor, we can calculate

A21 = A1AT

1 =

P 28 +

(duz

dr

)2

0 0

0 P 28 P8

duz

dr

0 P8duz

dr

(duz

dr

)2

, (7.9)

tr(A21) = 2

[P 2

8 +(

duz

dr

)2]

= P9. (7.10)

Equation (1.21) for n = 1 gives second Rivlin-Ericksen tensor

A2 =∂A1

∂t+(V ·∇)A1 +[A1(∇V)+(∇V)T A1]. (7.11)


∂t= 0, and

(V ·∇)A1 =

−2P8uθr

0 0

0 2P8uθr

uθr

duz

dr0

uθr

duz

dr0

, (7.12)

144

A1(∇V) =

P8duθdr

+(

duz

dr

)2

0 0

0 −P8uθr

0

0 −uθr

duz

dr0

, (7.13)

(∇V)T A1 = (A1∇V)T =

P8duθdr

+(

duz

dr

)2

0 0

0 −P8uθr

−uθr

duz

dr0 0 0

. (7.14)

Therefore,

A2 =

P9 0 0

0 0 0

0 0 0

, (7.15)

we get tr(A21) = tr(A2). With the help of equations (7.8) and (7.15), we obtained

A1A2 =

0 0 0

P8P9 0 0

P9duz

dr0 0

, (7.16)

A2A1 =

0 P8P9 P9duz

dr0 0 0

0 0 0

, (7.17)

then

A1A2 +A2A1 =

0 P8P9 P9duz

drP8P9 0 0

P9duz

dr0 0

, (7.18)

145

and equations (7.8) and (7.10) implies

tr(A21)A1 =

0 P8P9 P9duz

drP8P9 0 0

P9duz

dr0 0

. (7.19)

For n = 2 we obtain A3 from equation(1.21) as

A3 =∂A2

∂t+(V ·∇)A2 +[A2(∇V)+(∇V)T A2]. (7.20)


∂t= 0, and

(V ·∇)A2 =

0 P9uθr

0

P9uθr

0 0

0 0 0

, (7.21)

A2(∇V) =

0 −P9uθr

0

0 0 0

0 0 0

, (7.22)

(∇V)T A2 = (A2∇V)T =

0 0 0

−P9uθr

0 0

0 0 0

, (7.23)

then

A3 =

0 0 0

0 0 0

0 0 0

. (7.24)

146

Using equations (7.8 - 7.24) in equation (1.19), the non-zero components of extra stress

tensor, S are:

Srr =(

α1 +α2

2

)P9, (7.25)

Srθ = = Sθr = µ+(β2 +β3)P9P8, (7.26)

Srz = = Szr = µ+(β2 +β3)P9 duz

dr, (7.27)

Sθθ = α2P 28 , (7.28)

Sθz = Szθ = α2P8duz

dr, (7.29)

Szz = = α2

(duz

dr

)2

. (7.30)

Therefore momentum equations (7.3 - 7.5) become

ρ(−u2

θr

)=

1r

ddr

[r

(2α1 +α2)

((duθdr

− uθr

)2

+(

duz

dr

)2)]

− α2

r

(duθdr

− uθr

)2

− ∂P∂r

, (7.31)

0 =1r2

ddr

[r2

µ+2(β2 +β3)

((duθdr

− uθr

)2

+(

duz

dr

)2)(

duθdr

− uθr

)], (7.32)

0 =1r

ddr

[r

µ+2(β2 +β3)

((duθdr

− uθr

)2

+(

duz

dr

)2)

duz

dr

]−ρg. (7.33)

Introducing dimensionless parameters,

r∗ =rr1

, v∗ =uθ

Ωr1, w∗ =

uz

Ωr1and P∗ =

Pρ(Ωr1)2 , (7.34)

in equations (7.31 - 7.33) we get

147

− v∗2

r∗=

1ρr2

1(2α1 +α2)

1r∗

ddr∗

[r∗

(dv∗

dr∗− v∗

r∗

)2

+(

dw∗

dr∗

)2]

− α2

ρr21

1r∗

(dv∗

dr∗− v∗

r∗

)2

− ∂P∗

∂r∗, (7.35)

0 =d

dr∗

[r∗2

(dv∗

dr∗− v∗

r∗

)+ β∗r∗2

(dv∗

dr∗− v∗

r∗

)2

+(

dw∗

dr∗

)2(

dv∗

dr∗− v∗

r∗

)], (7.36)

0 =d

dr∗

[r∗

dw∗

dr∗+ β∗r∗

(dv∗

dr∗− v∗

r∗

)2

+(

dw∗

dr∗

)2

dw∗

dr∗

]− r∗λ∗, (7.37)

and boundary conditions (7.1) becomes

v∗ = 1 w∗ = 0 at r∗ = 1,

v∗ = 0 w∗ = 0 at r∗ = δ,(7.38)

where β∗ =2(β2 +β3)Ω2

µand λ∗ =

ρgr1

µΩare dimensionless parameters and δ =

r2

r1> 1.

Dropping “∗ ”, and simplifying equations (7.35 - 7.38) we get

− v2

r=

1ρr2

1(2α1 +α2)

1r

ddr

[r

(dvdr− v

r

)2

+(

dwdr

)2]

− α2

ρr21

1r

(dvdr− v

r

)2

− ∂P∂r

, (7.39)

0 =ddr

[r2

(dvdr− v

r

)+ β

r5

(ddr

(vr

))3

+ r3(

dwdr

)2 (ddr

(vr

))], (7.40)

0 =ddr

[r

dwdr

+ β

r3

(ddr

(vr

))2 dwdr

+ r(

dwdr

)3]

−λr, (7.41)

v = 1 w = 0 at r = 1,

v = 0 w = 0 at r = δ.(7.42)

148

Since our concentration is on velocity profiles in θ and z−dirctions which can be obtained

from equations (7.40) and (7.41) together with boundary conditions (7.42), which are

second order nonlinear coupled ordinary differential equations, the exact solutions seem

to be difficult, so we used HPM to solve these equations.


We construct a homotopy as f (r, p) : Ω× [0,1] −→ ℜ and g(r, p) : Ω× [0,1] −→ ℜ

satisfying: (see section:1.8.2),

G1( f )−G1(v0)+ pG1(v0)

+p

[β

ddr

r5

(ddr

(fr

))3

+ r3(

dgdr

)2 (ddr

(fr

))]= 0, (7.43)

G2(g)−G2(w0)+ pG2(w0)

+p

[β

ddr

r3

(ddr

(fr

))2 dgdr

+ r(

dgdr

)3−λr

]= 0, (7.44)

where G1 = r2 d2

dr2 + rddr− 1 and G2 = r

d2

dr2 +ddr

are differential operators and let us

assume that

v0 = X1r +X2

r, (7.45)

w0 = Z1(r2−1)+Z2 lnr, (7.46)

are initial guess approximations. On substituting series (1.44) equations (7.43) and (7.44)

become

G1

(∞

∑i=0

pi fi

)−G1(v0)+ pG1(v0)+ p

β

ddr

r5

(ddr

(1r

∞

∑i=0

pi fi

))3

+ r3

(ddr

∞

∑i=0

pigi

)2 (ddr

(1r

∞

∑i=0

pi fi

))

= 0, (7.47)

149

G2

(∞

∑i=0

pigi

)−G2(w0)+ pG2(w0)+ p

β

ddr

r3

(ddr

(1r

∞

∑i=0

pi fi

))2

ddr

(∞

∑i=0

pigi

)+ r

(ddr

(∞

∑i=0

pigi

))3−λr

= 0, (7.48)

and the boundary conditions (7.42) take the form

∞

∑i=0

pi fi = 1,∞

∑i=0

pigi = 0, at r = 1, (7.49)

∞

∑i=0

pi fi = 0,∞

∑i=0

pigi = 0, at r = δ. (7.50)

Equating the equal powers of p from equations (7.47 - 7.50) we obtain the system of

equations as:


p0 : G1( f0)−G1(v0) = 0, (7.51)

G2(g0)−G2(w0) = 0, (7.52)

f0(1) = 1 g0(1) = 0, f0(δ) = g0(δ) = 0, (7.53)

give

f0 = X1r +X2

r, (7.54)

g0 = Z1(r2−1)+Z2 lnr, (7.55)

which match to the solutions of Newtonian case.

150


p1 : G1( f1)+G1(v0)

+βddr

r5

(ddr

(f0

r

))3

+ r3(

dg0

dr

)2 (ddr

(f0

r

))= 0, (7.56)

G2(g1)+G2(w0)

+βddr

r3

(ddr

(f0

r

))2 dg0

dr+ r

(dg0

dr

)3−λr = 0, (7.57)

f1(1) = g1(1) = 0, f1(δ) = g1(δ) = 0, (7.58)

result in

f1 = β(

X3

r5 +X4

r3 +X5

r+X6r +X7r lnr

), (7.59)

g1 = β(

Z3

r4 +Z4

r2 +Z5 +Z6 lnr +Z7r2 +Z8r4)

, (7.60)

which are first component solutions of velocity profiles in θ and z− directions.


p2 : G1( f2)+ βddr

3r5

(ddr

(f0

r

))2 (ddr

(f1

r

))+2r3

(dg0

dr

)

(dg1

dr

)(ddr

(f0

r

))+ r3

(dg0

dr

)2 (ddr

(f1

r

))= 0, (7.61)

151

G2(g2)+ βddr

r3

(ddr

(f0

r

))2 dg1

dr+2r3

(ddr

(f0

r

))

(ddr

(f1

r

))dg0

dr+3r

(dg0

dr

)2 (dg1

dr

)= 0, (7.62)

f2(1) = g2(1) = 0, f2(δ) = g2(δ) = 0, (7.63)

give the second component solutions of azimuthal and axial velocity profiles as

f2 = β2(

X8

r9 +X9

r7 +X10

r5 +X11

r3 +X12

r+X13r +X14r lnr

+ X15r3) , (7.64)

g2 = β2(

Z9

r9 +Z10

r8 +Z11

r7 +Z12

r6 +Z13

r5 +Z14

r4 +Z15

r3 +Z16

r2

+Z17

r+Z18 +Z19 lnr +Z20 lnr2 +Z21r +Z22r2 +Z23r3

+ Z24r4 +Z25r6)

, (7.65)

where Xi, Z j, i = 1, · · · ,15, j = 1, · · · ,25 are constant coefficients given in Appendix V.

Solutions obtained by HPM are

v =∞

∑i=0

pi fi = f0 + p f1 + p2 f2 + · · · , (7.66)

and

w =∞

∑i=0

pigi = g0 + pg1 + p2g2 + · · · (7.67)

As p→ 1, approximate solutions become

v = limp−→1

f = f0 + f1 + f2 + · · · , (7.68)

152

equation (7.68) gives the velocity profile in θ−direction as

v = X1r +X2

r+ β

(X3

r5 +X4

r3 +X5

r+X6r +X7r lnr

)+ β2

(X8

r9 +X9

r7 +X10

r5

+X11

r3 +X12

r+X13r +X14r lnr

+ X15r3) , (7.69)

and

w = limp−→1

g = g0 +g1 +g2 + · · · , (7.70)

which gives

w = Z1(r2−1)+Z2 lnr + β(

Z3

r4 +Z4

r2 +Z5 +Z6 lnr +Z7r2 +Z8r4)

+ β2(

Z9

r9 +Z10

r8 +Z11

r7 +Z12

r6 +Z13

r5 +Z14

r4 +Z15

r3 +Z16

r2

+Z17

r+Z18 +Z19 lnr +Z20 lnr2 +Z21r +Z22r2 +Z23r3

+ Z24r4 +Z25r6)

, (7.71)

is the velocity profile in z−direction.

7.2.4 Stresses

Using equations (7.66) and (7.67) in equations (7.26), (7.27) and (7.29) we obtain the

solution for shear stresses

S∗rθ = S∗θr =(

1+ βΠ5

)[X1−X2

r2 + β(−5X3

r6 − 3X4

r4 −X5

r2 +X6 +X7 +X7 lnr)

+ β2(−9X8

r10 − 7X9

r8 − 5X10

r6 − 3X11

r4 −X12

r2 +X13 +X14 +X14 lnr +3r2X15

)

153

− 1r

X1r +

X2

r+ β

(X3

r5 +X4

r3 +X5

r+X6r +X7r lnr

)+ β2

(X8

r9 +X9

r7

+X10

r5 +X11

r3 +X12

r+X13r +X14r lnr +X15r3

)], (7.72)

S∗rz = S∗zr =(

1+ βΠ5

)[2rZ1 +

Z2

r+ β

(−4Z3

r5 − 2Z4

r3 +Z6

r+2rZ7 +4r3Z8

)

+ β2(−9Z9

r10 −8Z10

r9 − 7Z11

r8 − 6Z12

r7 − 5Z13

r6 − 4Z14

r5 − 3Z15

r4

− 2Z16

r3 −Z17

r2 +Z19

r+

2Z20 lnrr

+Z21 +2rZ22 +3r2Z23

+ 4r3Z24 +6r5Z25

)], (7.73)

S∗θz = S∗zθ =α2Ω

µ

[X1−X2

r2 + β(−5X3

r6 − 3X4

r4 −X5

r2 +X6 +X7 +X7 lnr)

+ β2(−9X8

r10 − 7X9

r8 − 5X10

r6 − 3X11

r4 −X12

r2 +X13 +X14 +X14 lnr +3r2X15

)

− 1r

X1r +

X2

r+ β

(X3

r5 +X4

r3 +X5

r+X6r +X7r lnr

)+ β2

(X8

r9 +X9

r7

+X10

r5 +X11

r3 +X12


)]

[2rZ1 +

Z2

r+ β

(−4Z3

r5 − 2Z4

r3 +Z6

r+2rZ7 +4r3Z8

)

+ β2(−9Z9

r10 −8Z10

r9 − 7Z11

r8 − 6Z12

r7 − 5Z13

r6 − 4Z14

r5 − 3Z15

r4

− 2Z16

r3 −Z17

r2 +Z19

r+

2Z20 lnrr

+Z21 +2rZ22 +3r2Z23

+ 4r3Z24 +6r5Z25

)], (7.74)

where

Π5 =[(

X1−X2

r2 + β(−5X3

r6 − 3X4

r4 −X5

r2 +X6 +X7 +X7 lnr)

+ β2(−9X8

r10 − 7X9

r8 − 5X10

r6 − 3X11

r4 −X12

r2 +X13 +X14 +X14 lnr +3r2X15

)

− 1r

X1r +

X2

r+ β

(X3

r5 +X4

r3 +X5

r+X6r +X7r lnr

)+ β2

(X8

r9 +X9

r7

+X10

r5 +X11

r3 +X12


))2

154

+(

2rZ1 +Z2

r+ β

(−4Z3

r5 − 2Z4

r3 +Z6

r+2rZ7 +4r3Z8

)+ β2

(−9Z9

r10

− 8Z10

r9 − 7Z11

r8 − 6Z12

r7 − 5Z13

r6 − 4Z14

r5 − 3Z15

r4 − 2Z16

r3 −Z17

r2 +Z19

r

+2Z20 lnr

r+Z21 +2rZ22 +3r2Z23 +4r3Z24 +6r5Z25

))2]

. (7.75)

The shear stresses exerted by the fluid on the screw can be calculated by putting r = 1 in

equations (7.72 - 7.74).

S∗rθ|r=1 = S∗wrθ,

S∗rz|r=1 = S∗wrz,

S∗θz|r=1 = S∗wθz.

Using equations (7.66) and (7.67) in equations (7.25), (7.28) and (7.30) give the expressions

for Normal stresses

S∗rr =(2α1 +α2)Ω

µΠ5, (7.76)

S∗θθ =α2Ω

µ

[X1−X2

r2 + β(−5X3

r6 − 3X4

r4 −X5

r2 +X6 +X7 +X7 lnr)

+ β2(−9X8

r10 − 7X9

r8 − 5X10

r6 − 3X11

r4 −X12

r2 +X13 +X14 +X14 lnr +3r2X15

)

− 1r

X1r +

X2

r+ β

(X3

r5 +X4

r3 +X5

r+X6r +X7r lnr

)+ β2

(X8

r9 +X9

r7

+X10

r5 +X11

r3 +X12


)]2

, (7.77)

S∗zz =α2Ω

µ

[2rZ1 +

Z2

r+ β

(−4Z3

r5 − 2Z4

r3 +Z6

r+2rZ7 +4r3Z8

)

+ β2(−9Z9

r10 −8Z10

r9 − 7Z11

r8 − 6Z12

r7 − 5Z13

r6 − 4Z14

r5 − 3Z15

r4

− 2Z16

r3 −Z17

r2 +Z19

r+

2Z20 lnrr

+Z21 +2rZ22 +3r2Z23

+ 4r3Z24 +6r5Z25

)]2, (7.78)

155

where S∗i j =Si j

µΩ, i, j = x,θ,z and i 6= j are the dimensionless stresses.


