modelling of non-newtonian fluid problems and their solutions

MODELLING OF NON-NEWTONIAN FLUID

PROBLEMS AND THEIR SOLUTIONS

By

Rehan Ali Shah

CIIT/SP09-PMT-011/ISB

Ph.D. Thesis

in

Mathematics

COMSATS Institute of Information Technology,

Islamabad, Pakistan

Fall, 2011

ii

COMSATS Institute of Information Technology



A thesis presented to

COMSATS Institute of Information Technology, Islamabad

in partial fulfillment

of the requirement for the degree of

Ph.D. (Mathematics)

By

Rehan Ali Shah


Fall, 2011

iii



___________________________________________

A post graduate thesis submitted to the Department of Mathematics as partial

fulfillment of the requirement for the award of degree of Ph.D. (Mathematics).

Name Registration Number

Rehan Ali Shah CIIT/SP09-PMT-011/ISB

Supervisor

Prof. Dr. Tahira Haroon

Department of Mathematics,

Islamabad Campus,

COMSATS Institute of Information Technology (CIIT).

January, 2012

iv

Final Approval

This thesis titled



By

Rehan Ali Shah


Has been approved

For the COMSATS Institute of Information Technology, Islamabad

External Examiner: ___________________________________

Prof. Dr. Tahir Mahmood

Department of Mathematics, IUB, Bahawalpur

External Examiner: ___________________________________

Prof. Dr. Siraj-ul-Islam

Department of Basic Sciences & Islamiat, UET, Peshawar

Supervisor: ______________________________________


Department of Mathematics, CIIT, Islamabad

HoD: ______________________________________________

Dr. Moiz-ud-Din Khan


Dean, Faculty of sciences ____________________________________

Dr. Arshad Saleem Bhatti

v

Declaration

I, Rehan Ali Shah registration number CIIT/SP09-PMT-011/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: _________________

___________________

Rehan Ali Shah


vi

Certificate

It is certified that Rehan Ali Shah registration number CIIT/SP09-PMT-011/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 Ph.D. degree.

Date: _________________

Supervisor:

_____________________


Department of Mathematics,

CIIT, Islamabad

Head of Department:

_____________________________

Dr. Moiz-ud-Din Khan

Associate Professor,


vii

Dedicated

To

My parents, wife, children

And

Prof. Dr. A.M. Siddiqui

viii

ACKNOWLEDGEMENTS

Primarily and foremost, all praise for ALMIGHTY ALLAH, the benevolent and

merciful, the creator of the universe, who provided me the apt ability, strength and

courage to complete the work presented here. Many many thanks to Him as He blessed us

with the Holy Prophet, HAZRAT MUHAMMUD (PBUH) for whom the whole

universe is created and Who enabled us to Worship only to one God. He (PBUH) brought

us out of darkness and enlightened the way to heaven.

I would like to acknowledge the help I have received from many people throughout my

studies. First of all, I want to express my most sincere thanks to my devoted ex-

supervisor, Dr. Saeed Islam for his many valuable ideas, encouraging discussions,

capable guidance and consideration throughout the course of this research. Without his

patient guidance, it was not possible for me to complete this work. I would like to take

this opportunity to express my heartiest gratitude to him for all his help and sincere

friendship in all my years here.

A special thanks to my supervisor Prof. Dr. Tahira Haroon chairperson department of

Mathematics, COMSATS Institute of Information Technology for her valuable

assistance, heartfelt guidance, cooperation and for taking the time to examine this work

and for all of her valuable suggestions, without which this work is incomplete.

I also wish to express my deepest gratitude to Prof. Dr. Abdul Majeed Siddiqui

working as a Professor in Pennsylvania State University, York Campus, USA. I am

indebted to him for his able guidance, immense encouragement, limitless patience which

enable me in broadening and improving my capabilities. I learn a lot from him and owe

him deep thanks for helping in research work.

I would also like to express my appreciation for the Higher Education Commission

of Pakistan for providing me full financial support under the Indigenous 5000

Scholarship Batch-IV without which this study is not possible for me.

Thanks to COMSATS Institute of Information Technology, HOD department of

Mathematics and Dr. S. M. Junaid Zaidi, Rector CIIT, for offering me all essential

facilities and research environment at CIIT Islamabad. I am grateful particularly to Prof.

ix

Dr Saleem Asghar, for helping me to learn perturbation theory. I express my deep sense

of gratitude to Dr Muhammad Akram for their valuable guidance.

I would certainly not be where I stand today without the continuous support, help and

most of all encouragement of my family. Especially my utmost appreciation to my

parents without their unwavering support, encouragement would not have been possible.

I can not finish without expressing my feelings for my wife and sweet daughters

Javeria, Adeena and Aleeza, who suffered a lot due to my involvement in Ph. D work.

Finally, I would like to acknowledge the pleasant moments shared with my friends

and all my well wishers, they have all been grate help to me in their sincere support. I

also express my regards to all my school and college teachers who motivated me to do

well in my studies.

Rehan Ali Shah


x

ABSTRACT

MODELLING OF NON-NEWTONIAN FLUID PROBLEMS

AND THEIR SOLUTIONS

The thesis presents the theoretical analysis of wire coating extrusion process inside

pressure type die. Efforts at obtaining better insight into the process must be mainly

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

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

process and of product quality, higher rates and more accurate and less costly die design.

In this thesis, two types of problems have been studied, (i) problems within the die and

(ii) problems outside the die. The studies are performed with several elastic fluid models

such as Phan-Thien and Tanner, second grade, third grade, elastico-viscous and Oldroyd-

8-constant fluid models and for the inelastic power law fluid model. There are ten

chapters in this thesis.

Chapter 1 is introductory and discusses mainly mathematical modeling, wire coating

operation and the phenomena inbuilt to it, in detail. In addition, it discusses the physical

properties of the non-Newtonian fluids that have been considered here. Finally, it deals

with the literature related to the analysis of coating process.

Chapter 2 is concerned with the study of non-isothermal PTT fluid in wire coating

analysis in a finite length pressure type die. The analysis is carried out by neglecting the

exit and entrance effects. The expressions for axial velocity, average velocity, volume

flow rate, shear and normal stresses, thickness of coated wire, force on the total surface of

wire and the temperature distribution are obtained. The effects of the Deborah number,

Brinkman number, elongation parameter and the ratio of the pressure drop to that of

velocity of fluid are discussed. A domain for 2

cDe is found such that outside this domain,

the shear and normal stresses show insignificant effects.

Chapter 3 is devoted to the study of wire coating for heat transfer flow of a viscoelastic

PTT fluid with slip boundary conditions. The investigations are carried out by

considering nonzero pressure gradient in the axial direction. The wall shear stress, flow

analysis and the role of slip parameter are the areas of investigation. The effect of 2

cDe

xi

and the slip parameter on the velocity of melt polymer, volume flow rate, thickness of

coated wire, shear and normal stresses and on temperature distributions are studied. It is

observed that the shear stress across the gap must follows a linear variation irrespective

of the constitutive equation but its magnitude depends on the model parameters. In case

of normal stress, this reduction is in the form of parabolic and the profiles overshoot at

the centre of the annulus.

Chapter 4 is to explore the wire coating analysis in a pressure type die by considering

third grade fluid for constant and variable viscosity depends on temperature. For

temperature dependent viscosity, two models are under discussion (i) Reynolds model

and (2) Vogel’s model. The coupled momentum and energy equations are solved with the

help of regular perturbation method. The non-Newtonian behavior of the fluid is

discussed with the influence of perturbation parameter. Also, the solution of the problem

is discussed for different Reynolds and Vogel’s model parameters.

Chapter 5 is targeted to study the wire coating with a bath of Oldroyd 8-constant fluid

taking into account the effect of pressure variation in the axial direction. The influence of

pseudoplastic and dilatant parameters is investigated on the flow behavior such as

velocity, average velocity, volume flow rate and shear stress of the fluid and on the

temperature distributions. Also the influence of pressure gradient and the drag flow are

examined. Furthermore, the effect of viscosity parameter 0 is discussed on shear stress.

The aim of chapter 6 is to investigate an unsteady flow of a second grade fluid in a

cylindrical die of finite length. In this problem, wire is dragged in a pool of melt polymer

in the axial direction inside the die. The pressure gradient along the flow direction is

assumed to be zero. The flow phenomena satisfying the continuity equation are modeled

mathematically with the help of Navier Stokes equations and solutions for velocity

distribution is derived in two different cases (i) when the wire is dragged in the molten

polymer and (ii) when the wire is dragged with cosine oscillation in the melt polymer in a

die. An exact solution is obtained in case (i) and an Optimal Homotopy Asymptotic

Method (OHAM) is applied for handling solution of the problem in case (ii). The velocity

field has been examined with passage of time and the effect of oscillation is investigated

in the region of fluid flow. The stability analysis of this technique is discussed on some

examples related to the problem under discussion.

xii

Chapter 7 gives an analytical investigation of post-treatment of wire coating with heat

transfer analysis. The fluid is considered as third grade fluid. The investigation is

performed by considering the slippage exists at the contact surfaces of wire, polymer and

the gas. The mathematical model is derived for the fluid flow in a die. The governing

equations are solved for the velocity field and temperature distribution using the regular

Perturbation Method (PM) and OHAM. The explicit expressions for the flow rate,

average velocity, force on total surface of wire and thickness of coated wire are derived.

The solutions are examined under the effect of various parameters.

The analysis of post-treatment of wire coating with heat transfer analysis is studied in

chapter 8. The fluid is assumed to be satisfies the power law model. For temperature

distribution, three different cases have been discussed (i) temperature of the wire is

constant while it is varying linearly on the surface of the coated wire (ii) temperature of

the wire varying linearly while it is constant on the surface of the coated wire (iii)

temperature of the wire and the surface of coated wire are varying linearly at the same

temperature gradient. The analysis for velocity field, volume flow rate, average velocity,

shear rate, force on total surface wire and thickness of coated wire are carried out for the

power law index parameter n is or is not equal to 1. Temperature distribution is studied

separately in each of the three cases. The maximum temperature rise is investigated

which depends upon the non-dimensional parameter 0S .

Chapter 9 deals with the post-treatment of wire coating analysis with heat transfer

analysis. The fluid is assumed to be satisfies the elastico-viscous fluid model. The

pressure gradient is considered to be constant in the direction of drag of wire. The

analytical expressions for axial velocity, average velocity, volume flow rate, shear stress,

normal stress, thickness of coated wire, the force on the total wire and the temperature

distribution are derived by means of regular PM and Modified Homotopy Perturbation

Method (MHPM). The influences of elastic number ,eR cross-viscous number ,c

velocity ratio U and the non-dimensional parameter S are studied on the solutions of the

problem. It is concluded that an increase in the elastic number decreases, the flow rate

whereas thickness of coated wire and force on the total wire increases.

Chapter 10 is devoted to briefly review our main conclusions and future work directions.

xiii

TABLE OF CONTENTS

1. Introduction 1

1.1 Mathematical model 2

1.2 Wire coating process 2

1.2.1 Types of wire coating process 3

1.2.2 Designs of wire coating die 4

1.3 Constitutive equation 5

1.4 Non-Newtonian fluids 5

1.4.1 Time independent (Visco-inelastic) fluids 6

1.4.2 Time dependent fluids 7

1.4.3 Viscoelastic fluids 7

1.4.4 Brief comparison of non-Newtonian, Newtonian and

viscoelastic properties 8

1.5 Basic types of flows 8

1.6 Hamiltonian (quantum mechanics) 9

1.7 Perturbation theory 9

1.7.1 Perturbation theory (quantum mechanics) 9

1.7.2 Time-independent perturbation theory 10

1.7.3 Time-dependent perturbation theory 10

1.8 Reynolds model 10

1.9 Vogel’s model 10

1.10 Basic flow equations 10

1.10.1 Continuity equation 10

1.10.2 Equation of motion 11

1.10.3 Energy equation 12

1.11 Dimensionless numbers 12

1.11.1 Reynolds number 12

1.11.2 Prandtl number 12

xiv

1.11.3 Brinkman number 13

1.11.4 Elastic number 13

1.11.5 Cross-viscous number 13

1.11.6 Deborah number 13

1.12 Methods of solutions 14

1.12.1 Exact solution 14

1.12.2 Perturbation Method (PM) 14

1.12.3 Basic idea of Modified Homotopy Perturbation Method

(MHPM) 15

1.12.4 Basic idea of Optimal Homotopy Asymptotic Method

(OHAM) 16

1.13 Literature survey 19

2. Exact Solution of Non-Isothermal PTT Fluid in Wire Coating

Analysis 25

2.1 Formulation of the problem 26

2.2 Solution of the problem 31

2.3 Results and discussion 33

2.4 Conclusion 38

3. Wire Coating with Heat Transfer Analysis Flow of a Viscoelastic

PTT Fluid with Slip Conditions 39

3.1 Formulation and solution of the problem 40


3.3 Conclusion 53

4. Heat Transfer by Laminar Flow of a Third Grade Fluid in Wire

Coating Analysis with Temperature Dependent and Independent

Viscosity 55

4.1 Modeling of the problem 56

4.2 Perturbation solution 57

4.2.1 Constant viscosity case 57

xv

4.2.2 Temperature dependent viscosity 61

4.2.2a Reynolds model 61

4.2.2b Vogel’s model 65


4.4 Conclusion 76

5. Wire Coating Analysis with Oldroyd 8- Constant Fluid by

Optimal Homotopy Asymptotic Method 77

5.1 The basic equations and boundary conditions 78



5.4 Conclusion 91

6. Solution of Differential Equations Arising in Wire Coating

Analysis of Unsteady Second Grade Fluid 92

6.1 Problem formulation when the wire is translating only 93

6.1.1 Solution of the problem 95

6.2 Problem formulation when the wire is translating as

well as oscillating 97

6.2.1 Solution of the problem 98


6.4 Conclusion 111

7. Heat Transfer Analysis of a Third Grade Fluid in Post-treatment

Analysis of Wire Coating 112



7.2.1 Perturbation solution 117

7.2.2 Solution by optimal homotopy asymptotic method 120


7.4 Conclusion 125

xvi

8. Exact Solutions of a Power Law Fluid Model in Post-treatment

Analysis of Wire Coating with Linearly Varying Boundary

Temperature 126

8.1 Formulation and solution of the problem 127


8.3 Conclusion 139

9. Heat Transfer by Laminar Flow of an Elastico-Viscous Fluid in

Post-treatment Analysis of Wire Coating with Linearly Varying

Temperature along the Coated Wire 141



9.2.1 Perturbation solution 147

9.2.2 Solution by modified homotopy perturbation method 150


9.4 Conclusion 158

10. Conclusions and Future Work Directions 160

10.1 Conclusions 161

10.2 Future work directions 162

References 163

List of Publications/Submissions 171

Appendix A 172

Appendix B 175

Appendix C 176

xvii

LIST OF FIGURES

Figure 1.1. Typical wire coating process. 3

Figure 1.2. Schematic of wire coating dies: (a) pressure type die (b) tubing type die 4