The volume flow rate in θ−direction is zero as flight angle φ = 0, so an expression for net

flow rate through the channel can be obtained by integrating the axial component of the

velocity only over the cross section of the channel

Qz = 2π∫ r=r2

r=r1

uzrdr, (7.79)

equation (7.79) imply that

Q∗z = 2π

∫ δ

1wrdr, (7.80)

= 2π[(

1−2δ2 +δ4) Z1

4+

(1−δ2 +2δ2 lnδ

) Z2

4+β

(1− 1

δ2

)Z3

2+Z4 lnδ

− (1−δ2) Z5

2+

(1−δ2 +2δ2 lnδ

) Z6

4− (

1−δ4) Z7

4−

(1−δ6

) Z8

6

+ β2(

1− 1δ7

)Z9

7+

(1− 1

δ6

)Z10

6+

(1− 1

δ5

)Z11

5+

(1− 1

δ4

)Z12

4

+(

1− 1δ3

)Z13

3+

(1− 1

δ2

)Z14

2+

(1− 1

δ

)Z15 +Z16 lnδ− (1−δ)Z17

− (1−δ2) Z18

2+

(1−δ2 +2δ2 lnδ

) Z19

4− (

1−δ2 +2δ2 lnδ−2δ2 lnδ2) Z20

4

− (1−δ3) Z21

3− (

1−δ4) Z22

4−

(1−δ5

) Z23

5

−(

1−δ6) Z24

6− (

1−δ8) Z25

8

], (7.81)

where Q∗z =

Qz

Ωr31

is dimensionless volume flow rate.

156


Using equation (7.81) average velocity can be calculated as

w∗av =Q∗

z

π(δ2−1), (7.82)

where w∗av =uzav

Ωr1, is dimensionless average velocity.

7.3 Results and discussion

In this work, steady flow of an incompressible third-grade fluid is considered through

HSR with zero flight angle (see fig.7.1). We choose the cylindrical coordinate system

(r,θ,z) which is more suitable choice for the flow analysis in HSR. We assume that barrel

is fixed and screw is rotating with an angular velocity Ω. We obtained coupled second

order nonlinear inhomogeneous ordinary differential equations. Using HPM, expressions

for azimuthal and axial velocity components are derived. Expressions for shear stresses,

normal stresses, the shear stresses exerted by the fluid on the screw, volume flow rate and

average velocity are also calculated. The behavior of the velocity profiles observed for

different values of the non-Newtonian parameter β. Figures (7.2) and (7.3) are plotted for

azimuthal and axial components of velocities v, w respectively to show the effect of β < 1

on the velocity profiles, for the fixed ratio δ = 2 and non-dimensional number which has

gravitational effect λ = 0.4. It is observed that in both cases the velocity profiles strongly

depend on the β. As β increases a progressive increase observed in the velocity profiles

due to shear thinning. Velocities appeared to be negative due to the effect of λ. However

the magnitude of the velocities is larger near the screw. Figures (7.4 - 7.5) are sketched

for β > 1 it is observed that the behavior of the velocities is same as for β < 1 but the

magnitude of the velocity profiles is larger for β > 1.

157

Figure 7.2: Velocity profile v(r) for different values of β, keeping δ = 2.0 and λ = 0.4.

Figure 7.3: Velocity profile w(r) for different values of β, keeping keeping δ = 2.0 andλ = 0.4.

158

Figure 7.4: Velocity profile v(r) for different values of β, keeping δ = 2.0 and λ = 0.4.

Figure 7.5: Velocity profile w(r) for different values of β, keeping keeping δ = 2.0 andλ = 0.4.

159

7.4 Conclusion

The steady, homogeneous flow of an isothermal and incompressible third grade fluid is

investigated in HSR with zero flight angle. The assumption of zero flight angle reduces the

geometry of HSR in concentric annulus. It is also assumed that the concentric annulus is

vertical and the body force is gravity. The modeling of the third-grade fluid in the geometry

under consideration gives second order nonlinear coupled differential equations. Using

HPM the analytical expressions are obtained for the flow properties. It is noticed that

the zeroth component solution matches with solution of the Newtonian fluid. Graphical

representation shows that the velocity profiles strongly depend on parameters β and λ. The

effects of shear thinning observed with the increasing values of β and flow in downward

direction is due to λ.

160

Chapter 8

Homotopy Perturbation Solution for Flow of a

Third-Grade Fluid in Helical Screw Rheometer

161

A theoretical study of steady incompressible flow of a third-grade fluid in HSR with non-

zero flight angle is considered. The model developed in cylindrical coordinates pertains to

second order non linear coupled differential equations that are solved by HPM. Expressions

for velocity components in θ and z−direction are obtained. The volume flow rates are

calculated for the azimuthal and axial components of velocity profiles by introducing the

effect of flights. Shear and normal stresses, the shear stresses exerted by the fluid on the

screw and average velocity are also calculated. The velocity profiles are strongly dependent

on non-Newtonian parameter β and pressure gradients. Graphical representation shows that

increase in the value of these parameters increase velocity profiles.


Consider steady flow of an incompressible, homogeneous and isothermal third-grade fluid

through HSR with non-zero flight angle. The screwed channel is assumed to be bounded

by the barrel and screw root surfaces and by the two sides of a helical flight as shown in

Fig. 8.1. The geometry is approximated as a shallow infinite channel, by assuming the

width B of the channel large compared with the depth h i.e.,hB

<< 1. So that the side

effects can be ignored. No leakage of the fluid occurs across the flights. We choose the

cylindrical coordinate system (r,θ,z) which is more suitable choice for the flow analysis

in HSR. A congruent velocity distribution is assumed at parallel cross sections through the

channel. We also assumed that the flow is uniform, laminar and viscosity of the fluid is

constant. The barrel of radius r2 is assumed to be stationary and the screw of radius r1

rotates with angular velocity Ω. Let V = (ur,uθ,uz) be the velocity field, where ur, uθ and

uz are velocity components in r, θ and z−directions, respectively.

Geometry of the problem suggests the boundary conditions as

uθ = Ωr1, uz = 0, at r = r1,

uθ = 0, uz = 0, at r = r2.(8.1)

162

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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxx

xxxxxx

xxxxxxxxxxxxxxx

xx h

B

r

r1

2 φ

Flight angle Pitch Screw Flight

Root Dia of Screw

Barrel

z

r

Ω

Figure 8.1: Geometry of problem.

The assumptionhB

<< 1 and the congruent velocity distributions at parallel cross sections

imply∂P∂r

= 0, ur = 0 [3]. The flow is assumed fully developed and axisymmetric so that,

V = (0,uθ(r),uz(r)), S = S(r). (8.2)

Using velocity profile (8.2) equation (1.8) is satisfied identically and the momentum

equations (1.15 - 1.17) with the help of above assumptions reduce to

0 =1r

∂(rSrr)∂r

− Sθθr

, (8.3)

1r

∂P∂θ

=1r2

∂(r2Srθ)∂r

− (Sθr−Srθ)r

, (8.4)

∂P∂z

=1r

∂(rSrz)∂r

. (8.5)

Since the velocity profile is same as given in section 7.1 (equations (7.1) and (7.2)) so we

substitute (7.8 - 7.24) in equation (1.19), to obtain non-zero components of extra stress

tensor, S,

Srr =(

α1 +α2

2

)P9, (8.6)

Srθ = Sθr = µ+2(β2 +β3)P9P8, (8.7)

163

Srz = Szr = µ+2(β2 +β3)P9 duz

dr, (8.8)

Sθθ = α2P 28 , (8.9)

Sθz = Szθ = α2P8duz

dr, (8.10)

Szz = α2

(duz

dr

)2

. (8.11)

Using equations (8.6 - 8.11) in equations (8.3 - 8.5) we get

0 =1r

ddr

[r

(2α1 +α2)

((r

ddr

(uθr

))2

+(

duz

dr

)2)]

− α2

r

(r

ddr

(uθr

))2

, (8.12)

1r

∂P∂θ

=1r2

ddr

[r2

µ+2(β2 +β3)

((r

ddr

(uθr

))2

+(

duz

dr

)2)(

rddr

(uθr

))], (8.13)

∂P∂z

=1r

ddr

[r

µ+2(β2 +β3)

((r

ddr

(uθr

))2

+(

duz

dr

)2)

duz

dr

]. (8.14)

Since the right sides of equations (8.13) and (8.14) are functions of r alone and P 6= P(r),

this implies∂P∂θ


= constant. Our concentration is on azimuthal and axial

velocity components, so we will consider only equations (8.13) and (8.14).

Introducing dimensionless parameters

r∗ =rr1

, z∗ =zr1

, v∗ =uθ

Ωr1, w∗ =

uz

Ωr1, P∗ =

PµΩ

, (8.15)

164

in equations (8.13) and (8.14), yield after dropping “∗ ”,

ddr

[r2

(dvdr− v

r

)+ β

r5

(ddr

(vr

))3

+ r3(

dwdr

)2 (ddr

(vr

))]= rP,θ, (8.16)

ddr

[r

dwdr

+ β

r3

(ddr

(vr

))2 dwdr

+ r(

dwdr

)3]

= rP,z, (8.17)

where β =2(β2 +β3)Ω2

µ, P,θ =

∂P∂θ

and P,z =∂P∂z

, and boundary conditions (8.1) become

v = 1, w = 0, at r = 1,

v = 0, w = 0, at r = δ,(8.18)

where δ =r2

r1> 1.

Equations (8.16) and (8.17) are coupled second order nonlinear inhomogeneous ordinary

differential equations, the exact solution seems to be difficult. We use HPM to obtain

approximate solution.


As we have constructed homotopy in chapter 7, f (r, p) : Ω× [0,1] −→ ℜ and g(r, p) :

Ω× [0,1] −→ ℜ satisfying: (see section:1.8.2), the problem under consideration can be

written as

G1( f )−G1(v0)+ pG1(v0)

+p

[β

ddr

r5

(ddr

(fr

))3

+ r3(

dgdr

)2 (ddr

(fr

))− rP,θ

]= 0, (8.19)

G2(g)−G2(w0)+ pG2(w0)

+p

[β

ddr

r3

(ddr

(fr

))2 dgdr

+ r(

dgdr

)3− rP,z

]= 0, (8.20)

165

here G1 = r2 d2

dr2 + rddr−1 and G2 = r

d2

dr2 +ddr

are differential operators and

v0 =Θ1

r+Θ2r +Θ3r ln(r), (8.21)

w0 = Ψ1 +Ψ1r2 +Ψ2 ln(r), (8.22)

are initial guess approximations where Θ1, Θ2, Θ3, Ψ1 and Ψ2 are constant coefficients.

On substituting series (1.44) equations (8.19) and (8.20) become

G1

(∞

∑i=0

pi fi

)−G1(v0)+ pG1(v0)+ p

β

ddr

r5

(ddr

(1r

∞

∑i=0

pi fi

))3

+ r3

(ddr

∞

∑i=0

pigi

)2 (ddr

(1r

∞

∑i=0

pi fi

))− rP,θ

= 0, (8.23)

G2

(∞

∑i=0

pigi

)−G2(w0)+ pG2(w0)+ p

β

ddr

r3

(ddr

(1r

∞

∑i=0

pi fi

))2

ddr

(∞

∑i=0

pigi

)+ r

(ddr

(∞

∑i=0

pigi

))3− rP,z

= 0, (8.24)

and the boundary conditions (8.18) become

∞

∑i=0

pi fi = 1,∞

∑i=0

pigi = 0, at r = 1, (8.25)

∞

∑i=0

pi fi = 0,∞

∑i=0

pigi = 0, at r = δ. (8.26)

Now equating the equal powers of p. We get the following system of differential equations:

166


Zeroth order differential equations

G1( f0)−G1(v0) = 0, (8.27)

G2(g0)−G2(w0) = 0, (8.28)

together with boundary conditions

f0(1) = 1 g0(1) = 0, f0(δ) = g0(δ) = 0,

has the solution

f0 =Θ1

r+Θ2r +Θ3r ln(r), (8.29)

g0 = Ψ1 +Ψ1r2 +Ψ2 ln(r), (8.30)

where constant coefficients Θ1, Θ2, Θ3, Ψ1 and Ψ2 are given in Appendix VI. Equations

(8.29) and (8.30) are the solution for Newtonian case.


First order differential equations by equating equal powers of p are

G1( f1)+G1(v0)

+βddr

r5

(ddr

(f0

r

))3

+ r3(

dg0

dr

)2 (ddr

(f0

r

))− rP,θ = 0, (8.31)

G2(g1)+G2(w0)

+βddr

r3

(ddr

(f0

r

))2 dg0

dr+ r

(dg0

dr

)3− rP,z = 0, (8.32)

167

along with boundary conditions

f1(1) = g1(1) = 0, f1(δ) = g1(δ) = 0,

result in,

f1 = β(

Θ4

r5 +Θ5

r3 +Θ6

r+Θ7r +Θ8r ln(r)+Θ9r3

), (8.33)

g1 = β(

Ψ3

r4 +Ψ4

r2 +Ψ5 +Ψ6 ln(r)+Ψ7 ln(r)2 +Ψ8r2 +Ψ9r4)

, (8.34)

where Θi, Ψ j, i = 4, · · · ,9, j = 3, · · · ,9 are constant coefficients given in Appendix VI.


Second order differential equations

G1( f2)+ βddr

3r5

(ddr

(f0

r

))2 (ddr

(f1

r

))+2r3

(dg0

dr

)

(dg1

dr

)(ddr

(f0

r

))+ r3

(dg0

dr

)2 (ddr

(f1

r

))= 0, (8.35)

G2(g2)+ βddr

r3

(ddr

(f0

r

))2 dg1

dr+2r3

(ddr

(f0

r

))

(ddr

(f1

r

))dg0

dr+3r

(dg0

dr

)2 (dg1

dr

)= 0, (8.36)

together with boundary conditions

f2(1) = g2(1) = 0, f2(δ) = g2(δ) = 0.

Solving the above in conjunction with corresponding boundary conditions give

f2 = β2(

Θ10

r9 +Θ11

r7 +Θ12

r5 +Θ13

r3 +Θ14

r3 ln(r)+Θ15

r+

Θ16

rln(r)

168

+ Θ17r +Θ18r ln(r)+Θ19r ln(r)2 +Θ20r3 +Θ21r5)

, (8.37)

g2 = β2(

Ψ10

r9 +Ψ11

r8 +Ψ12

r7 +Ψ13

r6 +Ψ14

r5 +Ψ15

r5 ln(r)+Ψ16

r4 +Ψ17

r4 ln(r)

+Ψ18

r3 +Ψ19

r3 ln(r)+Ψ20

r2 +Ψ21

r2 ln(r)+Ψ22

r+

Ψ23

rln(r)+Ψ24

+ Ψ25 ln(r)+Ψ26 ln(r)2 +Ψ27 ln(r)3 +Ψ28r +Ψ29r2

+ Ψ30r2 ln(r)+Ψ31r3 +Ψ32r4 +Ψ33r6)

, (8.38)

where the constant coefficients Θi, Ψ j, i = 10, · · · ,21, j = 10, · · · ,33 are given in Ap-

pendix VI.