Figure 2.1. Schematic profile of wire coating in a pressure type die. 26

Figure 2.2. Dimensionless velocity profiles for different values of X at fixed values

of 2 10, 2.cDe 35

Figure 2.3. Dimensionless velocity profiles for different values of 2

cDe at fixed values of

0.5, 2.X 35

Figure 2.4. Dimensionless shear stress profiles for different values of 2

cDe at fixed

values of 1.5, 2.X 35

Figure 2.5. Dimensionless normal stress profiles for different values of 2

cDe at fixed


Figure 2.6. Dimensionless volume flow rates versus ratio of the radii for different values

of 2

cDe at fixed values of 0.65.X 36

Figure 2.7. Thickness of coated wire versus ratio of the radii for different values of

2

cDe at fixed values of 0.65.X 36

Figure 2.8. Force on the surface of the total wire for different values of 2

cDe at fixed

value of 0.5.X 37

Figure 2.9. Dimensionless temperature distributions for different values of Brinkman

number at fixed values of 20.5, 10, 2.cX De 37

Figure 2.10. Dimensionless temperature distributions for different values of 2

cDe at

fixed values of 0.2, 4, 2.X Br 37

Figure 2.11. Dimensionless temperature distributions for different values of X at fixed

values of 2 0.5, 2, 2.cDe Br 38

Figure 3.1. Dimensionless velocity profiles for different values of slip parameter at fixed

values of 21.5, 10, 2.cX De 47

xviii


cDe at fixed values

of 1, 2.5, 2.X 47

Figure 3.3. Dimensionless velocity profiles for different values of velocity ratio

X at fixed values of 2 7.5, 5, 2.cDe 48

Figure 3.4. Dimensionless shear stress profiles for different values of slip parameter

at fixed values of 22, 0.1, 2.cX De 48


cDe at fixed

values of 0.5, 0.2, 2.X 48


cDe at fixed

values of 0.5, 5, 2.X 49

Figure 3.7. Dimensionless normal stress profiles for different values of slip

parameter at fixed values of 21.5, 0.2, 2.cX De 49

Figure 3.8. Dimensionless volume flow rates versus for different values of slip

parameter at fixed values of 21, 0.5.cX De 50

Figure 3.9. Thickness of coated wire versus for different values of slip

parameter at fixed values of 21, 0.5.cX De 50

Figure 3.10. Effect of the slip parameter on the force of the total wire at fixed values of

20.5, 0.5.cX De 50

Figure 3.11. Dimensionless volume flow rates versus for different values 2

cDe at fixed


Figure 3.12. Thickness of coated wire versus for different values of 2

cDe at fixed


Figure 3.13. Effect of the slip parameter on the force of the total wire for different values

of 2

cDe at fixed values of 2.5, 2.X 51

Figure 3.14. Dimensionless temperature distributions for different values of slip

parameter at fixed values of 21.5, 10, 0.1, 2.cX Br De 52


xix

number at fixed values of 20.5, 2, 10, 10.cX De 52


cDe at

fixed values of 1.2, 10, 2, 2.X Br 52

Figure 3.17. Dimensionless temperature distributions for different values of X at

fixed values of 2 0.5, 2, 4.5, 2.cDe Br 53

Figure 4.1. Dimensionless velocity profiles in case of constant viscosity

when 5,2 Br for different perturbation parameter 0 . 72

Figure 4.2. Dimensionless temperature distribution in case of constant viscosity

when 02, 0.01 for various values of Brinkman number Br . 72

Figure 4.3. Dimensionless velocity profiles in case of Reynolds’s model

when 02, 10, 0.1Br for different values of m . 73

Figure 4.4. Dimensionless temperature distribution in case of Reynolds’s model

when 02, 0.1, 10m for various values of Brinkman number Br . 73

Figure 4.5. Dimensionless velocity profiles in case of Reynolds’s model when

10,10,2 mBr for different values of perturbation parameter 0 . 73

Figure 4.6. Dimensionless velocity profiles in case of Vogel’s model when

02, 10, 0.2, 0.05,Br B 20m for different values of 1 . 74

Figure 4.7. Dimensionless temperature distribution in case of Vogel’s model when

,2.0,5,2 BBr 0 0.05, 5m for different values of 1 . 74


2, 10,m 1 02, 0.05, 0.2B for various values of Brinkman

number Br . 74


2, 10,m 1 02, 0.05, 0.2B for various values of Brinkman

number Br . 75

Figure 4.10. Dimensionless velocity profiles in case of Vogel’s model when 2,

20, 5,Br m 3.0,51 B for different values of perturbation parameter 0 . 75


xx

,5,20,2 mBr 3.0,51 B for various values of perturbation

parameter 0 . 75

Figure 5.1. Wire coating die. 80

Figure 5.2. Wire coating process in a pressure type die.. 81

Figure 5.3. Dimensionless velocity profiles at different order of approximations

using OHAM when ,4.0,2.0 10.5, 0.002154869,C

2 0.0005341298C . 85

Figure 5.4. Dimensionless velocity profiles for different values of dilatant parameter

when 4.0 , 0.5 . 85

Figure 5.5. Dimensionless velocity profiles for different values of viscoelastic

parameter when 5.0 , 0.5 . 86

Figure 5.6. Dimensionless velocity profiles for different values of pressure gradient

when ,4.0 1 . 86

Figure 5.7. Profiles of shear stress for different values of parameter when ,2.0

2.00 , 0.5 . 86

Figure 5.8. Profiles of shear stress for different values of viscosity parameter 0

when 4.0,2.0 , 0.5 . 87

Figure 5.9. Profiles of shear stress for various values of the parameter when

2.0,25.00 , 0.5 . 87

Figure 6.1. Geometry of coating die. 93

Figure 6.2. Velocity profiles for ,01.0,5.0,2,2.0,02.0,2.0 11 aUw

1 20.5924838150, 0.09024558924C C . 106

Figure 6.3. Velocity profiles at different position of r when ,02.0,2.0 11

,2.0 ,01.0,5.0,2 aUw 1 0.5924838150,C 2 0.09024558924C . 106

Figure 6.4. Velocity distribution of fluid flow with passage of time t when

,5.0,8.0,2,2.0,02.0,2.0 11 aUw 1 0.3296806629,C

2 0.306008832C . 107

Figure 6.5. Velocity distribution of fluid flow at different time levels when ,2.0

xxi

11 0.02, 0.2, 2, 0.8, 0.5,wU a 1 0.3296806629,C

2 0.306008832C . 107

Figure 7.1. Polymer extrudate in wire coating. 114

Figure 7.2. Drag flow in wire coating. 115

Figure 7.3. Comparison of dimensionless velocity profiles using PM and OHAM when

0 1 20.6, 0.01, 0.001357286, 0.0027125721U C C . 123

Figure 7.4. Dimensionless velocity profiles for different values of the velocities ratio

U when .01.00 123

Figure 7.5. Dimensionless velocity profiles for different values of the dimensionless

parameter 0 when .2.0U 123

Figure 7.6. Comparison of dimensionless temperature distribution using PM and OHAM

when 0 1 20.7, 0.01, 10, 0.001473286, 0.0002569261U Br C C . 124

Figure 7.7. Dimensionless temperature distribution for different values of Brinkman

number Br when .7.0U 124

Figure 7.8. Dimensionless temperature distribution for different values of the velocity

ratioU when Brinkman .10Br 124

Figure 8.1. Drag flow in wire coating. 127

Figure 8.2. The velocity profiles for different values of n , when 2, 0.5.U 135

Figure 8.3. Dimensionless velocity profiles for different values of velocity ratio

U , when ,1.0n .2 135

Figure 8.4. The shear rate for different values of velocity ratio U

when ,1.0n .2 136

Figure 8.5. Force wF is plotted against U for different values of n when 2. 136

Figure 8.6. Radius of coated wire cR is plotted against n for different values of by

taking 1.2.U 136


taking 1.2.U 137

Figure 8.8. Volume flow rate is plotted against U for different values of power law

index n when 2. 137

xxii

Figure 8.9. The non-dimensional function G for different values of non-dimensional

parameter 0S when 2,05.0,5.0,5.0 HUn . 137

Figure 8.10. The non-dimensional function G for different values of H taking

15.0,5.0,5.0 0 SUn and .2 138

Figure 8.11. The non-dimensional function G for different values of n when

2,5,6.0,10 0 SUJ . 138

Figure 8.12. The non-dimensional function G for different values of H when

2,5.0,25,6.0,4 0 nSUJ . 138


2,10,6.0,5.0 0 SUJ . 139


parameter 0S when 2,4.0,3.0,2 nUJ . 139

Figure 9.1. Dimensionless velocity profiles for different values of radii ratio

U for fixed value of elastic number 0.2eR . 156

Figure 9.2. Dimensionless velocity profiles for different values of eR for fixed value of

.5.0U 156

Figure 9.3. Thickness of coated wire against elastic number eR for different values

when radii ratio 0.5.U 156

Figure 9.4. Force on the surface of coated wire against elastic number eR for different

values using radii ratio 1.2.U 157

Figure 9.5. Dimensionless temperature distribution for different values of d

0 when

0.1,eR ,5.0S 4.0U . 157

Figure 9.6. Dimensionless temperature distribution for different values of S when

0.1,eR 2.1U . 157

Figure 9.7. Dimensionless temperature distribution for different values ofU when

,5.00 d 0.3,eR 10S . 158

Figure 9.8. Dimensionless temperature distribution for different values of non-

dimensional parameter S when ,20 d 0.2, 0.4eR U . 158

xxiii

LIST OF TABLES

______________________________________________

Table 4.1. Shows velocity distribution at various order of approximations

when 02, 20, 0.01Br . 70

Table 4.2. Shows velocity distribution at various order of approximations when

02, 20, 0.3Br . 70


02, 0.5, 0.5Br . 71

Table 4.4. Shows temperature distribution at various order of approximations when

02, 20, 0.01Br . 71


02, 20, 0.3Br . 71


02, 0.5, 0.5Br . 72

Table 5.1. Shows variation of volume flow rate and average velocity for different values

of when ,3,2.0 0.5 . 88


of when ,5.0,2.0 0.5 . 88


of when 3,4.0 , 5.0 . 89


of pressure gradient when ,2.0,5.0 3 . 89

Table 5.5. Shows variation of the auxiliary constants 1C and 2C for different values

of when 0.5, 2, 2 . 90


of when 1.5, 2 , 0.8 . 90


of pressure gradient when 0.8, 0.1, 2 . 90

xxiv

Table 6.1. Shows velocity distribution for different values of time level t when

0.2, 11 0.02, 0.2, 2, 0.5, 0.01,wU a 1 0.5924838150,C

2 0.0902455892C . 103

Table 6.2. Shows velocity distribution of fluid flow at different time level t when

,01.0,5.0,2,2.0,02.0,2.0 11 aUw 1 0.5924838150,C

2 0.09024558924C . 103

Table 6.3. Shows velocity distribution of fluid flow at various order of approximations at

10t when ,01.0,5.0,2,2.0,02.0,2.0 11 aUw 1 0.5924838150,C

2 0.09024558924C . 104

Table 6.4. Shows velocity distribution of fluid flow at different values of time by using

0.2, 11 0.02, 0.2, 2, 0.8, 0.5,wU a 1 0.329680663,C

2 0.3060088.C 104

Table 6.5. Shows velocity distribution at different at different time level t when 0.2,

,02.0,2.0 11 2, 0.8, 0.5,wU a ,3296806629.01 C

306008832.02 C . 105

Table 6.6. Shows velocity distribution at various order of approximation at 5t when

,5.0,8.0,2,2.0,02.0,2.0 11 aUw 1 0.3296806,C

2 0.30600883.C 105

Table 6.7. Error between OHAM and exact solution up to third order of approximation at

2t (example 6.1). 108


different values of time level (example 6.1). 108


0.5t (example 6.2). 109

Table 6.10. Error between OHAM and exact solution up to third order of approximation

at different values of time level (example 6.2). 109


at 2t (example 6.3). 110

xxv


at different values of time level (example 6.3). 110

Table 9.1. Shows velocity distribution for different elastic number eR ,

when 2.1U . 158

Table 9.2. Shows shear rate distribution for different elastic number eR ,

when 2.1U . 159

Table 9.3. Shows temperature distribution for different elastic number eR , when

0 0.5d , 1.2, 20U S . 159

LIST OF ABBREVIATIONS

T stress tensor

p dynamic pressure

kronecker delta

viscosity coefficient

D rate of deformation tensor

u velocity vector

T transpose of the matrix

0 shear rate

fluid density

D Dt substantive derivative

t local derivative

I identity tensor

S shear stress tensor

z distance in the direction of flow

r transverse distance to flow direction

k thermal conductivity

pc specific heat

fluid temperature

xxvi

dissipation function

Re Reynolds number

Pr Prandtl number

Br Brinkman number

0 bulk fluid temperature

,w d temperature at surface of wire/ temperature at surface of die

eR elastic number

c coefficient of cross viscosity

1 elastic parameter

c cross-viscous number

cDe Deborah number

relaxation parameter

,zp pressure drop along axial direction

perturbation parameter

L linear operator

N non-linear operator

rg known analytical function

boundary of domain

differential operator

wR radius of wire

dR radius of die

L length of die

trS trace of the extra stress tensor S

constant viscosity coefficient

1A deformation rate tensor

S upper contra-variant convected derivative

constant pressure gradient

1P arbitrary constant of integration

xxvii

1V wire velocity

2V fluid velocity

avew average velocity

Q flow rate

cR thickness of wire

rzS shear stress

wF force on total wire

cU characteristic velocity scale

X ratio of characteristic velocity scale to that of velocity of wire

1P , 2 , ,a bP K K constants of integrations

radii ratio

slip parameter

1 2 1 2 3, , , , material constants

2 3,A A kinematic tensors

0 non-Newtonian parameter

BD, viscosity parameters

dilatant constant

pseudoplastic constant

a amplitude

frequency of oscillation

1R the radius of uncoated wire

2R radius of coated wire

: scalar invariant

0 consistency index

n power law index

temperature gradient

,G S dimensionless numbers

U velocity ratio

xxviii

i

j Kronecker delta

j

id rate of strain tensor

i

jS extra stress tensor

i

jS rate of stress tensor

P dimensionless pressure

K dimensionless constant of integration

d dimensionless temperature

i

j Cauchy stress tensor

1

Chapter 1

Introduction

2

1.1 Mathematical model

A model is simply a symbolic representation of some system. When this representation is

mathematical, we say, it is a mathematical model. Solution of the mathematical model

reveals the patterns of behavior of the modeled phenomenon. Modeling a system is a

complicated task, as most of the systems, we plan to model, are mostly complex in

nature. A system is almost open, i.e., the factors influencing it are frequent and are

affected by the surroundings; but only a closed system, i.e., the system, all of whose

components are specifically well-known, may be modeled. Any attempt to model a

system, depends on some in-built assumptions and some degree of approximations that

builds it theoretically closed. Therefore, we make attempts to model a particular

phenomenon of a system by ignoring the parameters with less influence on the

phenomenon.

1.2 Wire coating process

It is an important and oldest industrial process dating back to the 1840’s [1]. Polymer

extrudate is used to coat a wire for insulation, mechanical strength and environmental

safety. In coating process, either the liquid polymer is deposited continuously on moving

wire or the wire is dragged inside the die filled with coating liquid. Industrial coating

processes today use well refined apparatus and standardize experiments due to the risks

of the abuse of wire in electrical products. The experimental set-up of typical wire

coating process is shown in Fig.1.1 [2]. The uncoated wire unwinds at the payoff reel

passing through a straightener, a preheater, a cross head die in turn wire meeting the melt

polymer emerging from the extruder and gets coated. This coated wire passes through a

cooling trough, a capstan and a tester finally ending up on the rotating take-up reel.

3

Figure 1.1. Typical wire coating process.

The most important plastics resins used in coating of wires are plasticized polyvinyle

chloride (PVC), low density polyethylene (LDPE), high density polyethylene (HDPE),

nylon and polysulfone. The typical melt temperatures for PVC are 0185 C, for LDPE

0220 C, for HDPE 0260 C, 0285 C for nylon and 0365 C for polysulfone. Due to the threat

of thermal humiliation, PVC is sprint at lower velocities and temperatures as compared to

the other resins and as a result, it is commonly applied as covering other wires.

1.2.1 Types of wire coating process

Mostly, three different processes are used for wire coating namely, (i) coaxial extrusion

process (ii) dipping process and (iii) electro-statical deposition process. The coextrusion

process [2-4, 5] is an operation where either the polymer is extruded on axially moving

wire or the wire is dragged inside a die filled with molten polymer. In dipping process

coating [6, 7], the objects to be coated is initially immersed in a pool of coating fluid and

then withdrawn continuously results in adherence of a liquid film on the surface of

continuum. In electro-statical deposition process [8] of coating, the thermal treatment in

presence of electric field and a ray of non-penetrating electrons are applied. The first two

operations are quick but temporarily, the bonding exists between the wire and the

polymer is not so strong physically. On the other hand, the third method provides strong

bonding but relatively slow as compared to two previous processes. The efficiency of

coextrusion process can be improved by adopting hydrodynamic method [9, 10]. In this

method of coating the velocity of continuum and the melt polymer generates high

pressure in a specific region which produces strong bonding and also offers fast coating.

4

The coextrusion process is simple in applicability, time saving and economical process in

industrial point of view. Therefore many researchers [1-4, 5, 9-41] investigated the wire

coating phenomena using coextrusion process.

1.2.2 Designs of wire coating die

The designs of wire coating units are of fundamental importance as it strongly affects the

final materials. Generally, two types of dies are used in coating process (i) tubing dies

and (ii) pressure type dies shown systematically in Fig. 1.2. In tubing dies [34, 35, 36],

the polymer is extruded from annular die around the emerging wire. Usually vacuum is

applied inside the cone, to help draw the extrudate towards the wire and to stabilize the

shape of cone. In tube coating, the significance lies outside the coating unit where the

polymer meets up the wire. However, very few disclosures are present in the literature

with regard to studying the flow of melt polymer outside tubing type dies.

(a)

(b)

Figure 1.2. Schematic of wire coating dies: (a) pressure type die (b) tubing type die.

5

In pressure type dies [9, 10, 37 - 41], the melt polymer meets the wire inside the die, an

obscure flow field exists and its knowledge is essential for the design of enhanced dies

with most excellent performance. This type of die is like an annulus with inner surface is

the wire to be coated. Pressure type dies are commonly used for primary insulation of

wires while covering of earlier coated wires or a set of wires is best done with tubing

dies. For infinitesimal wires, the tubing dies are favorite to keep away from the enhance

stress necessary to pull the wire through the melt polymer in pressure dies. In pressure

type dies of wire coating the bond between the melt polymer and continuum can build

stronger as compared to tubing dies by applying high pressure gradient. The pressure dies

have received more consideration than the tubing dies in this area because of its

performance, and they will be the focus of this study.

1.3 Constitutive equation

A relation between two or more physical quantities that is specific to a material or

substance and approximate the response of that material to external forces [42]. In more

specific form, the constitutive equations are equations of state relating suitably defined

stress and deformation variables.

1.4 Non-Newtonian fluids

The theory of non-Newtonian fluids is most important branch of fluid mechanics because

of industrial and biological applications. A fluid of constant viscosity that exhibits a

linear relationship between the shear stress and shear rate is characterized to be a

Newtonian fluid [42, 43]. The constitutive equation of incompressible Newtonian fluid is

given by

2p T D , (1.1)

where T is the stress tensor, p the arbitrary isotropic pressure, the kronecker delta,

the viscosity coefficient that could vary with pressure and temperature and D is the rate

of deformation tensor defined in terms of velocity vector u as

1

2

T D u+ u , (1.2)

in which T is the transpose of the matrix.

6

But a constant viscosity relation is not always a Newtonian relation as the convected

Maxwell fluid, second grade fluid and Oldroyd fluid A and B are non-Newtonian in

nature but reveal a constant viscosity. On the other hand, the fluid that is not

characterized by Eq. (1.1) is known as the non-Newtonian fluids. In such fluids, viscosity

is dependent or independent on shear stress. Cornstarch dissolved in water is the most

common everyday example of a non-Newtonian fluid. If you hit a container full of

cornstarch, the atoms in the fluid rearrange in the form of solid due to stress such that

your hand will not go through if you set down your hand into the fluid slowly, though, it

will go through successfully. If you draw your hand out suddenly, it will once again act

similar to a solid and in this way you can exactly drag a pail of the fluid out of its

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

melts, drilling mud’s, clay coatings, yogurt, gravy, paints, gels, rubbers, soaps, inks, oils,

concrete, ketchup, pastes, suspensions, slurries, biological liquids such as blood and

foodstuffs [42-43]. These fluids are divided generally into three groups, namely; Time

Independent (Visco-Inelastic) Fluids, Time Dependent Fluids and Viscoelastic Fluids on

the basis of their non-linear relationship between shear stress and the rate of shear.

1.4.1 Time independent (Visco-inelastic) fluids

The fluids whose shear rate is a nonlinear function of the shear stress, independent of

shearing time is known as time independent fluids [44].

For such fluids:

g0 , (1.3)

in which 0 is the shear rate and is the corresponding shear stress at a point.

These fluids are further divided into three categories: Bingham plastics, Pseudoplastic

fluids and dilatant fluids

Bingham plastics

Bingham fluids exhibit a solid like configuration and flows unless sheared by an external

stress higher than a yield stress, which is a characteristic of material. paints, ketchup and

mayonnaise are examples of Bingham plastic [42, 43].

7

Pseudoplastic fluids

These are fluids that experience a decrease in viscosity when exposed to a shear stress

[42]. Another name for a shear thinning fluid is a pseudoplastic. Ketchup/tomato sauce,

nail polish, polymer solutions and molten polymers and modern paints are those

examples of pseudoplastic materials which are commonly used. Drilling mud, synovial

fluid and clays are those fluids which are specialy used.

Dilatant fluids

The dilatant materials are that those apparent viscosity increases with respect to shear

force and also the value of apparent viscosity at a given shear rate is free of the shear

history of the sample [44]. The most commonly used dilatant fluid is cornstarch

dissolved in water [45]. And the special used dilatants (non-Newtonian fluids) are silky

putty and ethylene glycol [45].

1.4.2 Time dependent fluids

More complex fluids, for which the relation between shear stress and shear rate depends

upon the duration of shearing and their kinematic history, are called time dependent

fluids. They can generally be classified into two classes [44].

(i) Thixotropic fluid

(ii) Rheopectic fluid

1.4.3 Viscoelastic fluids

The word “Viscoelastic” means the coupled presence of viscous and elastic properties in

a material. The fluids posses a definite level of elasticity and memory in addition to the

shear thinning or shear thickening viscosity are called the viscoelastic fluids [43]. All

liquids of polymeric origin (melts, solutions, suspensions, emulsions) are viscoelastic.

http://en.wikipedia.org/wiki/Paint

http://www.wisegeek.com/what-is-cornstarch.htm

8

1.4.4 Brief comparison of non-Newtonian, Newtonian and viscoelastic

properties

1.5 Basic types of flows

There are many types of fluid flows here; we discussed only few of them in which

we are interested.

Laminar flow

Laminar flow is that type of fluid flow in which the fluid travels smoothly or in regular

paths, with no disruption between the layers [50]. In laminar flow, sometimes called

streamline flow, the velocity, pressure and other flow properties at each point in the fluid

remain constant.

Steady and Unsteady flows

The flow for which, all fluid flow properties such as velocity, temperature, pressure and

density are independent of time are known steady flows. On the other hand, if all these

properties varies from point to point are referred as unsteady flows.

Incompressible flow

Flows in which the material density is constant within a fluid parcel is known as

incompressible flows [50]. The volume or densities of such fluids do not change when

squeeze them.

http://www.britannica.com/EBchecked/topic/328742/laminar-flow

http://en.wikipedia.org/wiki/Density

http://en.wikipedia.org/wiki/Fluid_parcel

9

Isothermal flow

Flow of fluid which remains at the same temperature while flowing in conduit is known

as isothermal flow [51]. Although the flow temperature remains constant, a change in

stagnation temperature occurs because of a change in velocity.

Non-Isothermal flows

The material properties, such as viscosity and density, change accordingly when a fluid is

subjected to a temperature change. This phenomenon mostly occurs in heat exchangers,

chemical reactors or in processes where components are cooled.

1.6 Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy

of the system. It is usually denoted by 0H or H . Because of its close relation to the time-

evolution of a system, it is of fundamental importance in most formulations of quantum

theory [52].

1.7 Perturbation theory

The mathematical methods used to obtain an approximate solution to a problem at hand,

that are complicated to be solved exactly. Generally, in such type of methods an iterative

procedure is involved, in which each new obtaining term contributing to the solution has

less significance than the last. Perturbation theory is valid if we can introduce a "small"

term to the mathematical equation. It gives an expression for the problem solution in the

form of a power series in some "small" parameter known as a perturbation series which

computes the difference from the exactly solvable problem. The methods employed for

this purpose form perturbation theory.

1.7.1 Perturbation theory (quantum mechanics)

Perturbation theory in quantum mechanics is a class of approximation techniques

associated to mathematical perturbation for changing a complicated system to a simpler

one. The main idea is based on to start a simple system which has a mathematical

solution, and adds an extra "perturbing" Hamiltonian presenting a weak disturbance to

the system. When the disturbance adds to a system is not much large, the different

physical quantities related to the system which is perturbed can, from considerations of

continuity, be represented as 'corrections' to the simple systems.

http://en.wikipedia.org/wiki/Temperature

http://en.wikipedia.org/wiki/Quantum_mechanics

http://en.wikipedia.org/wiki/Operator_%28physics%29

http://en.wikipedia.org/wiki/Energy

http://en.wikipedia.org/wiki/Formal_power_series

http://www.answers.com/topic/perturbation-theory

http://en.wikipedia.org/wiki/Perturbation_theory

http://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29

http://en.wikipedia.org/wiki/Continuity_%28mathematics%29

10

1.7.2 Time-independent perturbation theory

This theory is one of the categories of perturbation theory. Time-independent

perturbation theory was introduced by Erwin Schrödinger in 1926 [49]. The perturbation

Hamiltonian is stationary in this theory [49, 53].

1.7.3 Time-dependent perturbation theory

Theory of time-dependent perturbation was presented by Paul Dirac [54]. This theory

illustrates the importance of a time-dependent perturbation applied to a time-independent

Hamiltonian 0H , As the perturbed Hamiltonian is time-dependent, so its energy levels

and eigenstates will also time-dependent.

1.8 Reynolds model

Reynolds model is used for account of variable viscosity. In this model, an exponential

expression is used for variation of viscosity with temperature. Mathematically, it can be

represented as [55]

exp L ,

in which is the temperature and L is the embedding parameter.

1.9 Vogel’s model

Vogel’s model is also account for the variation of viscosity depends on temperature,

reference viscosity and the viscosity parameters.

Mathematically, it can be represented as [55]

exp w

A

B

,

in which is the temperature, is the reference viscosity and both A and B are the

viscosity parameters.

1.10 Basic flow equations

The fundamental equations governing the flow of an incompressible fluid with thermal

effects are the continuity equation, equation of motion and the energy equation.