Equations (7.68) and (7.70) imply that

v = (Θ1 + βΘ6 + β2Θ15)1r

+(Θ2 + βΘ7 + β2Θ17)r +(Θ3 + βΘ8 + β2Θ18)r ln(r)

+ (βΘ4 + β2Θ12)1r5 +(βΘ5 + β2Θ13)

1r3 +(βΘ9 + β2Θ20)r3

+ β2 Θ10

r9 + β2 Θ11

r7 + β2 Θ14

r3 ln(r)+ β2 Θ16

rln(r)

+ β2Θ19r ln(r)2 + β2Θ21r5, (8.39)

is the velocity profile in θ−direction, and

w = (Ψ1 + βΨ5 + β2Ψ24)+(Ψ1 + βΨ8 + β2Ψ29)r2 +(Ψ2 + βΨ6 + β2Ψ25) ln(r)

+(

βΨ3 + β2Ψ16

) 1r4 +(βΨ4 + β2Ψ20)

1r2 +(βΨ7 + β2Ψ26) ln(r)2

+ (βΨ9 + β2Ψ32)r4 + β2 Ψ10

r9 + β2 Ψ11

r8 + β2 Ψ12

r7 + β2 Ψ13

r6 + β2 Ψ14

r5

+ β2 Ψ15

r5 ln(r)+ β2 Ψ17

r4 ln(r)+ β2 Ψ18

r3 + β2 Ψ19

r3 ln(r)+ β2 Ψ21

r2 ln(r)

+ β2 Ψ22

r+ β2 Ψ23

rln(r)+ β2Ψ27 ln(r)3 + β2Ψ28r

+ β2Ψ30r2 ln(r)+ β2Ψ31r3 + β2Ψ33r6, (8.40)

is the velocity profile in z−direction.

169

8.2.4 Stresses

The shear stresses (8.7), (8.8) and (8.10) can be calculated using equations (8.39) and (8.40)

as given

S∗rθ = S∗θr =(

1+ βΠ6

)Θ2 +Θ3 + βΘ7 + βΘ8− 9β2Θ10

r10 − 7β2Θ11

r8 −5(

βΘ4 + β2Θ12

)

r6

−3(

βΘ5 + β2Θ13

)

r4 +β2Θ14

r4 − 3β2Θ14 lnrr4 − Θ1 + βΘ6 + β2Θ15

r2 +β2Θ16

r2

− β2Θ16 lnrr2 + β2Θ17 +Θ18β2 +

(Θ3 + βΘ8 + β2Θ18

)lnr +2β2Θ19 lnr

+ β2Θ19(lnr)2 +3r2(

βΘ9 + β2Θ20

)+5r4β2Θ21

− 1r

(Θ1 + βΘ6 + β2Θ15)

1r


+ (βΘ4 + β2Θ12)1r5 +(βΘ5 + β2Θ13)

1r3 +(βΘ9 + β2Θ20)r3

+ β2 Θ10

r9 + β2 Θ11

r7 + β2 Θ14

r3 ln(r)+ β2 Θ16

rln(r)

+ β2Θ19r ln(r)2 + β2Θ21r5]

, (8.41)

S∗rz = S∗zr =(

1+ βΠ6

)[−9β2z10

r10 − 8β2Ψ11

r9 − 7β2Ψ12

r8 − 6β2Ψ13

r7 − 5β2Ψ14

r6 +β2Ψ15

r6

− 5β2 lnrΨ15

r6 −4(

βΨ3 + β2Ψ16

)

r5 +β2Ψ17

r5 − 4β2Ψ17 lnrr5 − 3β2Ψ18

r4 +β2Ψ19

r4

− 3β2Ψ19 lnrr4 −

2(

βΨ4 + β2Ψ20

)

r3 +β2Ψ21

r3 − 2β2Ψ21 lnrr3 − β2Ψ22

r2 +β2Ψ23

r2

+β2Ψ23 lnr

r2 +Ψ2 + βΨ6 + β2Ψ25

r+

2lnr(

βΨ7 + β2Ψ26

)

r+

3β2(lnr)2Ψ27

r+ β2Ψ28 +2r

(Ψ1 + βΨ8 + β2Ψ29

)+ rβ2Ψ30 +2rβ2 lnrΨ30 +3r2β2Ψ31

+ 4r3(

βΨ9 + β2Ψ32

)+6r5β2Ψ33

], (8.42)

170

S∗θz = S∗zθ =α2Ω

µ

Θ2 +Θ3 + βΘ7 + βΘ8− 9β2Θ10

r10 − 7β2Θ11

r8 −5(

βΘ4 + β2Θ12

)

r6

−3(

βΘ5 + β2Θ13

)

r4 +β2Θ14

r4 − 3β2Θ14 lnrr4 − Θ1 + βΘ6 + β2Θ15

r2 +β2Θ16

r2

− β2Θ16 lnrr2 + β2Θ17 +Θ18β2 +

(Θ3 + βΘ8 + β2Θ18

)lnr +2β2Θ19 lnr

+ β2Θ19(lnr)2 +3r2(

βΘ9 + β2Θ20

)+5r4β2Θ21

− 1r

(Θ1 + βΘ6 + β2Θ15)

1r


+ (βΘ4 + β2Θ12)1r5 +(βΘ5 + β2Θ13)

1r3 +(βΘ9 + β2Θ20)r3 + β2 Θ10

r9

+ β2 Θ11

r7 + β2 Θ14

r3 ln(r)+ β2 Θ16

rln(r)+ β2Θ19r ln(r)2 + β2Θ21r5

]

[−9β2z10

r10 − 8β2Ψ11

r9 − 7β2Ψ12

r8 − 6β2Ψ13

r7 − 5β2Ψ14

r6 +β2Ψ15

r6

− 5β2 lnrΨ15

r6 −4(

βΨ3 + β2Ψ16

)

r5 +β2Ψ17

r5 − 4β2Ψ17 lnrr5 − 3β2Ψ18

r4 +β2Ψ19

r4

− 3β2Ψ19 lnrr4 −

2(

βΨ4 + β2Ψ20

)

r3 +β2Ψ21

r3 − 2β2Ψ21 lnrr3 − β2Ψ22

r2 +β2Ψ23

r2

+β2Ψ23 lnr

r2 +Ψ2 + βΨ6 + β2Ψ25

r+

2lnr(

βΨ7 + β2Ψ26

)

r+

3β2(lnr)2Ψ27

r+ β2Ψ28 +2r

(Ψ1 + βΨ8 + β2Ψ29

)+ rβ2Ψ30 +2rβ2 lnrΨ30 +3r2β2Ψ31

+ 4r3(

βΨ9 + β2Ψ32

)+6r5β2Ψ33

], (8.43)

where

Π6 =

Θ2 +Θ3 + βΘ7 + βΘ8− 9β2Θ10

r10 − 7β2Θ11

r8 −5(

βΘ4 + β2Θ12

)

r6

−3(

βΘ5 + β2Θ13

)

r4 +β2Θ14

r4 − 3β2Θ14 lnrr4 − Θ1 + βΘ6 + β2Θ15

r2

+β2Θ16

r2 − β2Θ16 lnrr2 + β2Θ17 +Θ18β2 +

(Θ3 + βΘ8 + β2Θ18

)lnr

171

+ 2β2Θ19 lnr + β2Θ19(lnr)2 +3r2(

βΘ9 + β2Θ20

)+5r4β2Θ21

− 1r

((Θ1 + βΘ6 + β2Θ15)

1r


+ (βΘ4 + β2Θ12)1r5 +(βΘ5 + β2Θ13)

1r3 +(βΘ9 + β2Θ20)r3 + β2 Θ10

r9

+ β2 Θ11

r7 + β2 Θ14

r3 ln(r)+ β2 Θ16


)2

+

−9β2Ψ10

r10 − 8β2Ψ11

r9 − 7β2Ψ12

r8 − 6β2Ψ13

r7 − 5β2Ψ14

r6 +β2Ψ15

r6 − 5β2 lnrΨ15

r6

−4(

βΨ3 + β2Ψ16

)

r5 +β2Ψ17

r5 − 4β2Ψ17 lnrr5 − 3β2Ψ18

r4 +β2Ψ19

r4 − 3β2Ψ19 lnrr4

−2(

βΨ4 + β2Ψ20

)

r3 +β2Ψ21

r3 − 2β2Ψ21 lnrr3 − β2Ψ22

r2 +β2Ψ23

r2 − β2Ψ23 lnrr2

+Ψ2 + βΨ6 + β2Ψ25

r+

2(

βΨ7 + β2Ψ26

)lnr

r+

3β2Ψ27(lnr)2

r+ β2Ψ28

+ 2r(

Ψ1 + βΨ8 + β2Ψ29

)+ rβ2Ψ30 +2rβ2Ψ30 lnr +3r2β2Ψ31

+ 4r3(

βΨ9 + β2Ψ32

)+6r5β2Ψ33

2]. (8.44)

The shear stresses exerted by the fluid on the screw can be calculated by putting r = 1 in

equations (8.41 - 8.43).




Using equations (8.39) and (8.40) we can obtained the normal stresses (8.6), (8.9) and

(8.11) as given

S∗rr =(2α1 +α2)Ω

µΠ6, (8.45)

172

S∗θθ =α2Ω

µ

Θ2 +Θ3 + βΘ7 + βΘ8− 9β2Θ10

r10 − 7β2Θ11

r8 −5(

βΘ4 + β2Θ12

)

r6

−3(

βΘ5 + β2Θ13

)

r4 +β2Θ14

r4 − 3β2Θ14 lnrr4 − Θ1 + βΘ6 + β2Θ15

r2

+β2Θ16

r2 − β2Θ16 lnrr2 + β2Θ17 +Θ18β2 +

(Θ3 + βΘ8 + β2Θ18

)lnr

+ 2β2Θ19 lnr + β2Θ19(lnr)2 +3r2(

βΘ9 + β2Θ20

)+5r4β2Θ21

− 1r

((Θ1 + βΘ6 + β2Θ15)

1r


+ (βΘ4 + β2Θ12)1r5 +(βΘ5 + β2Θ13)

1r3 +(βΘ9 + β2Θ20)r3 + β2 Θ10

r9

+ β2 Θ11

r7 + β2 Θ14

r3 ln(r)+ β2 Θ16


)]2

, (8.46)

S∗zz =α2Ω

µ

[−9β2Ψ10

r10 − 8β2Ψ11

r9 − 7β2Ψ12

r8 − 6β2Ψ13

r7 − 5β2Ψ14

r6 +β2Ψ15

r6 − 5β2 lnrΨ15

r6

−4(

βΨ3 + β2Ψ16

)

r5 +β2Ψ17

r5 − 4β2Ψ17 lnrr5 − 3β2Ψ18

r4 +β2Ψ19

r4 − 3β2Ψ19 lnrr4

−2(

βΨ4 + β2Ψ20

)

r3 +β2Ψ21

r3 − 2β2Ψ21 lnrr3 − β2Ψ22

r2 +β2Ψ23

r2 − β2Ψ23 lnrr2

+Ψ2 + βΨ6 + β2Ψ25

r+

2(

βΨ7 + β2Ψ26

)lnr

r+

3β2Ψ27(lnr)2

r+ β2Ψ28

+ 2r(

Ψ1 + βΨ8 + β2Ψ29

)+ rβ2Ψ30 +2rβ2Ψ30 lnr +3r2β2Ψ31

+ 4r3(

βΨ9 + β2Ψ32

)+6r5β2Ψ33

]2, (8.47)

where S∗i j =Si j


173


An expression for net flow rate through the channel can be obtained by integrating the axial

component of the velocity over the cross section of the channel

Qz = 2π∫ r=r2

r=r1

uzrdr. (8.48)

To introduce the effect of flights, Booy [3], used the condition that no net flow occurs

across the helical flight, since v and w are only dependent on r. Fig. 8.2 shows an

elemental volume in the channel between screw root and barrel, bounded by a helical

surface parallel to the screw flight and by θ =constant and z =constant planes [3]. The

flow rate through z−plane (plane perpendicular to the z-direction) AABB must equal to the

flow rate through the θ−plane BBCC. Since no flow can occur through any other surface.

The helical boundary AACC requires that,

A

A'

B

B'

C

C'

r2

r1

v

w

dz=r2 tan φ d θ

flightθr2 d

zdθ

Figure 8.2: Fluid element bounded by helical surface, root and barrel diameter and planesθ and z constant.

174

dz = r2 tanφdθ, (8.49)

where φ denotes the flight angle.

The flow rate through z−plane is,

dQz = dθ∫ r=r2

r=r1

uzrdr. (8.50)

The flow rate through the θ−plane can be expressed as,

dQθ = dz∫ r=r2

r=r1

uθdr = (r2 tanφ)dθ∫ r=r2

r=r1

uθdr. (8.51)

Both the flow rate are same implying that

∫ r=r2

r=r1

uzrdr = (r2 tanφ)∫ r=r2

r=r1

uθdr, (8.52)

and

Qθ = 2π(r2 tanφ)∫ r=r2

r=r1

uθdr. (8.53)

Volume flow rate in θ-direction

Volume flow rate (8.53) in non-dimensional form is

Q∗θ = 2πδ tanφ

∫ δ

1vdr, (8.54)

where Q∗θ =

Qθ

Ωr31

. Substituting the expression for v from (8.39) and dropping “*”, we get

Qθ = 2πδ tanφ

(Θ1 +βΘ6 +β2Θ15)lnδ+12(Θ2 +βΘ7 +β2Θ17)(δ2−1)

− 14(Θ3 +βΘ8 +β2Θ18)(δ2−1)+

12(L3 +βΘ8 +β2Θ18)δ2 lnδ

− 14(βΘ4 +β2Θ12)(

1δ4 −1)− 1

2(βΘ5 +β2Θ13)(

1δ2 −1)

175

+14(βΘ9 +β2Θ20)(δ4−1)−β2 Θ10

8(

1δ8 −1)−β2 Θ11

6(

1δ6 −1)

− β2 Θ14

4(

1δ6 −1)−β2Θ14

lnδ8δ2 +β2 Θ16

2lnδ2−β2 Θ19

2(δ2−1)

+ β2 Θ19

2δ2 ln(δ2)+β2 Θ21

6(δ6−1)

. (8.55)

Volume flow rate in z-direction

Non-dimensional volume flow rate in z-direction is

Q∗z = 2π

∫ δ

1wrdr, (8.56)

where Q∗z =

Qz

Ωr31

. Dropping “∗ ” we get

Qz = 2π

12(Ψ1 +βΨ5 +β2Ψ24)(δ2−1)+

14(Ψ1 +βΨ8 +β2Ψ29)(δ4−1)

− 14(Ψ2 +βΨ6 +β2Ψ25)(δ2−1)− 1

2(Ψ2 +βΨ6 +β2Ψ25)δ2 lnδ

− 12(βΨ3 +β2Ψ16)(

1δ2 −1)+(βΨ4 +β2Ψ20) lnδ− 1

2(βΨ7 +β2Ψ26)(δ2−1)

+12(βΨ7 +β2Ψ26)δ2 lnδ2 +

16(βΨ9 +β2Ψ32)(δ6−1)−β2 Ψ10

7(

1δ7 −1)

− β2 Ψ11

6(

1δ6 −1)−β2 Ψ12

5(

1δ5 −1)−β2 Ψ13

4(

1δ4 −1)−β2 Ψ14

3(

1δ3 −1)

− β2 Ψ15

9(

1δ3 −1)−β2 Ψ15

3lnδδ3 −β2 Ψ17

4(

1δ2 −1)−β2 Ψ17

2lnδδ2

− β2Ψ18(1δ−1)−β2Ψ19(

1δ−1)−β2Ψ19

lnδδ

+β2 Ψ21

2lnδ2 +β2Ψ22(δ−1)

− β2Ψ23(δ−1)+β2Ψ23δ lnδ− 34

β2Ψ27(δ2−1)+β2 Ψ27

2δ2 lnδ3

+ β2 Ψ28

3(δ3−1)−β2 Ψ30

16(δ4−1)+β2 Ψ30

4δ4 lnδ

+ β2 Ψ31

5(δ5−1)+β2 Ψ33

8(δ8−1)

. (8.57)

176


Using volume flow rate (8.55) or (9.70) average velocity can be computed as

w∗av =Q∗

iπ(δ2−1)

, i = θ or z, (8.58)

where w∗av =uiav

Ωr1, i = θ or z, is dimensionless average velocity.