1.10.1 Continuity equation

If is the fluid density, then the balance of mass flow u entering and leaving an

infinitesimal control volume is equal to the change in density [42, 43, 56-58]

http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger

http://en.wikipedia.org/wiki/Paul_Dirac

11

0.D

div divD t t

u u (1.4)

For incompressible fluid, the density of fluid is constant, so t

will be zero and the

continuity equation takes the following form

0udiv . (1.5)

1.10.2 Equation of motion

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

equilibrium with the stresses S , the body forces f giving a typical equation in the

vector form is [42, 43, 56-58]

.D

Dt

uT f , (1.6)

where Dt

D denotes the substantive acceleration, consists of the local derivative

t

and

the convective derivative .u , i.e.,

.u

tDt

D.

The Cauchy stress tensor T is defined as

SIT p , (1.7)

where p is the dynamic pressure, I the identity tensor and S is the shear stress tensor.

The equation of motion in vector form can be written as the set of three equations in

cylindrical coordinates as

r - component of momentum equation

21 1

,r r r z r

r

rS S S Su u v u v u pu w g

t r r r z r r r r z r

(1.8)

- component of momentum equation

2

2 2

1 1 1,

z r r

r

v v v v uv vu w

t r r r z

S S S Spr S g

r r r r z r

(1.9)

z - component of momentum equation

12

z

zzz

zr gz

SS

rrS

rrz

p

z

ww

w

r

v

r

wu

t

w

11. (1.10)

1.10.3 Energy equation

The energy equation is based on the physical principle of the first law of thermodynamics

that is “total energy is conserved” in a system. The fluid element has two contributions to

its energy, the internal energy due to random molecular motion and the kinetic energy

due to translation motion of the fluid elements. The sum of these two energies is the

“total energy”. The energy equation derived on this principle is given as [42, 43, 58]

2

p

Dc k

Dt

, (1.11)

where is the constant density, pc the specific heat, D Dt denotes the material

derivative, k the thermal conductivity, the fluid temperature and is the dissipation

function.

1.11 Dimensionless numbers

1.11.1 Reynolds number

Reynolds number is the ratio of the magnitude of inertial forces to the magnitude of

viscous forces in the flow [42, 43], i.e.,

2

1 1 1

1 1

ReV V R

V R

.

The above equation shows that if the viscous forces are high relative to the inertial forces,

then Re is relatively low and the flow tends to be laminar. Similarly, if the inertial forces

dominate the viscous forces, then the turbulent flow will develop.

1.11.2 Prandtl number

It is defined as a measure of the ratio of the viscous diffusivity to the thermal diffusivity.

Mathematically

Prpc

k

.

It is the measure of the ratio of the rate of spread of effects of momentum changes in the

flow to the spread of effects of temperature differences in the flow.

13

1.11.3 Brinkman number

Brinkman number is the measure of the ratio of the viscous heating relative to the

conductive heat transfer. It is significant, where a high change occurs in velocity over

short distances such as lubricant flow.

2

1

0

,w

VBr

k

where Br is the Brinkman number, the fluid's dynamic viscosity, 1V the fluid's

velocity, k the thermal conductivity of the fluid, 0 the bulk fluid temperature and w

is the wall temperature.

1.11.4 Elastic number

The elastic number turns up in the elastico-viscous liquid and is defined as [59-61]

2

1 1

2

1

.cc

VR

R

where 1 is the elastic parameter, c the cross-viscous coefficient and is the

coefficient of viscosity of the fluid.

1.11.5 Cross-viscous number

The Cross viscous number also turns up in the elastico-viscous liquid and is defined as

[59-61]

,2

1R

cc

where c is the cross-viscous coefficient and is the density of fluid.

1.11.6 Deborah number

Deborah number cDe is the measure of the ratio of the rate of pressure drop in the flow

to the viscosity and is defined as [62, 63]

1 ,,

8

z

c

R pDe

where is the relaxation parameter, ,zp the pressure drop along axial direction and is

the viscosity coefficient of the fluid. This number usually appears in the flow of Phan-

Thien and Tanner fluid in cylinders.

http://en.wikipedia.org/wiki/Dynamic_viscosity

http://en.wikipedia.org/wiki/Thermal_conductivity

14

1.12 Methods of solutions

1.12.1 Exact solution

A set of functions for the velocity components and the fluid pressure constitute an exact

solution of the flow equations. Also, the problems are satisfied for all values of the

involved independent variables and of a realistically imposed physical problem for all

values of the fluid flow effected parameter such as density, viscosity, elasticity, thermal

conductivity, elongation, specific heat and time dilation etc.

The exact solutions attains important feature for the following two reasons as reported by

Wang [64]:

(i) It represents actual fundamental fluid-dynamic flows. Also, with the help of these

solutions, one can study more closely the basic phenomenon described by the flow

equations.

(ii) The exact solutions help us in finding the accuracies of various approximate

techniques, whether they are numerical, asymptotic or empirical.

1.12.2 Perturbation Method (PM)

The exact solutions of the nonlinear problems arising in modeling some practical

phenomena are rare in literature. The perturbation methods [65-67] are extensively

applied for getting approximate solutions to nonlinear problems. The perturbation

methods need the existence of a small parameter in a given problem. These methods are

so controlling that somewhere the small parameter is unnaturally introduced into an

equation free from such parameter and then at the end of solution it settled equal to unity

to get the original problem’s solution. We suppose as a perturbation parameter and

expand the dependent variable say ,u y in the form

2 3

0 1 2 3, ...u y u u u u . (1.12)

After substituting equation (1.12) in the differential equation and equating the

coefficients of like powers of , we get various order linear problems. These problems

are then solved in conjunction, which provides solution of the nonlinear differential

equation.

15

1.12.3 Basic idea of Modified Homotopy Perturbation Method (MHPM)

The modified homotopy perturbation method [68, 69] is developed on the basis of

perturbation and the homotopy perturbation method. To explain this, we consider the

following nonlinear differential equation

rrgwNwL ,0 ,

r

n

uu ,0, , (1.13)

where L is linear operator, N is the nonlinear operator, rg is known analytical

function, is the boundary of the domain and is the boundary operator. Here, n

denotes differentiation along the normal drawn outwards from .

In modified homotopy perturbation method, the known analytical function rg is

replaced by an infinite series as [68, 69]:

,0

rgrgn

n

(1.14)

In this case, we construct a homotopy as

,:, 1,0 prw which satisfy the following equation

1,00,0

00

prgwNpupLuLwLn

n , (1.15)

where p is an embedding parameter, i.e., 1,0p and 0u is any selected initial guess

which satisfies the boundary conditions. Here, we seek the solution of the form [68, 69]

rwprwn

n

n

0

. (1.16)

The MHPM reduces to HPM on setting gg 1 and 0...320 ggg in Eq. (1.15). In

MHPM, 0g is combined with component 0w , 1g is combined with component 1w , 2g is

combined with component 2w and so on.

The approximate solution rw of the considered nonlinear differential equation can

readily be obtained as [68, 69]:

0 1 21

lim , ...p

w r w r p w w w

. (1.17)

16

This method is helpful to grip the complication involved in source terms. Also, the length

of calculations is reduced compared with the traditional perturbation method or

homotopy perturbation method.

1.12.4 Basic idea of Optimal Homotopy Asymptotic Method (OHAM)

To study the basic idea of optimal homotopy asymptotic method [70-76], we consider the

following nonlinear differential equation

,,0 rrgrw ,,, 0

r

dr

dww (1.18)

where is a differential operator and is a boundary operator, rw is the unknown

function, r denotes the spatial independent variable, is the boundary of the domain

and rg is a known analytic function. The operator can be written as

,NL (1.19)

where L and N are the linear and nonlinear operators respectively.

We construct a homotopy Rpr 1,0:, which satisfies

,, 01 rrwNrwLpHrgprLp

0,

,,

dr

prdpr

,

(1.20)

where Rr and 1,0p is an embedding parameter, pH is a nonzero auxiliary

function for p does not equal to zero and zero for p is equal to zero, and pr, is an

unknown function. For 0p , the homotopy given in Eq. (1.20) only recover the linear

part of solution of the non-linear boundary value problem in Eq. (1.18) and there is no

contribution from the non-linear part of this problem, i.e., rwr 00, ,

also we have,

000 0

0 ,,,

dr

dwwrgrL . (1.21)

Similarly, for 1p , this homotopy recover the nonlinear boundary value problem given

in Eq. (1.21). In this case the solution converges to the exact solution, i.e., rwr 1, .

Thus, as p varies from 0 to1 , the solution pr, approaches from rw0 to rw .

Next, we choose auxiliary function pH in the following form

17

...,3

3

2

2

1 CpCppCpH (1.22)

where ,...,,, 321 CCC are constants to be determined later.

2

1 2 ...H p pC p C , (1.23)

where 1 2,, ...C C are constants to be determined such that to minimize the solution error.

To obtain an approximate solution, we expand iCpr ,, in Taylor’s series about p in

the following manner, [70-76]:

k

k

k

ki pCCCCrwrwCpr ,...,,,,,, 321

1

0

, (1.24)

Substituting both Eq. (1.24) and the auxiliary function given in Eq. (1.23) into Eq. (1.20)

and equating the coefficients of like powers of p , we obtain the different order problems,

where the zeroth order problem is given in Eq. (1.21) and the first and second order

problems are as follows [70-76]:

0110011 ,,

dr

dwwrwNCrgrwL , (1.25)

2 1 2 0 0 1 1 1 1

22 0.

,

,

L w r L w r C N w r C L w r N w r

dwB w

dr

(1.26)

In general form rwk , is given by:

1 0 0

1

0 1 1

1

2,3,...,, ,...,

k k k

k

i k i k i k

i

L w r L w r C N w r

C L w r N w r w r w r k

0,

dr

dwwB k

k , (1.27)

where rwrwrwN kik 110 ,...,, is the coefficient of ikp in the expansion of

prN , about the embedding parameter p as:

.,...,,,, 210

1

00

ik

ik

ik

ik pwwwwNrwNprN

. (1.28)

Here, it has been submitted that the convergence of the series given in Eq. (1.24) depends

upon the order of the problem and the auxiliary constants ,..., 21 CC .

18

If this series is convergent at 1p , one has [70]:

i

i

im CCCCrwrwCCCCrw ,...,,,,,...,,,,~321

1

0321

. (1.29)

On substituting Eq. (1.29) into Eq. (1.18), it results the expression for residual in the

following form [48]:

mmm CCCCrwNrgCCCCrwLCCCCrR ,...,,,,~,...,,,,~,...,,,, 321321321 . (1.30)

If we set 0R in Eq. (1.30), then the nonlinear differential Eq. (1.18) gives the exact

solution. But generally it doesn’t happen in nonlinear problems.

There are many methods such as Galerkin’s method, least squares Method, collocation

method, and Ritz method which can be used to determine the optimal values of

miiC ,...,3,2,1 . Here, method of least squares has applied to locate the optimal values

of auxiliary constants as [70]:

b

a

mm drCCCrRCCCJ ,...,,,,...,, 21

2

21 , (1.31)

and

0...21

mC

J

C

J

C

J, (1.32)

where a and b are the properly chosen numbers from the domain that locate the auxiliary

constants which minimize the residual defined in Eq. (1.30). Moreover, the system of Eq.

(1.32) is solved for optimal values of auxiliary constants which are responsible for the

approximate solution of the nonlinear differential Eq. (1.18).

Marinca et al. [70-74], Saeed et al. [75] and Javed et al. [76] successfully applied this

technique on solving different nonlinear boundary value problems of physical and

engineering interest and obtained satisfactory results. This method gives series form

solution which converges to the exact solution as the terms in the auxiliary function

increased. Moreover, the convergence can also be control by the proper selection of

auxiliary constants which optimize the solution.

19

1.13 Literature survey

Newtonian fluids

Carley [12] investigated the fluid mechanics of wire coating with a constant pressure

gradient along axial direction. He derived the equations for drag and pressure flow. These

equations can be used to calculate the velocity distribution, pressure drop through the die.

Haas and Skewis [77] examine the work of Carley [12] with same assumptions. They

concluded that the roughness of coated wire can be removed by using tapered dies with

high speed wire moving inside. Bagley and Storey [15] presented numerical solutions in

the sort of non-dimensional parameters characterizing the wire velocity, die dimensions,

melt viscosity, shear rate and radial position. White and Tallmadge [27] developed theory

for cylinder withdrawal from Newtonian fluids which is applicabable for a limited speed

range while the industrial need is towards higher speeds Soroka et al. [28]. Moreover,

investigations of wire coating process were carried out by White and Tallmadge [78],

Spiers et al. [79], Soroka and Tallmadge [80], Esmail and Hummel [81], Middleman [22]

in his book, Tadmor and Gogos [1], Han and Rao [2], Marvidis and Hrymak [82] and

Mitsoulis [83]. Carley et al. [21] found the first no-isothermal analysis without LAT.

They showed that the polymer flow is affected by the viscous dissipation. But their

analysis was restricted in the die region; therefore they neglected the annual and outlet

zones.

Non-Newtonian fluids

Paton et al. [13] and McKelvey [14] extended the work of Carley [12] using the same

assumptions and examined the flow of inelastic power law fluid inside an annulus by

considering drag and pressure driven flow independently. They reported the expression

for flow rate and shear rate. Fenner and Williams [16] performed an analysis of the

polymer flow in the tapering section of the pressure type die using power law fluid

model. They derived numerical solution for the pressure and velocity distributions inside

the die. Fenner offers further explanation in his next paper [17] and it is generally this

kind of investigation employed by the wire coating production for tapered pressure dies.

Kasajima and Ito [18] carried out an analysis of the drag flow outside the die including

20

thermal effects. They derived the expression for velocity field, shear rate, flow rate and

temperature distribution. Winter [19] extended the idea of Kasajima and Ito and studied

the non-isothermal analysis both inside and outside the die.

Tadmor and Bird [20] investigated the effect of viscoelasticity on the eccentricity of the

wire. They evaluated that the lateral forces acting on the surface of wire tend to stabilize

it into a concentric position. Caswell and Tanner [3] made first attempt to analyze the

isothermal flows of power law fluids inside the wire coating dies without lubrication

approximation theory (LAT). They illustrated some attractive aspects, such as

recirculating region and to avoid them and also becomes successful in determination of

the free surface at the die exist for both tubing and pressure type dies with the help of

finite element method. Hashmi et al. [23] presents analytical results for plasto-

hydrodynamic pressure in wire drawing through a stepped die, by taking temperature

independent viscosity. Later, Akter and Hashmi [37] extended previous work Hashmi et

al. [23] and include the effect of temperature dependent viscosity and pressure in the

hydrodynamic pressure unit. They obtained theoretical results for different wire speeds

and geometry parameters and compared the results with the previous solutions. Akter and

Hashmi [37] developed theoretical result in cartesian co-ordinates for pressure

distribution and plasto-hydrodynamic die-less wire coatings in a conical die. Later, Akter

and Hashmi [38] extended the previous work and studied wire coating using cylindrical

co-ordinates for analysis and derived pressure distributions for different wire speeds and

compared it with theoretical ones [37]. Dijksman and Savenije [30] investigated the wire

coating using Newtonian and non-Newtonian liquids in converging type dies by

introducing a special toroidal co-ordinate system and they successfully obtained the

expression for velocity distribution, shear rate, volume flow rate and pressure distribution

in radial direction. Wagner and Mitsoulis [25] and Mitsoulis [26] examined the effects of

viscoelasticity, slip at the wall and the thermal effects for various conditions such as, melt

polymers and die designs used in high speed wire coating operations. Shu et al. [31-33]

reported theoretical investigations by melt polymer flow by considering melt polymer as

a non-Newtonian power law fluid. Their studies only organize to deal with multi-

interface regions. In which Shu et al. [31] introduced a new numerical technique for fixed

grid size to solve the multi-interface problems more efficiently rather than the variable

21

grid size. Moreover, Shu et al. [33] examined the thermal multi-interface flow of

polymers in circular and annulus dies using finite element method. Roy and Dutt [29]

further investigated and extend the work of White and Tallmadge [27] with the concept

that many fluids used in coating industries are non-Newtonian in nature. They assumed

that the polymer obey the power law fluid model and suggested the theory for wire

coating by withdraw from a polymer and tested with experimental data on Newtonian and

pseudoplastic fluids. Kyle [34, 35] provides a numerical solution to the melt flow in high

speed tube coating with vacuum. He performed his analysis for a nonlinear visco-elastic

liquid, the Phan-Thien and Tanner model. He finds that various parameters affect the

melt cone shape and predict the steady state contact length. The critical region in tube

coating is the melt cone, where the melt experiences high stresses [34]. It is to be noted

that Kyle [34, 35] can not produce universal plots (or equations) for tube coating problem

solving. This was further investigated by Hade and Giacomin [36] for an isothermal flow

and produces such plots and also provides a numerical solution for the melt cone shape.

Moreover, they manipulate the solution to relate operating parameters to the process

variables. Sajid et al. [39] studied the flow of Oldroyd 8- constant fluids through a

uniform pressure type die. Recently, Siddiqui et al. [40, 41], studied the wire coating

extrusion in a uniform pressure type die in flow of third and fourth grade fluids

respectively.

The flow of non-Newtonian fluids has attained substantial importance owing of its

applications in different branches of science and engineering: Particularly in chemical

industries, bio-engineering and material processing. Some studies [84-98] have been

presented involving non-Newtonian fluids. It is a well known reality that the

characteristics of non-Newtonian fluids are relatively different when compared with the

viscous liquids. Therefore, the Navier–Stokes equations are incompatible to explain the

behavior of these fluids. Similar to viscous fluids, it is complicated to propose single

mathematical model that posses all properties of such fluids. In view of that, various

models have been planned to describe the behavior of these fluids. Amongst there are

fluids of the differential type of grade n (Truesdell and Noll [99]), such as second grade

fluid, third grade fluid, fourth grade fluid, elastico viscous fluid, Maxwell fluid, Oldroyd

B fluid, Oldroyd 8- constant fluids, Phan Thien and Tanner fluid, power law fluids etc,

22

and the first grade fluid are the viscous fluid. The constitutive equations of non-

Newtonian fluids make the governing equations more complicated involving number of

parameters and the exact solutions are even rare in literature for these equations. These

equations are typically handled with numerical, iterative and asymptotic methods.

Asymptotic methods have then proved to be a powerful tool to find approximate solution

of those equations which arising during modeling of non-Newtonian fluids. A

comprehensive review on these types of methods is given by He [100]. To solve realistic

problems, different perturbation techniques have been extensively used in the field of

science and engineering [101 and the references therein]. The traditional perturbation

method [65-67] is mostly applied for evaluating approximate solutions to nonlinear

equations having a small parameter. However they can not be applied to all nonlinear

problems as the small parameter does not exist generally in natural problems. Therefore

in a couple of years, a number of new methods have been established to remove the

“small parameter” assumption, such as the artificial parameter method proposed by Liu

[102], the momotopy analysis method by Liao [103], the homotopy perturbation method

introduced by He [104-106], the modified homotopy perturbation method [68] by Odibat

and the optimal homotopy asymptotic method (OHAM) by Marinca et al. [70].

Many fluids used in processing, manufacturing and chemical industry are considerably

non-Newtonian in nature. Industrial applications demand concentrated study of this kind

of fluid. Therefore it is essential to extend the theoretical analysis in more concise way

for wire withdrawal from Newtonian fluids to cater for materials of this type. Thus in our

studies, we only focus on these fluids, especially on elastic non-Newtonian fluids. In this

perspective, we present chapters 2-10.

In second chapter, we studied the wire coating process for heat transfer flow of a

viscoelastic PTT fluid. The exact solutions for velocity and temperature distribution are

derived. Moreover, the expression for thickness of coated wire, shear and normal stress

are also reported. It is concluded that the thickness of coated wire increases with increase

of Deborah number and elongational parameter. It is also found that the temperature

distribution also rises with increase of Brinkman number, Deborah number and the

elongational parameter.

23

In third chapter, the polymer is considered to be satisfies the PTT fluid mdel. The volume

flow rate, thickness of coated wire, force on total wire, stresses and temperature

distribution are derived. The above theoretical analysis is performed in a more realistic

situation that the polymer slips at the boundary of coating die. In this analysis it is

observed that the only temperature distribution decreases with increase of slip parameter

and other reported results show increase with increase of slip parameter. This analysis

explores an important result that without changing of coating materials, the thickness of

coated wire can be increased because the slippage of the polymer at the boundary of die

is not an expensive procedure.

In fourth chapter, we examined the wire coating in a canonical type die using cylindrical

co-ordinates and consider that the polymer satisfies the third grade fluid model. Here we

take constant viscosity and viscosity as a function of temperature and discussed two cases

for temperature dependent viscosity (i) Reynolds model and (2) Vogel’s model. The

theoretical results are derived and investigated under the influence of non-dimensional

parameters. It is observed, that as the perturbation parameter increases the non-

Newtonian behavior of the fluid also increases. It is concluded that temperature

distribution in both Reynolds’s model and Vogel’s model is directly related with

Brinkman number.

In fifth chapter, we investigated the wire coating in a canonical die with a bath of

Oldroyd 8- constant fluid taking the effect of pressure variation in axial direction. The

velocity distribution, shear stress, volume flow rate and average velocity is evaluated and

the effect of velocity distribution on changing the thickness of polymer in a die is studied

with OHAM. Also the influence of Dilatant constant , the Pseudoplastic constant and

the pressure gradient is studied on velocity distribution and shear stress. Further more the

shear stress is examined with varying the viscosity parameter 0 of the melt polymer.

Sixth chapter of this thesis examined the problem arises from wire coating process. Exact

and OHAM solutions are obtained for unsteady second grade isothermal flow in straight

annular die. The exact mathematical tools are used for the problem when the wire is only

dragged in the pool of melt polymer in the axial direction inside the die and OHAM is

applied when the wire is dragged as well as oscillating. For stability measurements of the

OHAM, some time dependent linear and nonlinear problems having exact solutions are

24

solved. The velocity profile of the melt polymer has been reported at different times. It is

concluded that as we move from the surface of wire towards the boundary of die, the

amplitude of oscillation decreases and the velocity of fluids decreases from point to point.

In seventh chapter, we considered the study of wire coating problem outside the die with

heat transfer analysis for the purpose of cooling the coating material. The polymer

considered are satisfies the third grade fluid model. The velocity and temperature

distribution are derived using approximate analytical methods OHAM and PM. It is

concluded that the velocity of the fluid decreases as the non-dimensional parameter 0

increases and vice versa. Furthermore it has been found that the temperature distribution

strongly depends on the Brinkman number so that it increases as the Brinkman number

increases. Moreover, it has been the part of this study that the temperature decreases as

the ratio of the fluid velocities at the surface of continuum to that of fluid velocity at the

surface of coated wire increases.

In eighth chapter of this thesis, the steady flow of power law fluid model in post-

treatment of wire coating with linearly varying boundary temperature is considered. The

exact solution of the drag flow of the coated wire outside the die with heat transfer

analysis has been carried out. As a consequence, velocity field, volume flow rate, average

velocity, shear rate, thickness of coated wire, force on total surface wire and temperature

distribution have been derived exactly for the power law index parameter n is or is not

equal to 1. It is concluded that the polymer velocity reduces with increase of power law

index n . Moreover, the effect of linearly varying wall temperature along the direction of

flow results the highest temperature in the centre line of the fluid domain depends on the

non-dimensional parameter 0S .

In chapter nine, we analyzed the post-treatment of wire coating with heat transfer. The

polymer selected for coating is assumed to be obeys the elastico-viscous fluid model. The

velocity and temperature distribution have been obtained by means of PM and MHPM.

The variation of the polymer velocity, flow rate, thickness of coated wire, shear stress,

the total force on the surface of coated wire and temperature distribution are sketched and

investigated by varying various emerging parameters.

In chapter 10, conclusions and problems for future work are given.

25

Chapter 2

Exact Solution of Non-Isothermal PTT Fluid in Wire

Coating Analysis

26

In this chapter, the problem of heat transfer analysis is considered in wire coating

analysis. The flow is assumed to be viscoelastic obeying the non-linear rheological

constitutive equation of PTT fluid. Exact analytic expressions for axial velocity, shear

stress, normal stress, average velocity, volume flux and temperature distribution are

obtained. Thickness of coated wire and force on the total wire are also calculated. The

effects of different emerging parameters on the solution are discussed. The results

corresponding to Maxwell and linear viscous model can easily be obtained by setting

and equal to zero respectively.

2.1 Formulation of the problem

The geometry under consideration is shown systematically in Fig. 2.1, where a stationary

pressure type die of length L and radius dR kept at temperature d is filled with an

incompressible viscoelastic PTT fluid of constant density. The wire at temperature w

and of radius wR is dragged along the centre line of the die with velocity wU . The fluid is

acted upon by a constant pressure gradient dp dz in the axial direction. The wire and die

are concentric and the coordinate system is chosen at the centre of the wire, in which r is

taken perpendicular to the direction of fluid flow and z is taken in the direction of fluid

flow.

Figure 2.1. Schematic profile of wire coating in a pressure type die.

27

Assuming that the flow is steady, laminar, axisymmetric and neglecting the entrance and

exit effects. The fluid velocity and temperature distributions are considered as

0, 0, w ru , rS S and r . (2.1)

The model adopted here to illustrate the viscoelastic behavior of the fluid is PTT model

which may be expressed as [59, 60, 107-111],

1f tr

S S AS , (2.2)

in which is the constant viscosity coefficient of the fluid, is the relaxation time, trS

is the trace of the extra stress tensor S and 1A the deformation rate tensor given by

1

T A L L , (2.3)

where superscript T stands for the transpose of matrix.

The upper contra-variant convected derivative

S in Eq. (2.2) is defined as

TD

Dt

SS u S S u . (2.4)

The function f is given by Tanner [112] as

1f tr tr

S S . (2.5)

In Eq. (2.5), f trS is the stress function in which is related to the elongation

behavior of the fluid. For 0 , the model reduces to the well-known Maxwell model and

for 0 , the model reduces to Newtonian one.

Boundary conditions for the problem on the velocity field and temperature distribution

are

w ww R U , 0dw R , (2.6)

w wR , d dR . (2.7)

Using Eq. (2.1), the continuity Eq. (1.5) is identically satisfied and from Eqs. (1.6) and

(2.2) – (2.5), we arrive at

0

r

p, (2.8)

0

p, (2.9)

28

1

rz

p drS

z r dr

, (2.10)

2

2

10rz

d d dwk S

dr r dr dr

, (2.11)

2zz r z

dwf tr S S

drS , (2.12)

rz

dwf tr S

drS , (2.13)

dr

dwS zr . (2.14)

From Eqs. (2.8) and (2.9), it is concluded that p is a function of z only. Assume that the

pressure gradient along the axial direction is constant. Thus, we have dp dz , where

is constant.

Integrating Eq. (2.10) with respect to r , we get

1

2rz

PS r

r

, (2.15)

where 1P is an arbitrary constant of integration.

Using Eq. (2.15) in Eq. (2.13), we obtain

1

2

dw

drf trP

rr

S . (2.16)

Combining Eqs. (2.12), (2.13) and (2.15), we obtain the explicit expression for normal

stress component zzS as

2

122

zz

PS r

r

, (2.17)

According to Eq. (2.16) and definition of f trS given in Eq. (2.5), we have

112

z z

PdwS r

dr r

. (2.18)

29

Inserting Eq. (2.17) in Eq. (2.18), we obtain an analytical expression for axial velocity

gradient as

32

1 1

3

12

2 2

P Pdwr r

dr r r

. (2.19)

Here we list the basic formulas related the wire coating analysis for future use in our

work.

The average velocity of polymer is

2 2

2d

w

R

ave

d w R

w r w r drR R

. (2.20)

At some control surface downstream, the volume flow rate of coating is

2 2w c wQ U R R , ` (2.21)

where cR is the radius of the coated wire.

The volume flow rate is

2

d

w

R

R

Q rw r dr . (2.22)

The thickness of the coated wire can be obtained from Eqs. (2.21) and (2.22) as

1

2

2 2.

d

w

R

c w

w R

R R r w r drU

(2.23)

The force on the wire is calculated by determining the shear stress at the surface of wire

given by

1 .2w

w

rz r R

r R

PS r

r

(2.24)

The force on the total wire surface is

2w

w w rz r RF R LS

. (2.25)

Introduce the following dimensionless parameters

30

11 2

2, , , ,w

w w d w w

Pr wr w P

R U R

2

, , , 1c c w dc

w w d w w

U U U RDe X Br

R U k R

, (2.26)

in which 2 8c wU R is the characteristic velocity scale, cDe is the characteristic

Deborah number based on the velocity scale cU , X has the physical meaning of a non-

dimensional pressure gradient and Br is the Brinkman number.