In this chapter we have considered steady flow of an incompressible third-grade fluid

in HSR with non-zero flight angle (see fig.8.1), using cylindrical coordinates. For

the geometry of HSR we obtained coupled second order nonlinear inhomogeneous

ordinary differential equations. Using HPM, expressions for azimuthal and axial velocity

components are derived. The volume flow rates, shear stresses, normal stresses, the shear

stresses exerted by the fluid on the screw and average velocity are also calculated. Here

we discussed the effect of involved flow parameters on the velocity profiles with the help

of graphical representation. Figure (8.3) is plotted for the velocity v for different values of

fluid parameter β, steadily increase observed in the velocity from screw toward barrel and

the velocity attains maximum values in between the channel which show shear thinning

due to increases in the value of β. Figure (8.4) is sketched for the velocity profile w for

different values of β, the velocity profile is seem to be parabolic in nature. The velocity w

takes the fluid toward the exit. The velocity profile w is in close resemblance with velocity

profile s as discussed in chapter 2.

177

Figure 8.3: Velocity profile v(r) for different values of β, keeping P,θ =−4.0 , P,z =−4.0and δ = 2.

Figure 8.4: Velocity profile w(r) for different values of β, keeping P,θ =−4.0, P,z =−4.0and δ = 2.

178

Figure (8.5 - 8.6) are shown for the velocity v for different values of pressure gradients P,θ

and P,z respectively, it can be seen that velocity v increases with the increase in pressure

gradients. It is noticed that P,z resist the velocity v as graphs show the smaller magnitude

of v for P,z. Similarly figures (8.7 - 8.8) are plotted for the velocity w for different values

of P,θ and P,z. With the increase in the value of P,θ and P,z, increase in the w is observed,

however the effect of P,θ is observed less on w which show P,θ try to resist the flow in axial

direction. The figures (8.10 - 8.11) are sketched to see the effect of β on shear stresses

Srθ and Srz. Graph of Srθ result in unexplainable behavior, while the graph of Srz gives the

points where the velocity w attains maximum values in the channel. In this work the flow

pattern is in the forward direction due to the non-zero fight angle.

Figure 8.5: Velocity profile v(r) for different values of P,θ, keeping β = 0.4, P,z = −4.0and δ = 2.

179

Figure 8.6: Velocity profile v(r) for different values of P,z, keeping β = 0.4, P,θ = −4.0and δ = 2.

Figure 8.7: Velocity profile w(r) for different values of P,θ, keeping β = 0.4, P,z = −4.0and δ = 2.

180

Figure 8.8: Velocity profile w(r) for different values of P,z, keeping β = 0.4, P,θ = −4.0and δ = 2.

Figure 8.9: Variation of shear stress Srθ for different values of β, keeping P,θ =−2.0, P,z =−2.0 and δ = 2.

181

Figure 8.10: Variation of shear stress Srθ for different values of β, keeping P,θ =−2.0, P,z =−2.0 and δ = 2.

Figure 8.11: Variation of shear stress Srz for different values of β, keeping P,θ =−2.0, P,z =−2.0 and δ = 2.

182

8.4 conclusion

The steady flow of an isothermal, homogeneous and incompressible third-grade fluid is

investigated in HSR. We choose the cylindrical coordinate system (r,θ,z) which seems to

be a more natural choice due to the geometry of HSR. The model developed in cylindrical

coordinates pertains to second order non linear coupled differential equations. Using HPM

the analytical expressions are obtained for the flow properties i.e., velocities, volume flow

rates, shear and normal stresses, the shear stresses exerted by the fluid on the screw and

average velocity. Graphical discussion is given for the velocity profiles and shear stresses.

It is observed that fluid velocity can be controlled with the proper choice of the values of the

non-Newtonian parameter and pressure gradients. The difference between the flow pattern

of this work and in the work as discussed in chapter 7 is due to the flights of non-zero flight

angle.

183

Chapter 9

Co-rotational Maxwell Fluid Analysis in Helical Screw

Rheometer Using Adomian Decomposition Method

184

This chapter aims to study the steady incompressible flow of co-rotational Maxwell fluid

in HSR (see fig. 8.2). The rheological constitutive equation for co-rotational Maxwell

fluid model gives the second order nonlinear inhomogeneous coupled differential equations

which could not be solved explicitly. An iterative procedure, ADM is used to obtain the

analytical solution. Expressions for velocity components in θ and z−direction, shear and

normal stresses, the shear stresses exerted by the fluid on the screw and average velocity are

obtained. The volume flow rates are also calculated for the azimuthal and axial components

of velocity field by introducing the effect of flights. The results have been discussed with

the help of graphs as well. We observe that the velocity profiles are strongly dependent

on non-dimensional parameter α and pressure gradients P,θ and P,z. Velocity increases

progressively with the increase in value of involved parameters.


Steady, laminar flow of an incompressible isothermal co-rotational Maxwell fluid is

considered in HSR. Geometry of the problem and all other assumptions are same as

given in section 8.1. To calculate the components of extra stress tensor, S = S(y) for co-

rotational Maxwell fluid we proceed as follows: Using velocity profile (8.2) and equations

(1.18), (7.6) and (7.7) in equation (1.27) where∂S∂t

= 0 and

(V ·∇)S =

−2uθr

Srθuθr

(Srr−Sθθ) −uθr

Sθzuθr

(Srr−Sθθ) 2uθr

Sθruθr

Srz

−uθr

Sθzuθr

Srz 0

, (9.1)

(∇V)T S =

−uθr

Sθr −uθr

Sθθ −uθr

Sθz

duθdr

Srrduθdr

Srθduθdr

Srz

duz

drSrr

duz

drSrθ

duz

drSrz

, (9.2)

185

S(∇V) =

−uθr

Srθduθdr

Srrduz

drSrr

−uθr

Sθθduθdr

Sθrduz

drSθr

−uθr

Szθduθdr

Szrduz

drSzr

, (9.3)

therefore

∇S =

0 −SrrP8 −Srrduz

dr−SrrP8 −2SrθP8 −SrzP8−Srθ

duz

dr−Srr

duz

dr−SrzP8−Srθ

duz

dr−2Szr

duz

dr

. (9.4)


A1S =

SθrP8 +Szrduz

drSθθP8 +Szθ

duz

drSθzP8 +Szz

duz

drSrrP8 SrθP8 SrzP8

Srrduz

drSrθ

duz

drSrz

duz

dr

, (9.5)

SA1 =

SrθP8 +Srzduz

drSrrP8 Srr

duz

drSθθP8 +Szθ

duz

drSθrP8 Sθr

duz

drSzθP8 +Szz

duz

drSzrP8 Szr

duz

dr

, (9.6)

then

A1S+SA1 =

2(

SrθP8 +Srzduz

dr

)P10P8 +Szθ

duz

drSθzP8 +P11

duz

dr

P10P8 +Szθduz

dr2SrθP8 SrzP8 +Srθ

duz

drSθzP8 +P11

duz

drSrzP8 +Srθ

duz

dr2Szr

duz

dr

, (9.7)

where P10 = Srr +Sθθ and P11 = Srr +Szz.

On substituting equations (7.8), (1.18), (9.4) and (9.7) in equation (1.26) we obtain non-

zero components of extra stress tensor, S as

Srr = −η0λ1

(duθdr

− uθr

)2

+(

duz

dr

)2

P12, (9.8)

186

Srθ = Sθr = η0

(duθdr

− uθr

)P12, (9.9)

Srz = Szr = η0duz

drP12, (9.10)

Sθθ = η0λ1

(duθdr

− uθr

)2

P12, (9.11)

Sθz = Szθ = η0λ1duz

dr

(duθdr

− uθr

)P12, (9.12)

Szz = η0λ1

(duz

dr

)2

P12, (9.13)

where

P12 =1

1+λ21

(duθdr

− uθr

)2

+(

duz

dr

)2 . (9.14)

Using velocity profile (8.2) equation of continuity (1.8) is satisfied identically and

substitution of components of extra stress tensor (9.8 - 9.13) in equations (8.3 - 8.5) result

in

0 =1r

ddr

r

(duθdr

− uθr

)2

+(

duz

dr

)2

1+λ21

(duθdr

− uθr

)2

+(

duz

dr

)2

− 1r

(duθdr

− uθr

)2

1+λ21

(duθdr

− uθr

)2

+(

duz

dr

)2

, (9.15)

1r

∂P∂θ

= η01r2

ddr

r2

(duθdr

− uθr

)

1+λ21

(duθdr

− uθr

)2

+(

duz

dr

)2

, (9.16)

187

∂P∂z

= η01r

ddr

r

duz

dr

1+λ21

(duθdr

− uθr

)2

+(

duz

dr

)2

. (9.17)

Equations (9.16) and (9.17) implies∂P∂θ


= constant. Our concentration

is on azimuthal and axial velocity components, so we will consider only equations (9.16)

and (9.17). Introducing dimensionless parameters

r∗ =rr1

, z∗ =zr1

, v∗ =uθ

Ωr1, w∗ =

uz

Ωr1and P∗ =

Pη0Ω

, (9.18)

in equations (9.16) and (9.17) we get

1r∗2

ddr∗

r∗2

(dv∗

dr∗− v∗

r∗

)

1+(Wi∗)2

(dv∗

dr∗− v∗

r∗

)2

+(

dw∗

dr∗

)2

=1r∗

∂P∗

∂θ, (9.19)

1r∗

ddr∗

r∗

dw∗

dr∗

1+(Wi∗)2

(dv∗

dr∗− v∗

r∗

)2

+(

dw∗

dr∗

)2

=∂P∗

∂z∗, (9.20)

where Wi∗ = λ1Ω is the Weissenberg number. Dropping “∗ ” equations (9.19) and (9.20),

give

1r2

ddr

r2

(dvdr− v

r

)

1+(Wi)2

(dvdr− v

r

)2

+(

dwdr

)2

=1r

∂P∂θ

, (9.21)

1r

ddr

r

dwdr

1+(Wi)2

(dvdr− v

r

)2

+(

dwdr

)2

=∂P∂z

, (9.22)

188

Integrating equation (9.21) and (9.22) with respect to “r” and assume that α = (Wi)2,

P,θ =12

∂P∂θ

and P,z =12

∂P∂z

, we get

ddr

(vr

)=

(P,θr

+C1

r3

)[1+ α

r2

(ddr

(vr

))2

+(

dwdr

)2]

, (9.23)

dwdr

=(

P,zr +C2

r

)[1+ α

r2

(ddr

(vr

))2

+(

dwdr

)2]

, (9.24)

where C1 and C2 are constants of integration.

The resultant equations (9.23) and (9.24) are coupled first order nonlinear inhomogeneous

ordinary differential equations, with boundary conditions (8.18). To solve this system of

equations, we use ADM in the following section.


ADM (discussed in sec: 1.8.3) describes that equations (9.23) and (9.24), will take the form

Lr

(vr

)=

(P,θr

+C1

r3

)[1+ α

r2

(ddr

(vr

))2

+(

dwdr

)2]

, (9.25)

Lr(w) =(

P,zr +C2

r

)[1+ α

r2

(ddr

(vr

))2

+(

dwdr

)2]

, (9.26)

where Lr =ddr

is the invertible differential operator.

Applying L−1r to both sides of equations (9.25) and (9.26), respectively result in

vr

= C3 +L−1r

(P,θ

r+

C1

r3

)

+ αL−1r

(P,θr

+C1

r3

)r2

(ddr

(vr

))2

+(

dwdr

)2

, (9.27)

w = C4 +L−1r

(P,zr +

C2

r

)

+ αL−1r

(P,zr +

C2

r

)r2

(ddr

(vr

))2

+(

dwdr

)2

, (9.28)

189

where C3 and C4 are also constant of integrations can be determined using boundary

conditions. Writing v, w , C1 and C2 in component form as

v =∞

∑n=0

vn, w =∞

∑n=0

wn, C1 =∞

∑n=0

C1,n and C2 =∞

∑n=0

C2,n.

Thus (9.27) and (9.28) take the form

∞

∑n=0

vn = C3r + rL−1r

P,θr

+

∞∑

n=0C1,n

r3

+ αrL−1

r

P,θr

+

∞∑

n=0C1,n

r3

r2

ddr

∞∑

n=0vn

r

2

+

ddr

(∞

∑n=0

wn

)2

, (9.29)

∞

∑n=0

wn = C4 +L−1r

P,zr +

∞∑

n=0C2,n

r

+ αL−1

r

P,zr +

∞∑

n=0C2,n

r

r2

ddr

∞∑

n=0vn

r

2

+

ddr

(∞

∑n=0

wn

)2

, (9.30)

and the nonlinear terms in the form of Adomian polynomials are

∞

∑n=0

An =

P,θr

+

∞∑

n=0C1,n

r3

r2

ddr

∞∑

n=0vn

r

2

+

ddr

(∞

∑n=0

wn

)2 , (9.31)

190

∞

∑n=0

Bn =

P,zr +

∞∑

n=0C2,n

r

r2

ddr

∞∑

n=0vn

r

2

+

ddr

(∞

∑n=0

wn

)2 , (9.32)

thus (9.29) and (9.30) respectively reduce to

∞

∑n=0

vn = C3r + rL−1r

P,θr

+

∞∑

n=0C1,n

r3

+ αrL−1

r

(∞

∑n=0

An

), (9.33)

∞

∑n=0

wn = C4 +L−1r

P,zr +

∞∑

n=0C2,n

r

+ αL−1

r

(∞

∑n=0

Bn

). (9.34)

The boundary conditions will take the form

∞∑

n=0vn = 1,

∞∑

n=0wn = 0, at r = 1,

∞∑

n=0vn = 0,

∞∑

n=0wn = 0, at r = δ.

(9.35)

Here we assumed that(

P,θr

+C1,0

r3

)and

(P,zr +

C2,0

r

)are forcing functions and

∞∑

n=1C1,n

r3

and

∞∑

n=1C2,n

r

are remainders of the linear part [44]. We device the recursive relation in

equations (9.33 - 9.35) as:

v0 = C3r + rL−1r

(P,θr

+C1,0

r3

), (9.36)

w0 = C4 +L−1r

(P,zr +

C2,0

r

), (9.37)

191

v0 = 1, w0 = 0, at r = 1,

v0 = 0, w0 = 0, at r = δ,(9.38)

vn+1 = rL−1r

(C1,n+1

r3

)+ αrL−1

r (An) , n≥ 0, (9.39)

wn+1 = L−1r

(C2,n+1

r

)+ αL−1

r (Bn) , n≥ 0, (9.40)

∞∑

n=1vn = 0,

∞∑

n=1wn = 0, at r = 1,

∞∑

n=1vn = 0,

∞∑

n=1wn = 0, at r = δ.

(9.41)

Then

v =∞

∑n=0

vn, and w =∞

∑n=0

wn, (9.42)

are the solutions for the v and w− components of velocities in θ and z−directions,

respectively.


Equations (9.36) and (9.38) give the zeroth component solutions

v0 = P,θr ln(r)− N1

r+N2r, (9.43)

w0 = M1(r2−1)+M2 ln(r), (9.44)

are the solutions for Newtonian fluid, where N1, N2, M1 and M2 are constant coefficients

(Appendix VII).


Equations (9.39 - 9.41) for n = 0 give the first components problems which are

v1 = rL−1r

(C1,1

r3

)+ αrL−1

r (A0) , (9.45)

192

w1 = L−1r

(C2,1

r

)+ αL−1

r (B0) . (9.46)

With the help of equations (9.31) and (9.32) taken as

A0 =(

P,θr

+C1,0

r3

)r2

(ddr

(v0

r

))2

+(

dw0

dr

)2

, (9.47)

B0 =(

P,zr +C2,0

r

)r2

(ddr

(v0

r

))2

+(

dw0

dr

)2

, (9.48)

along with the boundary conditions

v1 = 0, w1 = 0, at r = 1,

v1 = 0, w1 = 0, at r = δ.(9.49)

Then, the solutions become

v1 = α(− N3

6r5 −N4

4r3 +(N8− N5

2)1r

+N6r ln(r)+N7

2r3−N9r

), (9.50)

w1 = α(−M3

4r4 −M4

2r2 +(M5 +M8) ln(r)+M6

2r2 +

M7

4r4 +M9

), (9.51)

where Ni, M j, i = 3, · · · ,9, j = 3, · · · ,9 are constant coefficients (Appendix VII).