Using these new variables, Eqs. (2.11) and (2.19) after dropping the asterisks take the

following form

2 3 2

1 1

2 2 3 2

1 1 3

14 4 128 384

1 1384 128 ,

c c

c c

dwrX P X X De r X P De r

dr r

X P De X P Der r

(2.27)

2

14 0.d d dw

r BrX r Pdr dr dr

(2.28)

The boundary conditions given by Eqs. (2.6) and (2.7) become

11 w and 0w , (2.29)

01 and 1 . (2.30)

The non-dimensional expressions for average velocity, volume flow rate, thickness of

coated wire, shear stress and force on the total surface of wire are

2 2

2

12

ave d w

ave

w w

w R Rw r w r dr

R U

, (2.31)

2

12 w w

QQ r w r dr

R U

, (2.32)

1

2

1

1 2 ,cc

w

RR r w r dr

R

(2.33)

11

1

4(1 ),rzrz r

c w r

SS P

U R

(2.34)

1.

8

ww rz r

FF S

L (2.35)

31

The stress components given in Eqs. (2.15) and (2.17) can also be presented in non-

dimensional form by using the various non-dimensional parameters mentioned above as

follows:

14( ),rzrz

c w

S PS r

U R r

(2.36)

2132( ) .

z z

z z

c c w

S PS r

De U R r

(2.37)

2.2 Solution of the problem

To obtain the solution for velocity field, we integrate Eq. (2.27) with respect to r and

after considerable simplification, we find that

2 2 4 2 2

1 1

2 2 3 2

1 1 22

2 4 ln 32 192

1384 ln 64 ,

c c

c c

w r Xr P X r X De r XP De r

X P De r X P De Pr

(2.38)

where 2P is another constant of integration to be determined. The expression given in Eq.

(2.38) represents the general solution for velocity field. Now we proceed to find the

constants involved in the velocity field. For this, we insert the boundary conditions given

in Eq. (2.29) into Eq. (2.38), after considerable simplification we end up with a cubic

equation

3 2

1 0 1 1 1 2 0P B P B P B , (2.39)

with coefficients

02 10 1 2

3 3 3

, , ,AA A

B B BA A A

where

2 2 4

0 2 1 16 1 1,cA X De

2 2

1 4 ln 48 1 ,cA X De

2

2 384 ln ,cA X De 2

3 2

164 1 .cA X De

The real root to the cubic Eq. (2.39) can be obtained explicitly by the formula for third

order algebraic equations [58, 60] as

32

1 3 1 3 01 ( ) ( )

3

BP sign M M sign N N , (2.40)

with

2 31 2 1 2

2 2

0 0 1 01 2

, , ,2 2 4 27

2, .

3 3 27

S S S RM T N T T

B B B BR B S B

.

The second constant of integration 2P appearing in the axial velocity profile can be

determined from Eq. (2.38) by setting the no-slip condition at the die wall as

2 32 2 4 2 2 2 2 1

2 1 1 1 2

322 2 ln 16 96 192 ln c

c c c

De PP X P De De P P De

. (2.41)

Substituting the constants from Eqs. (2.40) and (2.41), the particular solution for velocity

profile is obtained which depends strongly on elongation behavior of the fluid, Deborah

number and the ratio between the wire (uncoated) and die.

Now using the particular solution for velocity field into Eq. (2.32) and solving the

resulting equation, we obtain the analytical expression for volume flow rate as

2 2 2 2 4

1 1 2 1

2 6 3 2 2 2 2

1 1 1

1 196 1 48 1

2 2

161 64 ln 2 96 ln .

3

c c

c c c

Q X P P De P P De

De P De P P De

(2.42)

The force on the total wire can be determined from Eqs. (2.34) and (2.35) as

14(1 ).wF P (2.43)

Here we conclude an important result that the force on the wire is constant at any point of

the wire for fixed values of the parameters involved in 1C .

Now setting the dimensionless volume flow rate in Eq. (2.33), we end up with an

expression for the thickness of the coated wire as

1 2

2 2 2 2 4

1 1 2 1

2 6 3 2 2 2 2

1 1 1

1 11 2 96 1 48 1

2 2.

161 64 ln 2 96 ln

3

c c

c

c c c

X P P De P P De

R

De P De P P De

(2.44)

33

Now substituting the expression for velocity profile in the energy equation (2.28) and

solving the resulting equation corresponding to boundary conditions given by Eq. (2.30),

we obtain the temperature distribution as

2 6 6 4 41 2

2 23 54

2 2

ln ln16 1 1 1 1

6 ln 4 ln

ln 1 ln 11 1 ln ln ln 1 1 ,

2 ln 2 2 ln

D Dr rr Br X r r

D DDr rr r r

r

(2.45)

where the values of the constants 1D , 2D , 3D , 4D and 5D are

2

1

16,

3cD De 2 2

2 1 1

32 124 ,

3 3c cD De P De P 2 1

3 196 ,2

c

PD De P

2 3 2

4 1 1128 ,cD De P P 2 4

5 116 .cD De P

2.3 Results and discussion

The effect of viscoelastic parameter 2

cDe , the dimensionless number X and Brinkman

number Br are discussed. Figs. 2.2 and 2.3 present the velocity profiles as a function of

r for several values of dimensionless number X and 2

cDe . In Fig. 2.2, we varied the

ratio between the pressure drop and the speed of wire ,c wX U U

i.e., 1, 0.25, 0.5, 0.8X , and fixed 2 10, 2cDe . The figure shows that the rise in

pressure gradient increases the speed of flow. The Fig. 2.3 is sketched for

2 0.1, 1, 5, 10cDe by fixing 0.5, 2X . It is obvious from Figs. 2.2 and 2.3 that the

velocity increases with an increase in dimensionless parameter X and 2

cDe

respectively. For low elasticity 2 0.1cDe the velocity disparity in Fig. 2.3 diverges

little from the Newtonian one, however when 2

cDe is increased, these profiles turn into

more flattened showing the shear-thinning effect. It can be seen that as is reduced, the

profiles turn to the Newtonian one and the result is therefore independent of cDe . Figs.

2.4 and 2.5 are plotted for variation of shear stress and normal stress respectively for

different values of 2

cDe . It is to be noted that both the stresses increase with increase in

parameter 2

cDe but this increase is insignificant for small and large values of 2

cDe .

34

Moreover, the normal stress profiles in Fig. 2.5 over shoot at the centre of annulus. The

normal stress (Fig. 2.5) decreases for 1, 1.42 and increases for 1.42 at fixed

values of 2.5, 2X . The maximum values of the normal stress exist at the

boundaries. Fig. 2.6 is plotted for variation of volume flow rate for various values of

2

cDe fixing 0.65X . Here, it is observed that the flow rate is same for1 1.4 , for

different values of 2

cDe and increases with increase of 2

cDe when 1.4 . The

expression in Eq. (2.45) representing the thickness of coated wire is plotted in Fig. 2.7 for

various values of 2

cDe when 0.65X . These curves are of particular interest which

shows that the curves give same thickness when 1 1.3 and increases with an increase

of 2

cDe for 1.3 . The considerable sighting from Fig. 2.7 is, however, that near the

continuum (wire), there is no influence of 2

cDe . The dimensionless force on the total

wire versus is sketched in Fig. 2.8 for different values of 2

cDe when 0.5X . Here, it

is observed that the force gradually increases when we increase 2

cDe but this increase

becomes comparatively small near to the die wall. In Figs. 2.9-2.11, we plotted the

dimensionless temperature profiles r versus with selected sets of parameters. It

can be observed that the temperature profile attains its maximum value at the centre of

the annular gap for different values of Br and 2

cDe , then it decreases as to meet the far

field boundary conditions for fixed parameters. Comparing four curves in each figure, we

find that the temperature at fixed location increases with the Brinkman number Br , 2

cDe

and X . However, this rise is relatively small in Figs. 2.11 for increasing X .

35

Figure 2.2. Dimensionless velocity profiles for different values of X at fixed values of

2 10, 2.cDe


cDe at fixed values of

0.5, 2.X


cDe at fixed

values of 1.5, 2.X

36


cDe at fixed

values of 2.5, 2.X

Figure 2.6. Dimensionless volume flow rates versus ratio of the radii for different values

of 2

cDe at fixed values of 0.65.X

Figure 2.7. Thickness of coated wire versus ratio of the radii for different values of

2


37

Figure 2.8. Force on the surface of the total wire for different values of 2

cDe at fixed

value of 0.5.X


number at fixed values of 20.5, 10, 2.cX De


cDe at

fixed values of 0.2, 4, 2.X Br

38


values of 2 0.5, 2, 2.cDe Br

2.4 Conclusion

Analytical solutions are derived for the axisymetric flow of nonlinear viscoelastic PTT

fluids in wire coating analysis. Expressions are presented for the radial variation of the

axial velocity and the temperature distribution. In engineering point of view, some results

were also derived such as for the flow rate, average velocity, shear stress, normal stress,

thickness of coated wire and force on the total wire. It was found that the climax axial

velocity takes place at the centre of the annulus and it depends upon the parameters X

and 2

cDe . Moreover, the velocity increases with increasing value of these parameters. It

is also found that the shear stress and normal stress increases with increasing 2

cDe in the

range 210 100cDe . The flow rate, thickness of coated wire and the force on the total

wire increase with increase of 2

cDe . The fluid temperature depends upon 2, cBr De and

X and it increases very quickly with increasing values of these parameters, especially for

Br and 2

cDe . The current investigation is more universal when compared with Maxwell

and linear viscous model. Our results respectively, reduce to Maxwell and linear viscous

model by setting and equal to zero.

39

Chapter 3

Wire Coating with Heat Transfer Analysis Flow of a

Viscoelastic PTT Fluid with Slip Conditions

40

This chapter extended the work presented in chapter 2 and the same problem is examined

with slip boundary conditions. Exact solutions for an incompressible viscoelastic PTT

fluid in a die are obtained. The effects of slip parameter as well as 2

cDe (viscoelastic

index) on the axial velocity, shear stress, normal stress, average velocity, volume flux and

temperature distribution are examined. The force on the surface of total wire and the

thickness of coated wire are also studied. The results reduce to the solution of no slip

boundary value problem by letting the slip parameter to be zero. Furthermore, the results

for Maxwell and viscous model can be recovered by setting and equal to zero

respectively.

3.1 Formulation and solution of the problem

The geometry of the problem is same as discussed in chapter 2. The following

assumptions are made during formulation of the problem:

The die is occupied by polymer melt.

The polymer flow is steady, laminar and axisymmetric.

Entrance and exit effects are neglected.

Slippage occurs along the contacting surfaces of the wire, polymer and the die.

We assume that

0, 0, w ru , ,rS S (3.1)

r . (3.2)

The universal appearance of the constitutive equation presenting the PTT fluid [57, 58,

107-111] is

2f tr

S S S D, (3.3)

where is the constant viscosity coefficient of the model, the relaxation time, tr S the

trace of the extra stress tensor S and D the deformation rate tensor given by

1,

2

TD u+ u (3.4)

where T denotes the transpose of matrix.

The upper contra-variant convected derivative

S in Eq. (3.3) is defined as

.TD

Dt

SS S S u u (3.5)

41

The stress function f has an exponential form but can be linearized when the

deformation rate is small [114]

1f tr tr

S S , (3.6)

where f tr S is the stress function and is the elongational parameter of the model.

For 0 , the model in Eq. (3.3) reduces to the well-known Maxwell model and for

0 , this model reduces to Newtonian one.

The boundary conditions for the stated problem arise from slip at the die and wire walls

on the velocity field and temperature distribution are given by [113]

w

w w r zr R

w R U S

, d

d r zr R

w R S

, (3.7)

w wR , d dR . (3.8)

Upon making use of velocity field, the continuity Eq. (1.5) is satisfied identically.

Substituting the velocity field in the Eqs. (1.6) and (3.3) – (3.6), we get the following

nonzero components of stress tensor S as

2z z r zSdw

f tr Sdr

S , (3.9)

r z

dwf tr S

drS . (3.10)

Consequently, the momentum equation in the absence of body forces and the energy

equation (1.11) take the form

0

r

p, (3.11)

0

p, (3.12)

1

r z

p dr S

z r dr

, (3.13)

2

2

10r z

d d dwk S

dr r dr dr

. (3.14)

From Eqs. (3.11) and (3.12), it is concluded that p is a function of z only. Assume that

the pressure gradient along the axial direction is constant. Thus, we have dp dz ,

where is constant.

42

The reduction of Eq. (3.13) to the first order is given by

2

ar z

KS r

r

, (3.15)

where aK is an arbitrary constant of integration.

Substitute Eq. (3.15) in Eq. (3.10), we obtain the stress function in the following form:

2a

dw

drf trK

rr

S . (3.16)

Similarly, making use of Eq. (3.10) and (3.15) in Eq. (3.9), we obtain the normal

component of shear stress given by

2

22

az z

KS r

r

. (3.17)

According to Eq. (3.16) and definition of f tr S given in Eq. (3.6), we have

12

az z

KdwS r

dr r

. (3.18)

After substitution of Eq. (3.17) in Eq. (3.18), we obtain 32

3

12

2 2

a aK Kdwr r

dr r r

. (3.19)

Introduce the following dimensionless parameters

2

2, , , , ,w w a

a

w w d w w

R Kr wr w K

R U R

2

, , , 1,c c w dc

w w d w w

U U U RDe X Br

R U k R

(3.20)

in which 2 8c wU R is the characteristic velocity scale, cDe is the characteristic

Deborah number based on the velocity scale cU , X has the physical meaning of a non-

dimensional pressure gradient and Br is the Brinkman number.

Thus in non-dimensional form, after dropping the asterisks, the differential equations for

velocity and temperature distribution become

2 3 2

2 2 3 2

3

14 4 128 384

1 1384 128 ,

a c a c

a c a c

dwrX K X X De r X K De r

dr r

XK De XK Der r

(3.21)

43

24 0a

d d dwr BrX r K

dr dr dr

. (3.22)

The boundary conditions presented in Eqs. (3.7) and (3.8) take the following form

1 1 4 1 aw X K , 4 aKw X

, (3.23)

01 , and 1 . (3.24)

and the system of Eqs.(2.21), (2.23) – (2.26) reduce to

2 2

2

12

ave d w

ave

w w

w R Rw r w r dr

R U

, (3.25)

2

12 w w

QQ r w r dr

R U

, (3.26)

1

2

1

1 2 ,cc

w

RR r w r dr

R

(3.27)

1

1

4(1 ),rzrz ar

c w r

SS K

U R

(3.28)

1.

8

ww rz r

FF S

L (3.29)

The non-dimensional form of the stress components given in Eq. (3.15) and (3.17) is

given by

4( ),arzrz

c w

KSS r

U R r

(3.30)

232( ) ,

z z az z

c c w

S KS r

De U R r

(3.31)

Now we solve Eq. (3.21) to get fluid velocity corresponding to slip boundary conditions

given in Eq. (3.23). Accordingly, we integrating Eq. (3.21) with respect to r , we obtain

2 2 4 2 2

2 2 3 2

2

2 4 ln 32 192

1384 ln 64 ,

a c a c

a c a c b

w r Xr K X r X De r XK De r

XK De r XK De Kr

(3.32)

where bK is another constant of integration to be determined. The expression for velocity

field given in Eq. (3.32) involves two constants aK and bK can be determined explicitly

by using the boundary conditions given in Eq. (3.23). If we insert the boundary

conditions given in Eq. (3.23) into Eq. (3.32), after simplification we end up with a cubic

equation:

44

3 2

0 1 2 0a a aK L K L K L , (3.33)

with coefficients

02 10 1 2

3 3 3

, , ,HH H

L L LH H H

where

2 2 4

0 2 1 16 1 4 1 1,cH X De

2 2

1

14 ln 48 1 2 1 ,cH X De

2

2 384 ln ,cH X De 2

3 2

164 1 .cH X De

The real root to the cubic Eq. (3.33) can be obtained explicitly by the formula for third

order algebraic equations [59, 60]:

1 3 1 3 0( ) ( )3

a

LK sign Y Y sign Z Z , (3.34)

with

2 22 31 2 1 2 0 0 1 0

1 2

2, , , ,

2 2 4 27 3 3 27

L L L LS S S RY T Z T T R L S L .

The second constant of integration bK appeared in the axial velocity profile can be

determined according to the procedure defined:

If both the boundaries are stationary or both moving, then use of any boundary condition

gives this constant. On the other hand, if one boundary is stationary and the other is

moving with constant speed, then the constant can be determined from the condition on

the stationary boundary, otherwise the solution will not satisfy both the boundary

conditions. Here we determine bK from Eq. (3.32) by setting the no-slip condition at the

die wall given as follows:

32 4 2 2 2

2

322 2 ln 16 96 192 ln 2a a

b a a a c

K KK X K K K De

.

(3.35)

Thus, the analytic expression for velocity profile is obtained by substituting aK and bK

in Eq. (3.32). Once using the non-dimensional velocity field in Eq. (3.26), we obtain the

non-dimensional volume flux as

45

2 2 2 2 4

2 6 3 2 2 2 2

1 196 1 48 1

2 2

161 64 ln 2 96 ln ,

3

a a c b a c

c a c a a c

Q X K K De K K De

De K De K K De

(3.36)

and the expression for the thickness of the coated wire is

1 2

2 2 2 2 4

2 6 3 2 2 2 2

1 11 2 96 1 48 1

2 2.

161 64 ln 2 96 ln

3

a a c b a c

c

c a c a a c

X K K De K K De

R

De K De K K De

(3.37)

The force on the total wire can be determined from Eq. (3.29) and (3.30) as

4(1 ).w aF K (3.38)

The above expression asserts an important result that the force on the wire is constant,

and depends upon the parameters involved in the constant aK .

Next, inserting w r from Eq. (3.32) with known values of aK and bK into Eq. (3.22)

and solving the resulting equation corresponding to the boundary conditions given in eq.

(3.24), we obtain the temperature field as

2 6 6 4 41 2

2 23 54

2 2

ln ln16 1 1 1 1

6 ln 4 ln

ln 1 ln 11 1 ln ln ln 1 1 ,

2 ln 2 2 ln

r rr Br X r r

r rr r r

r

(3.39)

where

2

1

16,

3cDe 2 2

2

32 124 ,

3 3c a c aDe K De K 2

3 96 ,2

ac a

KDe K

2 3 2

4 128 ,c a aDe K K 2 4

5 16 .c aDe K

We observe that the solution r given in Eq. (3.39) provides systematic explicit

expression for temperature distribution and is independent of Deborah number cDe ,

constant of integration aK , elongation parameter and strongly depend upon the

Brinkman number Br and the non-dimensional pressure gradient X .

46


Eqs. (3.21) and (3.22) along with the boundary conditions given in Eqs. (3.23) and (3.24)

are solved exactly for the velocity of fluid and temperature distribution respectively.

Also, the volume flow rate, average velocity, thickness of coated wire, shear stress,

normal stress and force on wire are derived by using velocity field. Figs. 3.1-3.3 show the

variation of velocity with r for several values of the slip parameter , 2

cDe and the

velocity ratio X respectively. We observe from these figures that the velocity increases

with increase in these parameters. In addition, the main contribution on the velocity field

is seen in Fig. 3.2 when 2

cDe increases. For low elasticity 2 0.01cDe , the velocity

deviation in Fig. 3.2 differs slightly from the Newtonian one but on increasing 2

cDe , the

velocity profiles become more flattened representing the effect of shear thinning. In Fig.

3.3, we varied the ratio between the pressure drop and the speed of wire c wX U U ,

i.e., 0.2, 0.5, 1, 1.5X and fixed 2 7.5, 2, 5cDe . It is to be noted that the

pressure gradient increases the speed of polymer flow. Figs. 3.4-3.7 are plotted for shear

and normal stresses for different values of slip parameter and 2

cDe . It is observed

from Figs. 3.4 and 3.7 that the slip parameter reduces both the stresses. Small effect of

the slip parameter is observed on shear and normal stresses. The shear stress follows a

linear change across the cavity depends on the involved parameters. In case of normal

stress, this reduction is in the form of parabolic and the profiles over shoots at the centre

of the annulus. Common to both the stresses is the finding that the effects of slip

parameter and 2

cDe are proportional to the value of 2

cDe : for small values of 2

cDe ,

say 2

cDe ≤ 0.1 and for large values of 2 100cDe , the shear and normal stresses show

very small variations respectively at fixed values of 2, cX De and . In order to illustrate

the influence of slip parameter and 2

cDe on the non-dimensional volume flow rate Q ,

the radius of the coated wire cR and the force on the total wire wF , Figs. 3.8–3.13 are

made respectively. In these figures, the slip parameter and 2

cDe is varied in the wide

47

range [0, 10] and (0, 1000] respectively. It is observed that the higher values of and

2

cDe lead to increasing Q , cR and wF profiles.

Figure 3.1. Dimensionless velocity profiles for different values of slip parameter at fixed

values of 21.5, 10, 2.cX De


cDe at fixed values

of 1, 2.5, 2.X

48

Figure 3.3. Dimensionless velocity profiles for different values of velocity ratio X at

fixed values of 2 7.5, 5, 2.cDe

Figure 3.4. Dimensionless shear stress profiles for different values of slip parameter at

fixed values of 22, 0.1, 2.cX De


cDe at fixed

values of 0.5, 0.2, 2.X

49


cDe at fixed

values of 0.5, 5, 2.X

Figure 3.7. Dimensionless normal stress profiles for different values of slip parameter

at fixed values of 21.5, 0.2, 2.cX De

50

Figure 3.8. Dimensionless volume flow rates versus for different values of slip

parameter at fixed values of 21, 0.5.cX De

Figure 3.9. Thickness of coated wire versus for different values of slip parameter at

fixed values of 21, 0.5.cX De

Figure 3.10. Effect of the slip parameter on the force of the total wire at fixed values

of 20.5, 0.5.cX De

51

Figure 3.11. Dimensionless volume flow rates versus for different values 2

cDe at fixed

values of 0.2, 10.X

Figure 3.12. Thickness of coated wire versus for different values of 2

cDe at fixed

values of 0.2, 10.X

Figure 3.13. Effect of the slip parameter on the force of the total wire for different

values of 2


52

Figure 3.14. Dimensionless temperature distributions for different values of slip

parameter at fixed values of 21.5, 0.1, 2, 10.cX De Br


number at fixed values of 20.5, 2, 10, 10.cX De


cDe at

fixed values of 1.2, 10, 2, 2.X Br

53


values of 2 0.5, 2, 2, 4.5.cDe Br

The expression for temperature r given by Eq. (3.39), is plotted in Figs. 3.14-3.17 in

which Fig. 3.14 is plotted for different values of slip parameter when

21.5, 0.1, 2, 10cX De Br , Fig. 3.15 for various values of Brinkman number

when 20.5, 2, 10, 10cX De , Fig. 3.16 for various values of 2

cDe

when 1.2, 10, 2, 2X Br and Fig. 3.17 is plotted for various values of

dimensionless parameter X when 2 0.5, 5, 2, 4.5cDe Br . This study tells that

the temperature increases with increasing values of the parameters 2, cBr De , X and

decreases with increase of slip parameter [107].

3.3 Conclusion

Effects of the slip parameter on the wire coating operation are discussed. It is noticed that

the variation in the fluid velocity, temperature distribution, volume flow rate, thickness of

coated wire and the force on the surface of wire with 2, cBr De , X and is quite

interesting. It is found that the velocity, volume flow rate, thickness of wire and the force

on the wire increases with increase of while the temperature distribution decreases with

increase of . Further, it was found that the shear stress and normal stress decreases with

increase of only for a limiting value of the boundary slip coefficient. The current

investigation is more universal when compared with Maxwell and linear viscous model.

54

The results reduce to no slip when the slip parameter is vanished. Also, it respectively,

reduces to Maxwell and linear viscous model by setting and equal to zero.

55

Chapter 4

Heat Transfer by Laminar Flow of a Third Grade Fluid in

Wire Coating Analysis with Temperature Dependent and

Independent Viscosity

56

In this chapter, the analysis of wire coating is performed using melt polymer satisfies

third grade fluid model. Constant viscosity and temperature dependent viscosity cases are

treated separately. The Reynolds and Vogel’s model [55] are account for the variable

viscosity. Investigation is carrying out by means of perturbation method. In each case,

solutions for the fluid velocity and temperature distribution have been derived

respectively. The influence of non-Newtonian parameter , Reynolds model parameter

m and Vogel’s model viscosity parameters 1 and B are investigated on velocity and

temperature distributions solution and are shown graphically.

4.1 Modeling of the problem

The geometry under consideration is shown systematically in Fig. 2.1, where the wire of

radius wR and temperature w is translating with velocity wU in a bath of third grade

fluid inside a stationary pressure type die of finite length L having radius dR and

temperature d . The coordinate system is chosen at the same way as discussed in chapter

2. The applicable equations are the differential equations of continuity, momentum and

energy listed in chapter 1, with certain modifications appropriate to our problem which

are as follows:

The flow is steady, unidirectional and axisymetric. The velocity and temperature fields

are defined as

rw,0,0u , rSS , r . (4.1)

Boundary conditions are

wUw , w at wRr ,

0w , d at dRr . (4.2)

For third grade fluid S , is defined as [58-60, 116]

123122123112211 AAAAAAAAAAS tr , (4.3)

in which is the coefficient of viscosity of the fluid, 32121 ,,,, are the material

constants and 321 ,, AAA are the line kinematic tensors defined by [56]

LLA T

1 , (4.4)

11 1 , 2,3T n

n n n

Dn

Dt

AA A L LA (4.5)

57

where the superscript T denotes the transpose of the matrix.

Under the above consideration of velocity field, the continuity Eq. (1.5) is satisfied

identically and Eq. (4.3) gives the nonzero components of the extra stress tensor S as

2

212

dr

dwSrr , (4.6)

2

2

dr

dwS zz , (4.7)

3

322

dr

dw

dr

dwS zr . (4.8)

Substituting the velocity field and the stress components given in Eq. (4.6-4.8) with the

incorporate of incompressibility and neglecting gravity, the balance of momentum given

in Eq. (1.6) becomes as follows

2

21

12

dr

dwr

dr

d

rr

p , (4.9)

0

p, (4.10)

3

3221

dr

dwr

dr

d

dr

dwr

dr

d

rz

p . (4.11)

Consider the flow is only due to the drag of wire, so the pressure gradient in the axial

direction is to be taken zero.

Hence from Eq. (4.11), we have

01

2

3

32

dr

dwr

dr

d

rdr

dwr

dr

d , (4.12)

and the energy Eq. (1.11) becomes

2 42

2 32

12 0.

d d dw dwk

dr r dr dr dr

(4.13)

58

4.2 Perturbation solution

4.2.1 Constant viscosity case

In this case, the non-Newtonian parameter 0 will be assumed small to perform regular

perturbation.