For n = 1 equations (9.39 - 9.41) give the second components problems as

v2 = rL−1r

(C1,2

r3

)+ αrL−1

r (A1) , (9.52)

w2 = L−1r

(C2,2

r

)+ αL−1

r (B1) . (9.53)

From equations (9.31) and (9.32) Adomian polynomials can be written as,

A1 =(

P,θr

+C1,0

r3

)2r2 d

dr

(v0

r

) ddr

(v1

r

)+2

dw0

drdw1

dr

193

+C1,1

r3

r2

(ddr

(v0

r

))2

+(

dw0

dr

)2

, (9.54)

B1 =(

P,zr +C2,0

r

)2r2 d

dr

(v0

r

) ddr

(v1

r

)+2

dw0

drdw1

dr

+C2,1

r

r2

(ddr

(v0

r

))2

+(

dw0

dr

)2

, (9.55)

and boundary conditions

v2 = 0, w2 = 0, at r = 1,

v2 = 0, w2 = 0, at r = δ.(9.56)

have the solutions

v2 = α2(− N10

10r9 −N11

8r7 −N12

6r5 −N13

4r3 −N14

2r+N15r lnr +

N16r3

2

+N17r5

4+

N18

2

(r− 1

r

)+N19r

), (9.57)

w2 = α2(−M10

8r8 −M11

6r6 −M12

4r4 −M13

2r2 +(M14−M18) lnr +M15r2

2+

M16r4

4

+M17r6

6+M19

), (9.58)

where the constant coefficients Ni, M j, i = 10, · · · ,19, j = 10, · · · ,19 (Appendix VII).


Velocity profile in θ-direction

Adding equations (9.43), (9.50) and (9.57) give the solution for the azimuthal velocity

component

v = P,θr ln(r)− N1

r+N2r + α

(− N3

6r5 −N4

4r3 +(N8− N5

2)1r

+N6r ln(r)

+N7

2r3−N9r

)+ α2

(− N10

10r9 −N11

8r7 −N12

6r5 −N13

4r3 −N14

2r

194

+ N15r lnr +N16r3

2+

N17r5

4+

N18

2

(r− 1

r

)+N19r

). (9.59)


Combining equations (9.44), (9.51) and (9.58) give the solution for the axial velocity

component

w = M1(r2−1)+M2 ln(r)+ α(−M3

4r4 −M4

2r2 +(M5 +M8) ln(r)+M6

2r2

+M7

4r4 +M9

)+ α2

(−M10

8r8 −M11

6r6 −M12

4r4 −M13

2r2 +(M14−M18) lnr

+M15r2

2+

M16r4

4+

M17r6

6+M19

). (9.60)

9.2.5 Stresses

First derivatives of the equations (9.59) and (9.60) give the shear stresses as

S∗rθ = S∗θr =1

1+ αΠ7

P,θ(1+ lnr)+

N1

r2 +N2 + α(

5N3

6r6 +3N4

4r4 +(1+ lnr)N6

+3r2N7

2+

(N5

2−N8

)1r2 −N9

)+ α2

(9N10

10r10 +7N11

8r8 +5N12

6r6 +3N13

4r4 +N14

2r2

+ (1+ lnr)N15 +3r2N16

2+

5r4N17

4+

12

(1+

1r2

)N18 +N19

)− 1

r

(P,θr ln(r)

− N1

r+N2r + α

(− N3

6r5 −N4

4r3 +(N8− N5

2)1r

+N6r ln(r)+N7

2r3−N9r

)

+ α2(− N10

10r9 −N11

8r7 −N12

6r5 −N13

4r3 −N14

2r+N15r lnr +

N16r3

2+

N17r5

4

+N18

2

(r− 1

r

)+N19r

)), (9.61)

S∗rz = S∗zr =1

1+ αΠ7

2rM1 +

M2

r+ α

(M3

r5 +M4

r3 + rM6 + r3M7 +(M5 +M8)1r

)

+ α2(

M10

r9 +M11

r7 +M12

r5 +M13

r3 + rM15 + r3M16 + r5M17

+ (M14−M18)1r

), (9.62)

195

S∗θz = S∗zθ =√

α1+ αΠ7

P,θ(1+ lnr)+

N1

r2 +N2 + α(

5N3

6r6 +3N4

4r4 +(1+ lnr)N6

+3r2N7

2+

(N5

2−N8

)1r2 −N9

)+ α2

(9N10

10r10 +7N11

8r8 +5N12

6r6 +3N13

4r4 +N14

2r2

+ (1+ lnr)N15 +3r2N16

2+

5r4N17

4+

12

(1+

1r2

)N18 +N19

)− 1

r

(P,θr ln(r)

− N1

r+N2r + α

(− N3

6r5 −N4

4r3 +(N8− N5

2)1r

+N6r ln(r)+N7

2r3−N9r

)

+ α2(− N10

10r9 −N11

8r7 −N12

6r5 −N13

4r3 −N14

2r+N15r lnr +

N16r3

2+

N17r5

4

+N18

2

(r− 1

r

)+N19r

))2rM1 +

M2

r+ α

(M3

r5 +M4

r3 + rM6

+ r3M7 +(M5 +M8)1r

)+ α2

(M10

r9 +M11

r7 +M12

r5 +M13

r3

+ rM15 + r3M16 + r5M17 +(M14−M18)1r

), (9.63)

where

Π7 =[

P,θ(1+ lnr)+N1

r2 +N2 + α(

5N3

6r6 +3N4

4r4 +(1+ lnr)N6 +3r2N7

2

+(

N5

2−N8

)1r2 −N9

)+ α2

(9N10

10r10 +7N11

8r8 +5N12

6r6 +3N13

4r4 +N14

2r2

+ (1+ lnr)N15 +3r2N16

2+

5r4N17

4+

12

(1+

1r2

)N18 +N19

)− 1

r

(P,θr ln(r)

− N1

r+N2r + α

(− N3

6r5 −N4

4r3 +(N8− N5

2)1r

+N6r ln(r)+N7

2r3−N9r

)

+ α2(− N10

10r9 −N11

8r7 −N12

6r5 −N13

4r3 −N14

2r+N15r lnr +

N16r3

2+

N17r5

4

+N18

2

(r− 1

r

)+N19r

))2

+

2rM1 +M2

r+ α

(M3

r5 +M4

r3 + rM6 + r3M7

+ (M5 +M8)1r

)+ α2

(M10

r9 +M11

r7 +M12

r5 +M13

r3 + rM15 + r3M16

+ r5M17 +(M14−M18)1r

)2]

. (9.64)

196

Substituting r = 1 in equations (9.61 - 9.63), give shear stresses exerted by the fluid on the

screw as




The normal stresses are given as

S∗rr = −√

αΠ7

1+ αΠ7, (9.65)

S∗θθ =√

α1+ αΠ7

P,θ(1+ lnr)+

N1

r2 +N2 + α(

5N3

6r6 +3N4

4r4 +(1+ lnr)N6 +3r2N7

2

+(

N5

2−N8

)1r2 −N9

)+ α2

(9N10

10r10 +7N11

8r8 +5N12

6r6 +3N13

4r4 +N14

2r2

+ (1+ lnr)N15 +3r2N16

2+

5r4N17

4+

12

(1+

1r2

)N18 +N19

)− 1

r

(P,θr ln(r)

− N1

r+N2r + α

(− N3

6r5 −N4

4r3 +(N8− N5

2)1r

+N6r ln(r)+N7

2r3−N9r

)

+ α2(− N10

10r9 −N11

8r7 −N12

6r5 −N13

4r3 −N14

2r+N15r lnr +

N16r3

2+

N17r5

4

+N18

2

(r− 1

r

)+N19r

))2

, (9.66)

S∗zz =√

α1+ αΠ7

2rM1 +

M2

r+ α

(M3

r5 +M4

r3 + rM6 + r3M7 +(M5 +M8)1r

)

+ α2(

M10

r9 +M11

r7 +M12

r5 +M13

r3 + rM15 + r3M16 + r5M17

+ (M14−M18)1r

)2

, (9.67)

where S∗i j =Si j


197


Volume flow rate in θ-direction

Using equation (9.59), equation (8.54) gives the volume flow rate in θ-direction as

Qθ = 2πδ tanφ[(

K1

4+

αN6

4+

α2N15

4

)(1−δ2 +2δ2 lnδ

)− 12

(2N1 + αN5−2αN8

+ α2N14)

lnδ+12

(N2− αN9 + α2N19

)(δ2−1

)+

(αN3

24+

α2N12

24

)(1δ4 −1

)

+(

αN4

8+

α2N13

8

)(1δ2 −1

)+

(αN7

8+

α2N16

8

)(δ4−1

)

+ α2

N10

80

(1δ8 −1

)+

N11

48

(1δ6 −1

)

+N17

24

(δ6−1

)− N18

4(1−δ2 +2lnδ

)]. (9.68)

Volume flow rate in z-direction

Equation (8.56) with the help of equation (9.60) gives the volume flow rate in z-direction

as

Qz = 2π[

M1

4(1−2δ2 +δ4)+

14

(M2 + α(M5 +M8)+ α2(M14

− M18))(1−δ2 +2δ2 lnδ

)+

18

(αM3 + α2M12

)(1δ2 −1

)

− 12

lnδ(αM4 + α2M13

)+

18

(αM6 + α2M15

)(δ4−1

)

+1

24(αM7 + α2M16

)(δ6−1

)− 1

2(αM9 + α2M19

)(1−δ2)

+ α2

M10

48

(1δ6 −1

)+

M11

24

(1δ4 −1

)+

M17

48(δ8−1

)]. (9.69)

198


Volume flow rate (9.68) or (9.69) gives the average velocity as

w∗av =Q∗

iπ(δ2−1)

, i = θ or z, (9.70)

where w∗av =uiav

Ωr1, i = θ or z, is dimensionless average velocity.


This work considers the steady flow of an incompressible co-rotational Maxwell fluid

through HSR. The geometry is same as discussed in the chapter 8 (see fig.8.1). The

mathematical modeling in the geometry of HSR gives the coupled second order nonlinear

ordinary differential equations. Using ADM expressions for azimuthal and axial velocity

components are derived. The volume flow rates, shear stresses, normal stresses, the shear

stresses exerted by the fluid on the screw and average velocity are also derived. Here we

discussed the effect of involved dimensionless flow parameters on the velocity profiles with

the help of graphical representations. In figures (9.1 - 9.2), the pattern of velocity profiles

v and w for non-Newtonian parameter α, observed same as discussed for β in chapter 8,

only the larger magnitude velocities observed due more shear thinning with respect to α.

Figures (9.3 - 9.6) are sketched to note the effect of pressure gradients on azimuthal and

axial velocity components, keeping α fixed, same pattern is observed as given in chapter 8.

The figures (9.7 - 9.8) show the effect of α on shear stresses Srθ and Srz. The graphs of Srθ

and Srz give the points where the velocities v and w attains maximum values in the channel.

199

Figure 9.1: Velocity profile v(r) for different values of α, keeping P,θ =−2.0 , P,z =−2.0and δ = 2.

Figure 9.2: Velocity profile w(r) for different values of α, keeping P,θ =−2.0, P,z =−2.0and δ = 2.

200

Figure 9.3: Velocity profile v(r) for different values of P,θ, keeping α = 0.3, P,z = −2.0and δ = 2.

Figure 9.4: Velocity profile v(r) for different values of P,z, keeping α = 0.3, P,θ = −2.0and δ = 2.

201

Figure 9.5: Velocity profile w(r) for different values of P,θ, keeping α = 0.3, P,z = −2.0and δ = 2.

Figure 9.6: Velocity profile w(r) for different values of P,z, keeping α = 0.3, P,θ = −2.0and δ = 2.

202

Figure 9.7: Variation of shear stress Srθ for different values of α, keeping P,θ =−2.0, P,z =−2.0 and δ = 2.

Figure 9.8: Variation of shear stress Srz for different values of α, keeping P,θ =−2.0, P,z =−2.0 and δ = 2.

203

9.4 Conclusion

The steady, homogeneous flow of an isothermal and incompressible co-rotational Maxwell

fluid is investigated in HSR. Cylindrical coordinate system (r,θ,z) are used to develop the

model of co-rotational Maxwell fluid which results in the coupled second order nonlinear

ordinary differential equations which could not be solved explicitly. Using ADM analytical

expressions for azimuthal and axial velocity components, volume flow rates, shear stresses,

normal stresses, the shear stresses exerted by the fluid on the screw and average velocity are

derived. Graphical discussions are given for the velocity profiles and shear stresses. It is

observed that both strongly depend on involved flow parameters. It is noticed shear thinning

effects observed due to non-Newtonian parameter α. It is seen that shear thinning behavior

of co-rotational Maxwell fluid is larger than third-grade fluid as discussed in chapter 8.

204

Chapter 10

Conclusion

205

This thesis is focused to analyze the behavior of different types of non-Newtonian fluids in

helical screw rheometer. For this we did mathematical modeling and solutions are obtained

using exact and analytical methods.

In the first part of this thesis, i.e., chapters 2 - 6, the analyses of third-grade, co-rotational

Maxwell, Eyring, Eyring-Powell and Oldroyd 8−constant fluids are carried out in HSR,

using Cartesian co-ordinate system (x,y,z). The geometry of HSR is simplified in such

a way that the curvature of the screw channel is ignored, unrolled and laid out on a flat

surface. The surface of the barrel is flattened. The screw surface is supposed to be

lower plate and the barrel surface is assumed as upper plate. It is also assumed that the

lower plate is stationary and the upper plate is moving across the top of the channel with

velocity V at an angle φ to the direction of the channel (fig. 2.1). The developed nonlinear

ordinary differential equations for third grade, Eyring-Powell, Oldroyd 8−constant fluids

are solved using ADM, while co-rotational Maxwell fluid model is solved using PM.

Exact solution is obtained for Eyring fluid model. Expressions for velocities, shear and

normal stresses, shear stresses at barrel surface, shear forces exerted on the fluid, volume

flow rates and average velocity are derived for all above mentioned fluids. It is observed

that normal stresses do not contribute in case of Eyring and Eyring-Powell fluids. The

behavior of the velocity profiles are discussed through graphical representations. It is

found that the velocity profiles are strongly dependant on the involved dimensionless flow

parameters. It is also observed that fluid net velocity in the direction of the axis of the

screw is due to the pressure gradient. The effect of non-Newtonian parameters on the

behavior of shear stresses is also discussed with the help of graphs. It is also noticed

that the zeroth component solution of the analytical techniques PM and ADM results in

the solution of Newtonian case and the resultant velocity gives only the u component of

velocity for φ = 0 and w component of velocity for φ = 90. During this study we found

that, if we set β =αW 2

h2 in third-grade fluid solutions, we can calculate the solutions for

dilatant fluid, where α is dilatant constant. If we put Wi2 =βW 2

h2 in the co-rotational

Maxwell fluid solutions we can obtained the solutions for pseudoplastic fluid, where β is

206

pseudoplastic constant. The solutions for the Johnson-Tevaarwerk fluid can be obtained

from the solutions for Eyring fluid by relating dimensionless parameters of Eyring fluid

˜α and ˜β with material constants of Johnson-Tevaarwerk fluid. Letting˜β = 0 and ˜α =

1µ

,

all the solutions obtained for the Eyring-Powell fluid can be reproduce for the Newtonian

case. For ˜α =˜β = 0 flow is only due to drag of the plate (barrel) can be calculated in case

of Eyring-Powell model. Oldroyd 8-constant fluid solutions give the following result:

• When α = β we obtain solution for Newtonian case.

• The solutions for dilatant fluids can be calculated by setting β = 0, similarly α = 0

gives the solutions for pseudoplastic fluids.

• Setting α = µ(Wi)2(1− a2) and β = (Wi)2(1− a2), we can obtain the solution for

Johnson-Segalman fluid, where µ =µ

(µ+η)is dimensionless parameter and Wi =

mWh

denotes the Weissenberg number and µ, η are viscosities, m is the relaxation

time and a is slip parameter in the constitutive equation of Johnson-Segalman fluid.

In second part of the thesis, i.e., chapters 7 - 9, we used cylindrical co-ordinates system

(r,θ,z) to study the steady, laminar flow of incompressible, isothermal third-grade fluid

and co-rotational Maxwell fluid, which is more suitable choice for the flow analysis in

HSR.

In chapter 7, analysis of third-grade fluid is carried out in HSR with zero flight angle.

Chapter 8 and 9 consider the study of third-grade and co-rotational Maxwell fluids in HSR

with nonzero flight angles. Assuming that the barrel of radius r2 is stationary and the screw

of radius r1 rotates with angular velocity Ω (fig. 8.1). Using HPM and ADM expressions

for velocity profiles, shear and normal stresses, the shear stresses exerted by the fluid on

the screw, volume flow rates and average velocity are calculated. The behavior of the

velocity profiles is investigated through graphs. We found that velocity profiles strongly

depend upon involved non-Newtonian parameters. The effect of involved non-Newtonian

parameters on velocity profiles and shear stresses are given in results and discussion. It

is also observed that the zeroth component solution of both ADM and HPM provides the

207

solution for Newtonian fluid in HSR.

Thus the profound conclusion is that extrusion process depends on the involved flow

parameters.

208

Chapter 11

Appendices

209

11.1 Appendix I

A1 =12

(P,x +

3U2P,x

W 2 − 3UP2,x

W+

3P3,x

4+

2UP,z

W−2P,xP,z−

UP2,z

W+

34

P,xP2,z

),

A2 =16

(6UP2

,x

W−3P3

,x +4P,xP,z +2UP2

,z

W−3P,xP2

,z

), A3 =

112

(3P3

,x +3P,xP2,z),

A4 =12

(P,x +

9U4P,x

W 4 +10U2P,x

W 2 − 27U3P2,x

W 3 − 15UP2,x

W+

13P3,x

3+

21U2P3,x

W 2

− 27UP4,x

4W+

15P5,x

16+

8U3P,z

W 3 +8UP,z

W−10P,xP,z− 34U2P,xP,z

W 2 +25UP2

,xP,z

W

− 163

P3,xP,z−

5U3P2,z

W 3 − 17UP2,z

W+

383

P,xP2,z +

38U2P,xP2,z

3W 2 − 49UP2,xP2

,z

6W

+158

P3,xP2

,z +25UP3

,z

3W− 16

3P,xP3

,z−17UP4

,z

12W+

1516

P,xP4,z

),

A5 =16

(54U3P2

,x

W 3 +30UP2

,x

W−18P3

,x−90U2P3

,x

W 2 +87UP4

,x

2W− 15P5

,x

2+

68U2P,xP,z

W 2

− 108UP2,xP,z

W+

1043

P3,xP,z +

10U3P2,z

W 3 +34UP2

,z

W−54P,xP2

,z−54U2P,xP2

,z

W 2

+157UP2

,xP2,z

3W−15P3

,xP2,z−

36UP3,z

W+

1043

P,xP3,z +

53UP4,z

6W− 15

2P,xP4

,z +20P,xP,z

),

A6 =1

12

(18P3

,x +90U2P3

,x

W 2 − 90UP4,x

W+

45P5,x

2+

108UP2,xP,z

W−72P3

,xP,z +54P,xP2,z

+54U2P,xP2

,z

W 2 − 108UP2,xP2

,z

W+45P3

,xP2,z +

36UP3,z

W−72P,xP3

,z−18UP4

,z

W+

452

P,xP4,z

),

A7 =1

20

(60UP4

,x

W−30P5

,x +48P3,xP,z +

72UP2,xP2

,z

W−60P3

,xP2,z +48P,xP3

,z +12UP4

,z

W

− 30P,xP4,z), A8 =

130

(15P5

,x +30P3,xP2

,z +15P,xP4,z

),

210

B1 =12

(2

UP,x

W−P2

,x +3P,z +U2P,z

W 2 −2UP,xP,z

W+

34

P2,xP,z−3P2

,z +34

P3,z

),

B2 =16

(2P2

,x +4UP,xP,z

W−3P2

,xP,z +6P2,z−3P3

,z

), B3 =

112

(3P2

,xP,z +3P3,z),

B4 =12

(8U3P,x

W 3 +8UP,x

W−5P2

,x−17U2P2

,x

W 2 +25UP3

,x

3W− 17P4

,x

12+9P,z +

U4P,z

W 4

+10U2P,z

W 2 − 10U3P,xP,z

W 3 − 34UP,xP,z

W+

383

P2,xP,z +

38U2P2,xP,z

3W 2 − 16UP3,xP,z

3W

+1516

P4,xP,z−27P2

,z−15U2P2

,z

W 2 +25UP,xP2

,z

W− 49

6P2

,xP2,z +21P3

,z +13U2P3

,z

3W 2

− 16UP,xP3,z

3W+

158

P2,xP3

,z−27P4

,z

4+

15P5,z

16

),

B5 =16

(10P2

,x +34U2P2

,x

W 2 − 36UP3,x

W+

53P4,x

6+

20U3P,xP,z

W 3 +68UP,xP,z

W−54P2

,xP,z

− 54U2P2,xP,z

W 2 +104UP3

,xP,z

3W− 15

2P4

,xP,z +54P2,z +

30U2P2,z

W 2 − 108UP,xP2,z

W

+1573

P2,xP2

,z−90P3,z−

18U2P3,z

W 2 +104UP,xP3

,z

3W−15P2

,xP3,z +

87P4,z

2− 15P5

,z

2

),

B6 =1

12

(36UP3

,x

W−18P4

,x +54P2,xP,z +

54U2P2,xP,z

W 2 − 72UP3,xP,z

W+

452

P4,xP,z

+108UP,xP2

,z

W−108P2

,xP2,z +90P3

,z +18U2P3

,z

W 2 − 72UP,xP3,z

W

+ 45P2,xP3

,z−90P4,z +

45P5,z

2

),

B7 =1

20

(12P4

,x +48UP3

,xP,z

W−30P4

,xP,z +72P2,xP2

,z +48UP,xP3

,z

W−60P2

,xP3,z

+ 60P4,z−30P5

,z

),

B8 =1

30

(15P4

,xP,z +30 P2,xP3

,z +15P5,z

),

211

11.2 Appendix II

C0 =12

(P,x +

3U2P,x

W 2 − 3UP2,x

W+

3P3,x

4+

2UP,z

W−2P,xP,z−

UP2,z

W+

34

P,xP2,z

),

C1 =13

(3UP2

,x

W− 3P3

,x

2+2P,xP,z +

UP2,z

W− 3

2P,xP2

,z

), C2 =

14

(P3

,x +P,xP2,z),

C3 =12

(4U4P,x

W 4 +4U2P,x

W 2 − 17U3P2,x

W 3 − 9UP2,x

W+

17P3,x

6+

27U2P3,x

2W 2 − 17UP4,x

4W

+5P5

,x

8+

4U3P,z

W 3 +4UP,z


W 2 +16UP2

,xP,z

W− 10

3P3

,xP,z

− 3U3P2,z

W 3 − 11UP2,z

W+

496

P,xP2,z +

49U2P,xP2,z

6W 2 − 31UP2,xP2

,z

6W+

54

P3,xP2

,z

+16UP3

,z

3W− 10

3P,xP3

,z−11UP4

,z

12W+

58

P,xP4,z

),

C4 =13

(17U3P2

,x

W 3 +9UP2

,x

W−6P3

,x−30U2P3

,x

W 2 +57UP4

,x

4W− 5P5

,x

2+6P,xP,z

+22U2P,xP,z

W 2 − 36UP2,xP,z

W+

343

P3,xP,z +

3U3P2,z

W 3 +11UP2

,z

W−18P,xP2

,z

− 18U2P,xP2,z

W 2 +103UP2

,xP2,z

6W−5P3

,xP2,z−

12UP3,z

W

+343

P,xP3,z +

35UP4,z

12W− 5

2P,xP4

,z

),

C5 =14

(4P3

,x +20U2P3

,x

W 2 − 20UP4,x

W+5P5

,x +24UP2

,xP,z

W−16P3

,xP,z +12P,xP2,z

+12U2P,xP2

,z

W 2 − 24UP2,xP2

,z

W+10P3

,xP2,z +

8UP3,z

W−16P,xP3

,z−4UP4

,z

W+5P,xP4

,z

),

C6 =15

(10UP4

,x

W−5P5

,x +8P3,xP,z +

12UP2,xP2

,z

W−10P3

,xP2,z

+ 8P,xP3,z +

2UP4,z

W−5P,xP4

,z

),

C7 =16

(2 P5

,x +4 P3,xP2

,z +2 P,xP4,z

),

212

D0 =12

(2

UP,x

W−P2

,x +3P,z +U2P,z

W 2 −2UP,xP,z

W+

34

P2,xP,z−3P2

,z +34

P3,z

),

D1 =13

(P2

,x +2UP,xP,z

W− 3

2P2

,xP,z +3P2,z−

32

P3,z

),

D2 = P2,xP,z +P3

,z,

D3 =12

(4U3P,x

W 3 +4UP,x

W−3P2

,x−11U2P2

,x

W 2 +16UP3

,x

3W− 11P4

,x

12+4P,z +

4U2P,z

W 2

− 6U3P,xP,z

W 3 − 22UP,xP,z

W+

496

P2,xP,z +

49U2P2,xP,z

6W 2 − 10UP3,xP,z

3W+

58

P4,xP,z

− 17P2,z−

9U2P2,z

W 2 +16UP,xP2

,z

W− 31

6P2

,xP2,z +

27P3,z

2+

17U2P3,z

6W 2

− 10UP,xP3,z

3W+

54

P2,xP3

,z−17P4

,z

4+

5P5,z

8

),

D4 =13

(3P2

,x +11U2P2

,x

W 2 − 12UP3,x

W+

35P4,x

12+

6U3P,xP,z

W 3 +22UP,xP,z

W−18P2

,xP,z

− 18U2P2,xP,z

W 2 +34UP3

,xP,z

3W− 5

2P4

,xP,z +17P2,z +

9U2P2,z

W 2 − 36UP,xP2,z

W

+1036

P2,xP2

,z−30P3,z−

6U2P3,z

W 2 +34UP,xP3

,z

3W−5P2

,xP3,z +

57P4,z

4− 5P5

,z

2

),

D5 =14

(8UP3

,x

W−4P4

,x +12P2,xP,z +

12U2P2,xP,z

W 2 − 16UP3,xP,z

W+5P4

,xP,z

+24UP,xP2

,z

W−24P2

,xP2,z +20P3

,z +4U2P3

,z

W 2 − 16UP,xP3,z

W

+ 10P2,xP3

,z−20P4,z +5P5

,z

),

D6 =15

(2P4

,x +8UP3

,xP,z

W−5P4

,xP,z +12P2,xP2

,z +8UP,xP3

,z

W

− 10P2,xP3

,z +10P4,z−5P5

,z

),

D7 =16

(2P4

,xP,z +4P2,xP3

,z +2P5,z

).

213

11.3 Appendix III

ε0 = α tan2 φP,x + α2 tanφ(P,x)2 +

14

α3 (P,x)3 ,

ε1 = −2α2 tanφ(P,x)2− α3 (P,x)

3 , ε2 = α3 (P,x)3 ,

σ0 = αP,z− α2 (P,z)2 +

14

α3 (P,z)3 ,

σ1 = 2α2 (P,z)2− α3 (P,z)

3 , σ2 = α3 (P,z)3 ,

ε3 = α tan4 φP,x +3α2 tan3 φ(P,x)2 +

73

α3 tan2 φ(P,x)3 +

34

α4 tanφ(P,x)4

+5

48α5 (P,x)

5 ,

ε4 = −6α2 tan3 φ(P,x)2−10α3 tan2 φ(P,x)

3− 296

α4 tanφ(P,x)4− 5

6α5 (P,x)

5 ,

ε5 = 10α3 tan2 φ(P,x)3 +10α4 tanφ(P,x)

4 +52

α5 (P,x)5 ,

ε6 = −203


3α5 (P,x)

5 , ε7 =53

α5 (P,x)5 ,

σ6 =203

α4 (P,z)4− 10

3α5 (P,z)

5 , σ7 =53

α5 (P,z)5 ,

ε8 = α tan6 φP,x +6α2 tan5 φ(P,x)2 +

556

α3 tan4 φ(P,x)3 +

132

α4 tan3 φ(P,x)4

+11948

α5 tan2 φ(P,x)5 +

12

α6 tanφ(P,x)6 +

7144

α7 (P,x)7 ,

ε9 = −12α2 tan5 φ(P,x)2−40α3 tan4 φ(P,x)

3−43α4 tan3 φ(P,x)4

− 653

α5 tan2 φ(P,x)5− 97

18α6 tanφ(P,x)

6− 712

α7 (P,x)7 ,

ε10 = 40α3 tan4 φ(P,x)3 +90α4 tan3 φ(P,x)

4 +2053

α5 tan2 φ(P,x)5

+452

α6 tanφ(P,x)6 +

3512

α7 (P,x)7 ,

ε11 = −60α4 tan3 φ(P,x)4− 280

3α5 tan2 φ(P,x)

5− 4159


9α7 (P,x)

7 ,

ε12 =1403

α5 tan2 φ(P,x)5 +

1403

α6 tanφ(P,x)6 +

353

α7 (P,x)7 ,

ε13 = −563


3α7 (P,x)

7 , ε14 =289

α7 (P,x)7 ,

214

σ8 = αP,z−6α2 (P,z)2 +

556

α3 (P,z)3− 13

2α4 (P,z)

4 +11948

α5 (P,z)5

− 12

α6 (P,z)6 +

7144

α7 (P,z)7 ,

σ9 = 12α2 (P,z)2−40α3 (P,z)

3 +43α4 (P,z)4− 65

3α5 (P,z)

5 +9718

α6 (P,z)6

− 712

α7 (P,z)7 ,

σ10 = 40α3 (P,z)3−90α4 (P,z)

4 +2053

α5 (P,z)5− 45

2α6 (P,z)

6 +3512

α7 (P,z)7 ,

σ11 = 60α4 (P,z)4− 280

3α5 (P,z)

5 +415

9α6 (P,z)

6− 709

α7 (P,z)7 ,

σ12 =1403

α5 (P,z)5− 140

3α6 (P,z)

6 +353

α7 (P,z)7 ,

σ13 =563

α6 (P,z)6− 28

3α7 (P,z)

7 , σ14 =289

α7 (P,z)7 .

11.4 Appendix IV

H0 =12

(P,x +

3U2P,x

W 2 − 3UP2,x

W+

3P3,x

4+

2UP,z

W−2P,xP,z−

UP2,z

W+

34

P,xP2,z

),

H1 =13

(3UP2

,x

W− 3P3

,x

2+2P,xP,z +

UP2,z

W− 3

2P,xP2

,z

),

H2 =14

(P3

,x +P,xP2,z),

H3 =12

(4U4P,x

W 4 +4U2P,x

W 2 − 17U3P2,x

W 3 − 9UP2,x

W+

17P3,x

6+

27U2P3,x

2W 2 − 17UP4,x

4W

+5P5

,x

8+

4U3P,z

W 3 +4UP,z


W 2 +16UP2

,xP,z

W− 10

3P3

,xP,z

− 3U3P2,z

W 3 − 11UP2,z

W+

496

P,xP2,z +

49U2P,xP2,z

6W 2 − 31UP2,xP2

,z

6W+

54

P3,xP2

,z

+16UP3

,z

3W− 10

3P,xP3

,z−11UP4

,z

12W+

58

P,xP4,z

),

215

H4 =12

(P,x +

9U4P,x

W 4 +10U2P,x

W 2 − 27U3P2,x

W 3 − 15UP2,x

W+

13P3,x

3+

21U2P3,x

W 2

− 27UP4,x

4W+

15P5,x

16+

8U3P,z

W 3 +8UP,z


W 2 +25UP2

,xP,z

W

− 163

P3,xP,z−

5U3P2,z

W 3 − 17UP2,z

W+

383

P,xP2,z +

38U2P,xP2,z

3W 2 − 49UP2,xP2

,z

6W

+158

P3,xP2

,z +25UP3

,z

3W− 16

3P,xP3

,z−17UP4

,z

12W+

1516

P,xP4,z

),

H5 =13

(17U3P2

,x

W 3 +9UP2

,x

W−6P3

,x−30U2P3

,x

W 2 +57UP4

,x

4W− 5P5

,x

2+6P,xP,z

+22U2P,xP,z

W 2 − 36UP2,xP,z

W+

343

P3,xP,z +

3U3P2,z

W 3 +11UP2

,z

W−18P,xP2

,z

− 18U2P,xP2,z

W 2 +103UP2

,xP2,z

6W−5P3

,xP2,z−

12UP3,z

W+

343

P,xP3,z

+35UP4

,z

12W− 5

2P,xP4

,z

),

H6 =13

(27U3P2

,x

W 3 +15UP2

,x

W−9P3

,x−45U2P3

,x

W 2 +87UP4

,x

4W− 15P5

,x

4+10P,xP,z

+34U2P,xP,z

W 2 − 54UP2,xP,z

W+

523

P3,xP,z +

5U3P2,z

W 3 +17UP2

,z

W−27P,xP2

,z

− 27U2P,xP2,z

W 2 +157UP2

,xP2,z

6W− 15

2P3

,xP2,z−

18UP3,z

W

+523

P,xP3,z +

53UP4,z

12W− 15

4P,xP4

,z

),

H7 =14

(P3

,x +5U2P3

,x

W 2 − 5UP4,x

W+

5P5,x

4+

6UP2,xP,z

W−4P3

,xP,z +3P,xP2,z

+3U2P,xP2

,z

W 2 − 6UP2,xP2

,z

W+

52

P3,xP2

,z +2UP3

,z

W−4P,xP3

,z−UP4

,z

W+

54

P,xP4,z

),

H8 =15

(5UP4

,x

W− 5P5

,x

2+4P3

,xP,z +6UP2

,xP2,z

W−5P3

,xP2,z +4P,xP3

,z +UP4

,z

W− 5

2P,xP4

,z

),

H9 =16

(P5

,x +2P3,xP2

,z +P,xP4,z

),

216

L0 =12

(2

UP,x

W−P2

,x +3P,z +U2P,z

W 2 −2UP,xP,z

W+

34

P2,xP,z−3P2

,z +34

P3,z

),

L1 =13

(P2

,x +2UP,xP,z

W− 3

2P2

,xP,z +3P2,z−

32

P3,z

),

L2 =14

(P2

,xP,z +P3,z),

L3 =12

(4U3P,x

W 3 +4UP,x

W−3P2

,x−11U2P2

,x

W 2 +16UP3

,x

3W− 11P4

,x

12+4P,z +

4U2P,z

W 2

− 6U3P,xP,z

W 3 − 22UP,xP,z

W+

496

P2,xP,z +

49U2P2,xP,z

6W 2 − 10UP3,xP,z

3W+

58

P4,xP,z

− 17P2,z−

9U2P2,z

W 2 +16UP,xP2

,z

W− 31

6P2

,xP2,z +

27P3,z

2+

17U2P3,z

6W 2 − 10UP,xP3,z

3W

+54

P2,xP3

,z−17P4

,z

4+

5P5,z

8

),

L4 =12

(8U3P,x

W 3 +8UP,x

W−5P2

,x−17U2P2

,x

W 2 +25UP3

,x

3W− 17P4

,x

12+9P,z +

U4P,z

W 4

+10U2P,z

W 2 − 10U3P,xP,z

W 3 − 34UP,xP,z

W+

383

P2,xP,z +

38U2P2,xP,z

3W 2 − 16UP3,xP,z

3W

+1516

P4,xP,z−27P2

,z−15U2P2

,z

W 2 +25UP,xP2

,z

W− 49

6P2

,xP2,z +21P3

,z +13U2P3

,z

3W 2

− 16UP,xP3,z

3W+

158

P2,xP3

,z−27P4

,z

4+

15P5,z

16

),

L5 =13

(3P2

,x +11U2P2

,x

W 2 − 12UP3,x

W+

35P4,x

12+

6U3P,xP,z

W 3 +22UP,xP,z

W−18P2

,xP,z

− 18U2P2,xP,z

W 2 +34UP3

,xP,z

3W− 5

2P4

,xP,z +17P2,z +

9U2P2,z

W 2 − 36UP,xP2,z

W

+1036

P2,xP2

,z−30P3,z−

6U2P3,z

W 2 +34UP,xP3

,z

3W−5P2

,xP3,z +

57P4,z

4− 5P5

,z

2

),

L6 =13

(5P2

,x +17U2P2

,x

W 2 − 18UP3,x

W+

53P4,x

12+

10U3P,xP,z

W 3 +34UP,xP,z

W−27P2

,xP,z

− 27U2P2,xP,z

W 2 +52UP3

,xP,z

3W− 15

4P4

,xP,z +27P2,z +

15U2P2,z

W 2 − 54UP,xP2,z

W

+1576

P2,xP2

,z−45P3,z−

9U2P3,z

W 2 +52UP,xP3

,z

3W− 15

2P2

,xP3,z +

87P4,z

4− 15P5

,z

4

),

217

L7 =14

(2UP3

,x

W−P4

,x +3P2,xP,z +

3U2P2,xP,z

W 2 − 4UP3,xP,z

W+

54

P4,xP,z +

6UP,xP2,z

W

− 6P2,xP2

,z +5P3,z +

U2P3,z

W 2 − 4UP,xP3,z

W+

52

P2,xP3

,z−5P4,z +

5P5,z

4

),

L8 =15

(P4

,x +4UP3

,xP,z

W− 5

2P4

,xP,z +6P2,xP2

,z +4UP,xP3

,z

W−5P2

,xP3,z +5P4

,z−5P5

,z

2

),

L9 =16

(P4

,xP,z +2P2,xP3

,z +P5,z

).

11.5 Appendix V

X1 =1

1−δ2 , X2 =−δ2

1−δ2 , X3 =−43

X 32 , X4 =

−12

Z 22 X2,

X5 = 8ln(δ)Z 21 X 2

2 +1+δ2

2δ2 Z 22 X2− 4

3(1−δ6)

δ6 X 42 ,

X6 = −8ln(δ)Z 21 X 2

2 +1

2δ2 Z 22 X2 +

43

(1−δ4)δ6 X 4

2 ,

X7 = 8Z 21 X2, X8 =

−365

X 22 X3, X9 = 2Z2Z3X2− 3

4Z 2

2 X3−6X 22 X4,

X10 =163

Z1Z3X2 +43Z2Z4X2−4Z1Z2X3− −2

3Z 2

2 X4−4X 22 X5

X11 = 4Z1Z4X2−4Z1Z2X4−Z2Z6X2−6Z 21 X3− 1

2Z 2

2 X5 +3X 22 X7

X12 =−4(1+δ2 +δ4)

3δ4

(4Z1Z3X2 +Z2Z4X2 +3Z1Z2X3 +

12Z 2

2 X4 +3X 22 X5

)

+ 2δ2 (8Z1Z8X2−Z 2

1 X7)+

(1+δ2)(1+δ4)δ6

(2Z2Z3X2 +

34Z 2

2 X3 +6X 22 X4

)

+36(1+δ2 +δ4 +δ6 +δ8)

5δ8 X 22 X3 +

(1+δ2)δ2 (−4Z1Z4X2 +Z2Z6X2

+ 6Z 21 X3 +4Z1Z2X4 +

12Z 2

2 X5−3X 22 X7

)+

16δ2 ln(δ)(−1+δ2)

(Z1Z7X2

+ Z2Z8X2− 14Z1Z2X7 +

12Z 2

1 X5

),

218

X13 =4(1+δ2)

3δ4

(4Z1Z3X2 +Z2Z4X2−3Z1Z2X3− 1

2Z 2

2 X4−3X 22 X5

)

+36(1+δ2)(1+δ4)

5δ8 X 22 X3 +

(1+δ2 +δ4)δ6

(2Z2Z3X2− 3

4Z 2

2 X3 +6X 22 X4

)

− 2(1+δ2)(8Z1Z8X2−Z 2

1 X7)+

1δ2 (−4Z1Z4X2−Z2Z6X2

+ 6Z 21 X3 +4Z1Z2X4− 1

2Z 2

2 X5 +3X 22 X7

)+

16δ2 ln(δ)−1+δ2 (−Z1Z7X2

− Z2Z8X2− 12Z 2

1 X5 +14Z1Z2X7

),

X14 = 16Z1Z7X2 +16Z2Z8X2 +8Z 21 X5−4Z1Z2X7,

X15 = 16Z1Z8X2−2Z 21 X7.

Z1 =λ4, Z2 =

Z1(1−δ2)ln(δ)

, Z3 =32Z2X

22 , Z4 =

12

(Z 3

2 +16Z1X2

2),

Z5 =12

(4Z 3

1 +12Z 21 Z2−Z 3

2 −16Z1X2

2 −3Z2X2

2)

Z6 =(−1+δ2)

ln(δ)

(2(1+δ2)Z 3

1 +6Z 21 Z2 +

12δ2 Z 3

2 +8δ2 Z 2

1 X 22 +

3(1+δ2)2δ4 Z2X

22

)

Z7 = −6Z 21 Z2, Z8 =−2Z 3

1 , Z9 =−2027

Z 22 Z3,

Z10 =5X2

4(−2Z3X2 +3Z2X3) ,

Z11 =−6Z2

49(40Z1Z3 +3Z2Z4) ,

Z12 =19

(−3Z 22 Z3−16Z4X

22 −96Z1X2X3 +32Z2X2X4

),

Z13 =−325

(80Z 2

1 Z3 +24Z1Z2Z4−Z 22 Z6

),

Z14 =38

(−Z 22 Z4 +4Z6X

22 +8Z1Z2Z3 +32Z1X2X4 +8Z2X2X5

),

Z15 =−23

(12Z 2

1 Z4−2Z1Z2Z6 +Z 22 Z7

),

Z16 = 36Z 21 Z3 +

34Z 2

2 Z6 +8Z7X2

2 +6Z1Z2Z4 +16Z1X2X5−4Z2X2X7

Z17 = 12(Z 2

1 Z6−2Z1Z2Z7−3Z 22 Z8

),

219

Z18 = Z 22

(2927

Z3 +291392

Z4− 87100

Z6 +23Z7 +33Z8

)+

118

(45Z3 +32Z4

+ 27Z6−144Z7)X 22 +Z 2

1

(−132

5Z3 +8Z4−3Z6 +

572

Z7 +20Z8

)

− 136

(135X3 +128X4 +108X5−144X7)Z2X2 +(

9349

Z3− 10825

Z4

)Z1Z2

− 43

(8X3 +9X4 +12X5)Z1X2− 13

(4Z6−90Z7−441Z8)Z1Z2,

Z19 =12

(4−15δ3 +11δ5)

5δ5 ln(δ)Z 2

1 Z3−3(−80+49δ3 +31δ7)

49δ7 ln(δ)Z1Z2Z3

+

(20+9δ3−29δ9)

27δ9 ln(δ)Z 2

2 Z3−(−2+2δ3 +9δ3 ln(δ)2)

3δ3 ln(δ)(12Z 2

1 Z4

− 2Z1Z2Z6−Z 22 Z7)+

6(12−25δ3 +13δ5)

25δ5 ln(δ)Z1Z2Z4

+3(48+49δ3−97δ7)

392δ7 ln(δ)Z 2

2 Z4 +3(−4+δ+3δ3)

δ ln(δ)Z 2

1 Z6

+3(−4−25δ3 +29δ5)

100δ5 ln(δ)Z 2

2 Z6 +3(−19+16δ+3δ4)

2ln(δ)Z 2

1 Z7

+6(4−5δ+δ3)

δ ln(δ)Z1Z2Z7 +

4(−5+4δ3 +δ6)

ln(δ)Z 2

1 Z8

+3(−49+48δ+δ4)

ln(δ)Z1Z2Z8−

3(−12+11δ+δ3)

δ ln(δ)Z 2

2 Z8

− 5(−1+δ8)

4δ8 ln(δ)(2Z3X

22 −3Z2X2X3)−

16(−1+δ6)

9δ6 ln(δ)(Z4X

22 +6Z1X2X3

+ 2Z2X2X4)+3(−1+δ4)

2δ4 ln(δ)(Z6X

22 +8Z1X2X4 +2Z2X2X5)

+8(−1+δ2)

δ2 ln(δ)(Z7X

22 +2Z1X2X5)+

4(2δ2 ln(δ)2Z1−δ2Z2 +Z2

)

δ2 ln(δ)X2X7

− 16ln(δ)Z8X2

2 ,

Z20 = 36Z 21 Z4 +3Z 2

2 Z7 +16Z8X2

2 −6Z1Z2Z6−8Z1X2X7,

Z21 = −24Z1 (Z1Z7 +6Z2Z8) ,

Z22 = −9Z 21 Z6−6Z1Z2Z7 +3Z 2

2 Z8,

Z23 = −16Z 21 Z8, Z24 =

−32

(3Z 2

1 Z7 +2Z1Z2Z8), Z25 =−4Z 2

1 Z8.

220

11.6 Appendix VI

Θ1 =δ2(2+ ln(δ)P,θ)

2(−1+δ2), Θ2 =−(2+δ2 ln(δ)P,θ)

2(−1+δ2), Θ3 =

P,θ

2,

Θ4 = −4Θ31

3, Θ5 =−1

2Θ1(Ψ2

2−6Θ1Θ3),

Θ6 =4

3δ4 (1+δ2 +δ4)Θ31−2δ2Ψ2

1Θ3 +δ2 ln(δ)

(−1+δ2)(8Ψ2

1Θ1−4Ψ1Ψ2Θ3−Θ33)

+3(1+δ2)

δ2 Θ21Θ3 +

(1+δ2)2δ2 Ψ2

2Θ1,

Θ7 = − 12δ2 Ψ2

2Θ1 +2(1+δ2)

3δ4 (−2Θ31 +3δ4Ψ2

1Θ3)+3δ2 Θ2

1Θ3

+δ2 ln(δ)

(−1+δ2)(−8Ψ2

1Θ1 +4Ψ1Ψ2Θ3 +Θ33),

Θ8 = 8Ψ21Θ1−4Ψ1Ψ2Θ3−Θ3

3, Θ9 =−2Ψ21Θ3, Θ10 =−36

5Θ2

1Θ4,

Θ11 = 2Ψ2Ψ23Θ1− 3

4Ψ2

2Θ4 +6Θ1Θ3Θ4−6Θ21Θ5,

Θ12 =13(4Ψ2(Ψ4Θ1−Ψ3Θ3)+4Ψ1(4Ψ3Θ1−3Ψ2Θ4)−2Ψ2

2Θ5

− 3(3Θ23Θ4−8Θ1Θ3Θ5 +4Θ2

1Θ6)),

Θ13 =12(−Ψ2((2Ψ6 +Ψ7)Θ1 +2Ψ4Θ3)−12Ψ2

1Θ4−8Ψ1(−Ψ4Θ1

+ Ψ3Θ3 +Ψ2Θ5)−Ψ22Θ6−6Θ3(Θ3Θ5−2Θ1Θ6)−6Θ2

1Θ8),

Θ14 = −2Ψ2Ψ7Θ1,

Θ15 =1+δ2 +δ4

3δ4 (−16Ψ1Ψ3Θ1 +4Ψ2Ψ3Θ3 +12Ψ1Ψ2Θ4 +2Ψ22Θ5 +9Θ2

3Θ4

− 24Θ1Θ3Θ5−4Ψ2Ψ4Θ1 +12Θ21Θ6)+

(1+δ2)(1+δ4)δ6 (−2Ψ2Ψ3Θ1

− 9Θ1Θ3Θ4 +3Ψ2

2Θ4

4+6Θ2

1Θ5)+1+δ2

δ2 (−4Ψ1Ψ4Θ1 +Ψ2Ψ6Θ1 +Ψ2Ψ7Θ1

2

+ 4Ψ1Ψ3Θ3 +Ψ2Ψ4Θ3 +6Ψ21Θ4 +4Ψ1Ψ2Θ5 +3Θ2

3Θ5 +Ψ2

2Θ6

2−6Θ1Θ3Θ6

− 3Θ21Θ8)+16δ2Ψ1Ψ9Θ1−4δ2Ψ1Ψ8Θ3−4δ2(1+δ2)Ψ1Ψ9Θ3

− 4δ2Ψ2Ψ9Θ3 +36(1+δ2 +δ4 +δ6 +δ8)Θ2

1Θ4

5δ8 −2δ2Ψ21Θ8−2δ2(1+δ2)Ψ2

1Θ9

− 4δ2Ψ1Ψ2Θ9−3δ2Θ23Θ9, Θ16 = 2Ψ7(−4Ψ1Θ1 +Ψ2Θ3),

221

Θ17 =1+δ2

3δ4 (16Ψ1Ψ3Θ1 +4Ψ2Ψ4Θ1−4Ψ2Ψ3Θ3−12Ψ1Ψ2Θ4−9Θ23Θ4

− 2Ψ22Θ5 +24Θ1Θ3Θ5−12Θ2

1Θ6)+(1+δ2 +δ4)

δ6 (2Ψ2Ψ3Θ1− 3Ψ22Θ4

4

− 6Θ21Θ5 +9Θ1Θ3Θ4)+

1δ2 (4Ψ1Ψ4Θ1−Ψ2Ψ7Θ1

2−4Ψ1Ψ3Θ3−Ψ2Ψ4Θ3

− 6Ψ21Θ4−4Ψ1Ψ2Θ5−3Θ2

3Θ5−Ψ2

2Θ6

2+6Θ1Θ3Θ6 +3Θ2

1Θ8)

− 16(1+δ2)Ψ1Ψ9Θ1 +4(1+δ2)Ψ1Ψ8Θ3 +4(1+δ2 +δ4)Ψ1Ψ9Θ3

− Ψ2Ψ6Θ1 +4(1+δ2)Ψ2Ψ9Θ3− 36(1+δ2)(1+δ4)Θ21Θ4

5δ8

+ 2(1+δ2)Ψ21Θ8 +2(1+δ2 +δ4)Ψ2

1Θ9 +4(1+δ2)Ψ1Ψ2Θ9

+ 3(1+δ2)Θ23Θ9 +

2ln(δ)(−1+δ2)

(Ψ2Θ1

δ2 −Ψ2Θ3 +4Ψ1Θ1)Ψ7

+δ2 ln(δ)

(−1+δ2)(−16Ψ1Ψ8Θ1−16Ψ2Ψ9Θ1 +4Ψ1Ψ6Θ3 +4ln(δ)Ψ1Ψ7Θ3

+ 4Ψ2Ψ8Θ3−8Ψ21Θ6 +4Ψ1Ψ2Θ8 +3Θ2

3Θ8 +2Ψ22Θ9−24Θ1Θ3Θ9),

Θ18 = 4Ψ2(4Ψ9Θ1−Ψ8Θ3)+8Ψ21Θ6−4Ψ1(−4Ψ8Θ1 +Ψ6Θ3 +Ψ2Θ8)−2Ψ2

2Θ9

− 3Θ3(Θ3Θ8−8Θ1Θ9), Θ19 =−4Ψ1Ψ7Θ3,

Θ20 = −2Ψ21Θ8−4Ψ1(−4Ψ9Θ1 +Ψ8Θ3 +Ψ2Θ9)−Θ3(4Ψ2Ψ9 +3Θ3Θ9),

Θ21 = −2Ψ1(2Ψ9Θ3 +Ψ1Θ9).

Ψ1 =−P,z

4, Ψ2 =

P,z(1−δ2)4ln(δ)

, Ψ3 =32

Ψ2Θ21,

Ψ4 =Ψ3

22

+8Ψ1Θ21−4Ψ2Θ1Θ3,

Ψ5 = 2Ψ31 +6Ψ2

1Ψ2− 12

Ψ32−8Ψ1Θ2

1−32

Ψ2Θ21 +4Ψ2Θ1Θ3,

Ψ6 =(−1+δ4)

2ln(δ)(4Ψ3

1 +3Ψ2Θ2

1δ4 )+

(−1+δ2)δ2 ln(δ)

(6δ2Ψ21Ψ2 +

12

Ψ32 +8Ψ1Θ2

1−4Ψ2Θ1Θ3)

+ (8Ψ1Θ1−Ψ2Θ3)Θ3 ln(δ), Ψ7 =−8Ψ1Θ1Θ3 +Ψ2Θ23, Ψ8 =−6Ψ2

1Ψ2,

Ψ9 = −2Ψ31, Ψ10 =−20

27Ψ2

2Ψ3, Ψ11 =−52

Ψ3Θ21 +

154

Ψ2Θ1Θ4,

Ψ12 = −24049

Ψ1Ψ2Ψ3− 1849

Ψ22Ψ4,

Ψ13 = −13

Ψ22Ψ3− 16

9Ψ4Θ2

1 +329

Ψ3Θ1Θ3 +323

Ψ1Θ1Θ4− 83

Ψ2Θ3Θ4 +329

Ψ2Θ1Θ5,

Ψ14 = −485

Ψ21Ψ3− 72

25Ψ1Ψ2Ψ4 +

325

Ψ22Ψ6− 18

125Ψ2

2Ψ7, Ψ15 =625

Ψ22Ψ7,

222

Ψ16 = 3Ψ1Ψ2Ψ3− 38

Ψ22Ψ4 +

32

Ψ6Θ21 +Ψ7Θ2

1 +3Ψ4Θ1Θ3− 32

Ψ3Θ23−9Ψ1Θ3Θ4

+ 12Ψ1Θ1Θ5−3Ψ2Θ3Θ5 +3Ψ2Θ1Θ6,

Ψ17 = 3Ψ7Θ21, Ψ18 =−8Ψ2

1Ψ4 +43

Ψ1Ψ2Ψ6− 89

Ψ1Ψ2Ψ7− 23

Ψ22Ψ8,

Ψ19 =83

Ψ1Ψ2Ψ7,

Ψ20 = 36Ψ21Ψ3 +6Ψ1Ψ2Ψ4 +

34

Ψ22Ψ6 +

32

Ψ22Ψ7 +8Ψ8Θ2

1−4Ψ6Θ1Θ3−6Ψ7Θ1Θ3

− 2Ψ4Θ23−16Ψ1Θ3Θ5 +16Ψ1Θ1Θ6−4Ψ2Θ3Θ6−4Ψ2Θ1Θ8,

Ψ21 =32

Ψ22Ψ7−8Ψ7Θ1Θ3,

Ψ22 = 12Ψ21Ψ6 +24Ψ2

1Ψ7−24Ψ1Ψ2Ψ8−36Ψ22Ψ9, Ψ23 = 24Ψ2

1Ψ7,

Ψ24 = −1325

Ψ21Ψ3 +

9349

Ψ1Ψ2Ψ3 +2927

Ψ22Ψ3 +8Ψ2

1Ψ4− 7825

Ψ1Ψ2Ψ4 +291392

Ψ22Ψ4

− 3Ψ21Ψ6− 4

3Ψ1Ψ2Ψ6− 87

100Ψ2

2Ψ6− (42Ψ21−

89

Ψ1Ψ2 +339250

Ψ22)Ψ7

+ (572

Ψ21 +30Ψ1Ψ2 +

23

Ψ22)Ψ8 +(20Ψ2

1 +147Ψ1Ψ2 +33Ψ22)Ψ9 +(

52

Ψ3 +169

Ψ4

− 32

Ψ6−Ψ7−8Ψ8)Θ21− (

329

Ψ3 +3Ψ4−4Ψ6−6Ψ7)Θ1Θ3 +(32

Ψ3 +2Ψ4

+12

Ψ9)Θ23− (

323

Ψ1 +154

Ψ2)Θ1Θ4 +(9Ψ1 +83

Ψ2)Θ3Θ4

− (12Ψ1− 329

Ψ2)Θ1Θ5 +(16Ψ1 +3Ψ2)Θ3Θ5− (16Ψ1 +3Ψ2)Θ1Θ6

+ (4Θ3Θ6 +4Θ1Θ8)Ψ2 +Ψ1Θ3Θ9,

Ψ25 =1

δ9 ln(δ)(12(4+11δ5)δ4Ψ2

1Ψ3

5− 3Ψ1(588Ψ1 +31δ2Ψ2)δ7Ψ3

49

+3(80−49δ3)δ2Ψ1Ψ2Ψ3

49+

(20+9δ3−29δ9)δ9Ψ22Ψ3

27

− 4(−2+2δ3 +9δ3 ln(δ)2)δ6Ψ21Ψ4 +

6(12−25δ3 +13δ5)δ4Ψ1Ψ2Ψ4

25

+3(48+49δ3−97δ7)δ2Ψ2

2Ψ4

392+3(−4+δ+3δ3)δ8Ψ2

1Ψ6

+2(−2+2δ3 +9δ3 ln(δ)2)δ6Ψ1Ψ2Ψ6

3+

3(−4−25δ3 +29δ5)δ4Ψ22Ψ6

100+ 6(−4+7δ−3δ3 +(−4+3δ3) ln(δ))δ8Ψ2

1Ψ7

223

+4(2−2δ3−6ln(δ)+9δ3 ln(δ)3)δ6Ψ1Ψ2Ψ7

9

+3(12−125δ3 +113δ5−5(4+25δ3) ln(δ))δ4Ψ2

2Ψ7

250

+3(−19+16δ+3δ4)δ9Ψ2

1Ψ8

2+6(4−5δ+δ3)δ8Ψ1Ψ2Ψ8

− (−2+2δ3 +9δ3 ln(δ)2)δ6Ψ22Ψ8

3+4(−5+4δ3 +δ6)δ9Ψ2

1Ψ9

+ 3(−49+48δ+δ4)δ9Ψ1Ψ2Ψ9−3(−12+11δ+δ3)δ8Ψ22Ψ9

+ 2(3−3δ2 +4ln(δ))δ7Ψ7Θ1Θ3 +(−1+δ4−3ln(δ))δ5Ψ7Θ21)

+5(−1+δ8)4δ8 ln(δ)

(−2Ψ3Θ21 +3(−1+δ8)Ψ2Θ1Θ4)+

8(−1+δ6)9δ6 ln(δ)

(−2Ψ4Θ21

+ 4Ψ3Θ1Θ3 +12Ψ1Θ1Θ4−3Ψ2Θ3Θ4 +4Ψ2Θ1Θ5)+(−1+δ4)2δ4 ln(δ)

(3Ψ6Θ21

+ 6Ψ4Θ1Θ3−3Ψ3Θ23 +δ4Ψ9Θ2

3−18Ψ1Θ3Θ4 +24(−1+δ4)Ψ1Θ1Θ5

− 6(−1+δ4)Ψ2Θ3Θ5 +6(−1+δ4)Ψ2Θ1Θ6 +2δ4Ψ1Θ3Θ9)

+2(−1+δ2)

δ2 ln(δ)(4Ψ8Θ2

1−2Ψ6Θ1Θ3−Ψ4Θ23−8Ψ1Θ3Θ5 +8Ψ1Θ1Θ6−2Ψ2Θ3Θ6

− 2Ψ2Θ1Θ8)−16ln(δ)Ψ9Θ21 +8ln(δ)Ψ8Θ1Θ3− ln(δ)Ψ6Θ2

3

+13(3−2ln(δ)) ln(δ)Ψ7Θ2

3 +8ln(δ)Ψ1Θ3Θ6 +8ln(δ)Ψ1Θ1Θ8

− 2ln(δ)Ψ2Θ3Θ8 +8ln(δ)Ψ2Θ1Θ9,

Ψ26 = 36Ψ21Ψ4−6Ψ1Ψ2Ψ6 +3Ψ2

2Ψ8 +16Ψ9Θ21−8Ψ8Θ1Θ3 +Ψ6Θ2

3−Ψ7Θ23

− 8Ψ1Θ3Θ6−8Ψ1Θ1Θ8 +2Ψ2Θ3Θ8−8Ψ2Θ1Θ9,

Ψ27 = −4Ψ1Ψ2Ψ7 +23

Ψ7Θ23, Ψ28 =−24Ψ2

1Ψ8−144Ψ1Ψ2Ψ9,

Ψ29 = −9Ψ21Ψ6 +18Ψ2

1Ψ7−6Ψ1Ψ2Ψ8 +3Ψ22Ψ9, Ψ30 =−18Ψ2

1Ψ7,

Ψ31 = −16Ψ21Ψ9, Ψ32 =−9

2Ψ2

1Ψ8−3Ψ1Ψ2Ψ9− 12

Ψ9Θ23−Ψ1Θ3Θ9,

Ψ33 = −4Ψ21Ψ9.

224

11.7 Appendix VII

N1 =δ2(1+P,θ ln(δ))

1−δ2 , N2 =1+P,θδ2 ln(δ)

1−δ2 , M1 =P,z

2, M2 =

P,z(1−δ2)2ln(δ)

,

N3 = 4N31 , N4 = M2

2N1 +8P,θN21 , N5 = M2

2P,θ +4M1M2N1 +5P2,θN1,

N6 = 4M1M2P,θ +P3,θ +4M2

1N1, N7 = 4M21P,θ,

N8 =12

(N3(1+δ2 +δ4)

3δ4 +N4(1+δ2)

2δ2 +N5 +2N6δ2 ln(δ)

δ2−1+N7δ2

),

N9 =(

N3(1+δ2)6δ4 +

N4

4δ2 +N6δ2 ln(δ)

δ2−1+

N7

2(1+δ2)

),

M3 = 4M2N21 , M4 = M3

2 +4M2P,θN1 +4P,zN21 ,

M5 = 4M1M22 +M2P2

,θ +4P,θP,zN1, M6 = 4M21M2 +4M1M2P,z +P2

,θP,z, M7 = 4M21P,z,

M8 =1

4ln(δ)

(M3(

1δ4 −1)+2M4(

1δ2 −1)−4M5 ln(δ)

− 2M6(δ2−1)−M7(δ4−1)), M9 =

14

(M3 +2M4−2M6−M7) ,

N10 =(8N2

1 N3), N11 =

(4M2M3N1 +8K1N1N3 +8N2

1 N4),

N12 =(2K1M2M3 +8M1M3N1 +4M2M4N1 +2K2

1 N3 +8K1N1N4

+ 8N21 N5−24N2

1 N8),

N13 = (4K1M1M3 +2K1M2M4 +8M1M4N1 +4M2M5N1 +4M2M8N1

+ 2K21 N4 +8K1N1N5 +8N2

1 N6−2M22N8−24K1N1N8

),

N14 = (4K1M1M4 +2K1M2M5 +2K1M2M8 +8M1M5N1 +4M2M6N1

+ 8M1M8N1 +2K21 N5 +8K1N1N6 +8N2

1 N7−6K21 N8−8M1M2N8

),

N15 =(4K1M1M5 +2K1M2M6 +4K1M1M8 +8M1M6N1 +4M2M7N1 +2K2

1 N6

+ 8K1N1N7−8M21N8

),

N16 =(4K1M1M6 +2K1M2M7 +8M1M7N1 +2K2

1 N7), N17 = (4K1M1M7) ,

225

N18 =2δ2

1−δ2

(N10

10

(1− 1

δ10

)+

N11

8

(1− 1

δ8

)+

N12

6

(1− 1

δ6

)+

N13

4

(1− 1

δ4

)

+N14

2

(1− 1

δ2

)+N15 lnδ+

N16

2(δ2−1

)+

N17

4(δ4−1

)),

N19 =1

120(12N10 +15N11 +20N12 +30N13 +60N14−60N16−30N17) ,

M10 = (4M2N1N3) , M11 =(2M2

2M3 +2K1M2N3 +4K2N1N3 +4M2N1N4),

M12 =(2K2M2M3 +4M1M2M3 +2M2

2M4 +4M8N21 +2K1K2N3 +2K1M2N4

+ 4K2N1N4 +4M2N1N5−8M2N1N8) ,

M13 =(4K2M1M3 +2K2M2M4 +4M1M2M4 +2M2

2M5 +3M22M8 +4K1M8N1

+ 2K1K2N4 +2K1M2N5 +4K2N1N5 +4M2N1N6−4K1M2N8−8K2N1N8) ,

M14 =(4K2M1M4 +2K2M2M5 +4M1M2M5 +2M2

2M6 +K21 M8 +2K2M2M8

+ 8M1M2M8 +2K1K2N5 +2K1M2N6 +4K2N1N6 +4M2N1N7−4K1K2N8) ,

M15 =(4K2M1M5 +2K2M2M6 +4M1M2M6 +2M2

2M7 +4K2M1M8 +4M21M8

+ 2K1K2N6 +2K1M2N7 +4K2N1N7) ,

M16 = (4K2M1M6 +2K2M2M7 +4M1M2M7 +2K1K2N7) , M17 = (4K2M1M7) ,

M18 =1

lnδ

(M10

8

(1− 1

δ8

)+

M11

6

(1− 1

δ6

)+

M12

4

(1− 1

δ4

)+

M13

2

(1− 1

δ2

)

+ M14 lnδ+M15

2(δ2−1

)+

M16

4(δ4−1

)+

M17

6

(δ6−1

)),

M19 =(

M10

8+

M11

6+

M12

4+

M13

2−M15

2−M16

4−M17

6

),

226

Chapter 12

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