For better understanding and universal use, we transform the Eqs. (4.12) and (4.13) with

the corresponding boundary conditions given in Eq. (4.2) in terms of non-dimensionless

variables. For this purpose, we consider wR as characteristic length, wU as the

characteristic velocity, w and d as characteristic temperatures and introduce the

following dimensionless variables

,,,wd

w

ww U

ww

R

rr

0 2 3, 1d

w

R

R . (4.14)

The Eqs. (4.12), (4.13) and Eq. (4.2) after dropping the asterisks, become

032

32

2

2

02

2

dr

dw

dr

dw

dr

wdr

dr

dw

dr

wdr , (4.15)

11 w and 0w , (4.16)

021

4

0

2

2

2

dr

dwBr

dr

dwBr

dr

d

rdr

d , (4.17)

01 and 1 , (4.18)

where the Brinkman number

2

00 2

2

,w

d w w

w

UBr

k R

U

. (4.19)

To, approximate Eq. (4.15) subject to the boundary conditions given in Eq. (4.16), 0 is

considered to be a small perturbation parameter and the approximate velocity profile as

2

0 0 0 1 0 2, ...w r w r w r w r , (4.20)

is substituted into Eqs. (4.15) and (4.16) and collecting the order of 0 yields

20 0 0

0 2: 0

d w dwr

dr dr , (4.21)

0,11 00 ww , (4.22)

59

2 2 221 0 0 01 1

0 2 2: 2 6 0

dw dw d wd w dwr r

dr dr dr dr dr

, (4.23)

0,01 11 ww , (4.24)

2 2 22 22 0 0 0 02 2 1 1 1

0 2 2: 6 12 6 0

dw dw dw dwd w dw dw dw d wr r r

dr dr dr dr dr dr dr dr dr

, (4.25)

0,01 22 ww , (4.26)

Solving Eq. (4.21) corresponding to the conditions given in Eq. (4.22), we have

ln

ln10

rw . (4.27)

Substituting Eq. (4.27), into Eq. (4.23) and solving with respect to boundary conditions

given in Eq. (4.24), we obtain

2231

11

ln

ln11

ln

1

r

rw . (4.28)

Solving Eq. (4.25) with the help of Eqs. (4.27) and (4.28), finally one has

4

2

4

2

2

2

272

11ln

11lnln

11ln

11ln

ln

3

rr

rrw

. (4.29)

The second order approximation to the velocity field is

0

3 2 2

2 2220

7 2 2 4 4

ln 1 ln 11 1 1

ln lnln

3 1 1 1 1ln 1 ln 1 ln ln 1 ln 1 .

ln

r rw r

r

r rr r

(4.30)

Next, we evaluate the temperature distribution by the perturbation method using the non-

Newtonian parameter as the perturbation parameter.

The temperature distribution in terms of perturbation expansion can be represented as

follows:

2

0 0 0 1 0 2, ...r r r r . (4.31)

Substituting Eq. (4.31) into Eqs. (4.17) and (4.18), and collecting the coefficients of

0 1 2

0 0 0, , , we obtain

60

(4.32)

1,01 00 , (4.33)

421 0 01 1 1

0 2: 2 2 0

dw dwd d dwr rBr rBr

dr dr dr dr dr

, (4.34)

1 11 0, 0 , (4.35)

3 222 0 02 2 1 1 2

0 2: 8 2 0

dw dwd d dw dw dwr rBr rBr rBr

dr dr dr dr dr dr dr

, (4.36)

0,01 22 . (4.37)

The solution of system of Eqs. (4.32) and (4.33) is given by

rBrBr

rln2ln

ln2

ln20

. (4.38)

Substituting Eq. (4.38) into Eq. (4.34) and integrating twice, we obtain

22

2

51

11ln

11lnln2ln2ln

ln2 rrrr

Br

. (4.39)

Similarly, use of Eqs. (4.38) and (4.39) into Eq. (4.36) and solving corresponding to

boundary conditions given in Eq. (4.37) yields

2

2 8 2 2 2

2

2 2 4 4

1 17 1ln 8 14 ln 1 11 ln ln 1

4 ln

1 1 ln 1 18ln 1 1 12ln ln 1 3 ln 1 .

ln

Brr r r

rr

r r

(4.40)

Combining the solution to second order, one finally has

20

2 5 2

22

0

82 2 2 2

2 2 4

ln 1ln 2 ln ln 2 ln 2ln ln 1

2 ln 2 ln

1 1 17 1ln 1 ln 8 14 ln 1 11 ln ln 1

4 ln

1 1 ln 18ln 1 1 12ln ln 1

ln

Brrr Br Br r r r r

r r rr

rr

r

2

4

13 ln 1 .

r

(4.41)

220 0 0 0

0 2: 0,

d d dwr r Br

dr dr dr

61

4.2.2 Temperature dependent viscosity

For temperature dependent viscosity Reynolds and Vogel’s models are employed.

4.2.2a Reynolds model

In this case, Reynolds model [55] is used to account for the temperature dependent

viscosity.

Non-dimensionless momentum and energy equations with boundary conditions are

032

32

2

2

02

2

dr

dw

dr

d

dr

dw

dr

dw

dr

wdr

dr

dw

dr

wdr

, (4.42)

11 w and 0w , (4.43)

021

4

0

2

2

2

dr

dwBr

dr

dwBr

dr

d

rdr

d , (4.44)

01 , and 1 . (4.45)

The non-dimensional parameters are

,,,wd

w

ww U

ww

R

rr

1,320

w

d

R

R,

0

2

2

0

0

0

2

0 ,,

w

wwd

w

U

Rk

UBr ,

where 0 is a reference viscosity. For Reynolds model, the dimensionless viscosity

Lexp (4.46)

can be used for variation of viscosity with temperature. The approximate solution of Eqs.

(4.15) – (4.18) can be obtained by choosing the non-Newtonian parameter 0 as the

perturbation parameter and selecting 0L m .

Using Taylor series expansion, one has

0 01 ,d d

m mdr dr

. (4.47)

Inserting Eqs. (4.20), (4.31) and (4.47) in Eqs. (4.15) – (4.18), and separating at each

order of approximation yields

62

(4.48)

0,11 00 ww , (4.49)

(4.50)

1,01 00 , (4.51)

2 2 221 0 0 0 0 0 01 1

0 02 2

2

0 0

2

: 2 6

0,

dw dw d w dw dw dd w dwr r m mr

dr dr dr dr dr dr dr dr

d w dmr

dr dr

(4.52)

0,01 11 ww , (4.53)

4 221 0 0 01 1 1

0 02: 2 2 0,

dw dw dwd d dwr rBr rBr m rBr

dr dr dr dr dr dr

(4.54)

0,01 11 , (4.55)

2 2 22 22 0 0 0 02 2 1 1 1

0 2 2

2

0 0 01 1 1 11 0 0 2

: 6 12 6

0,



dw d dwdw dw d d wm m mr mr mr


(4.56)

0,01 22 ww , (4.57)

3 222 0 02 2 1 1 2

0 2

2

0 0 11 0

: 8 2

2 0,



dw dw dwmr Br mr Br

dr dr dr

(4.58)

0,01 22 . (4.59)

Solving these problems in conjunction with the corresponding boundary conditions, we

obtain

ln

ln10

rw , (4.60)

rBrBr

rln2ln

ln2

ln20

, (4.61)

20 0 0

0 2: 0,

d w dwr

dr dr

220 0 0 0

0 2: 0,

d d dwr rBr

dr dr dr

63

2

1 4 2 2

1 1 112ln 1 12ln 1 ln 6ln 2 ln ln ,

12 lnw r m r Br r Br

r

(4.62)

2

1 5 2

2 2 3 3

2 2 2

2

112ln 24 ln 24ln ln 1

24 ln

ln2 ln ln ln ln ln ln

ln

1ln ln 4 ln 8 ln 12ln ln 12ln 1 ,

Brr r r

rmBr r mBr r r

m r r rr

(4.63)

2

2 7 2 2 2 4

2

2 2

3 2 2

2 4 2

2

2

1 1 1 1 13ln 1 3ln 1 1 3ln ln 1

ln

5 1 1 1ln ln 1 ln ln 1

4 4

1 1 1 5 1ln ln 1 3 ln 1 ln 1

2 4

12 ln ln 1

w r rr

mBr r m Br r

mBr r mBrr r

m r

2

2 2

2 2 2 2

2 2 2

3 3 3

2 2 2 2

3 3

2 2

5 1 1ln ln

4

3 1 1 1 3 1ln ln 1 ln ln 1 ln ln 1

4 2 4

1 1 1 12 ln 1 ln 1 2 ln ln

1 1 12 ln ln 1 ln ln 1

2

mBr rr

mBr r m r mBr rr

m Br m rr r r

m r mBr rr

4

2

4 4 32

2

3 1ln 1

2

1 1 7ln 1 ln ln 2

4 48

mr

mBr m Br r Brr

2 4 22

2 4 52 2

6 2 2 5 22 2

2

2

1ln ln 8 ln ln 8ln 3 ln

24

1 1ln ln ln ln ln ln ln ln

4 24

1 7ln ln 20 ln ln

240 120

12 ln ln ln ln ,

m Br r Br r r Br r

m r Br r m Br r Br r

m r Br m Br r

m Br r rr

(4.64)

64

2 22

2 8 2 2

2

2 2 2

2 2 2

4 2

3

2

1 1 12880 ln 1 5040 ln 1

1440 ln

1 1 12880 ln 1 1 2340 ln ln 1

1 14320 ln ln 1 1800 ln ln 1

1120 ln ln 1 240 ln

Br r Br r

Br Br rr

Br r m Br r

m r m r

4 2

2 4

2 2 2

2 4

2 2 3 2

2 2

3 2 3

2 2

1 1ln 1 1080 ln 1

1 12340 ln 1 4320 ln ln 1

1 12880 ln ln 1 ln 1 480 ln ln 1 1

1 12880 ln 1 1440 ln 1

840

Brr

m Br Br rr r

mBr r r mBr r Br

mBr m Brr r

m

2 2 3 4

2 2

2 4 2 4

2 2

4

2 2 4

2 2 2 2 2

2 2

1 1ln ln 1 1440 ln 1

1 1240 ln 1 600 ln ln 1

11 28 17240 ln ln 13 360 ln ln 11

9 8 5360 ln ln 1 360 ln ln

Br r mBrr

m Br m Br rr

mBr r mBr r

m Br r m Br rr

2 2

2 3 2 5 22

2

3 5 2 4 42 2

7 2 3 4 42 2

44

3720 ln ln 5 14 ln ln 3 2 ln ln

30 ln ln 8 1 120 ln ln 2 1

3 ln ln 12 30 50 ln ln

r

mBr r m Br r Br Br r

m Br r Br Br m Br r Br

m Br r Br Br m Br r

2 3 2 5 22

2

3 5 2 4 42 2

7 2 2 6 22 2

3

2 2 2 2

3720 ln ln 5 14 ln ln 3 2 ln ln

30 ln ln 8 1 120 ln ln 2 1

3 ln ln 12 30 ln ln 11 120 60

36 24 16 12120 ln ln 12 11

mBr r m Br r Br Br r

m Br r Br Br m Br r Br

m Br r Br Br m Br r Br Br

mBr r Brr r

. (4.65)

65

4.2.2b Vogel’s model

In this case, the temperature dependent viscosity is taken as

w

B

Dexp . (4.66)

Using expansion, we have

21 1B

D , (4.67)

where

w

B

Dexp1 and BD, are viscosity parameters associated to Vogel’s

model [57].

The approximate solutions of Eqs. (4.15) – (4.18) can be obtained by selecting 0D b ,

where 0 is introduced a non-natural small parameter which can be eliminated at the end

to recurred the original parameter.

Moreover, inserting Eqs. (4.20), (4.31) and (4.67) in Eqs. (4.15) – (4.18), and separating

at each order of approximation, one obtains

20 0 0

0 2: 0

d w dwr

dr dr , (4.68)

0,11 00 ww , (4.69)

220 0 0 0

0 12: 0

d d dwr r Br

dr dr dr

, (4.70)

0 01 0, 1 , (4.71)

2 2 221 0 0 0 01 1 1

0 1 1 02 2 2

2

0 0 0 01 1

2 2 2

: 2 6

0,

dw dw d w dwd w dwr r b

dr dr dr dr dr B dr

dw d d w dbr br

B dr dr B dr dr

(5.72)

0,01 11 ww , (4.73)

66

4 221 0 0 01 1 1 1

0 02 2: 2 2 0

dw dw dwd d dwr rBr rBr m r Br

dr dr dr dr dr B dr

, (4.74)

1 11 0, 0 , (4.75)

2 2 22 22 0 0 0 02 2 1 1 10 2 2

2

0 0 01 1 1 1 11 0 02 2

: 6 12 6

0,



dw d dwdw dw d d wb m br

B dr dr dr dr dr dr dr

(4.76)

0,01 22 ww , (4.77)

3 222 0 02 2 1 1 2

0 2

2

0 01 11 02

: 8 2

2 0,



dw dw dwbrBr brBr

B dr dr dr

(4.78)

0,01 22 . (4.79)

Finally, solving the above problems in conjunction with corresponding boundary

conditions, we obtain

ln

ln10

rw , (4.80)

rBrBr

rln2ln

ln2

ln1120

, (4.81)

2 2

1 4 2 22

1

11 1

1 1 112 ln 1 12 ln 1

12 ln

ln6ln 2 ln ln ln ln ,

ln

w B B rrB

rBr r Br b r

(4.82)

67

2 2

1 5 22

2 2 3 32 2

1 1

2 2 2 2

1 2

112ln 24 ln 24ln ln 1

24 ln

ln2 ln ln ln ln ln ln

ln

14 ln ln ln 2 ln 3ln ln 12 ln 1 ,

Brr r r B

B

rb Br r bBr r r

b r r r Br

(4.83)

2

4 4 4

2 7 2 2 2 44 2

1

22 2 2 2

1 12 2

3 22 2 4 2 2

1 12 4

1 1 1 1 13 ln 1 3 ln 1 1 3 ln ln 1

ln

5 1 1 1ln ln 1 ln ln 1

4 4

1 1 1 5ln ln 1 3 ln 1

2 4

w B r B B rrB

bB Br r b B Br r

bB Br r B bBr

2

2

2 22 2 2

1 12 2 2

2 2 22 2 2

1 12 2

2 3 32 2 2

1 12 2 2

1ln 1

1 5 1 12 ln ln 1 ln ln

4

3 1 1 1ln ln 1 ln ln 1

4 2

3 1 1 1ln ln 1 2 ln 1 ln 1

4

2

Brr

bB r bB Br rr

bB Br r bB rr

bBr r bB B Brr r

3 32 2

1 12 2 2

3 42 2 2

1 12 2

4 4 32 2 2 3

1 12

2 4 22 3

1 1

1 1 1ln ln 2 ln ln 1

1 1 3 1ln ln 1 ln 1

2 2

1 1 7ln 1 ln ln 2

4 48

1ln ln 8 ln ln 8ln 3 ln

24

1

4

bB r bB rr r

bB Br r bBr

bB Br b Br r Brr

b Br r Br r r Br r

b

2 4 52 3 2 3

1 1 1 1

6 2 2 5 22 2 2 2

1 1

2

2

1ln ln ln ln ln ln ln ln

24

1 7ln ln 20 ln ln

240 120

12 ln ln ln ln ,

r Br r b Br r Br r

b r Br b Br r

b Br r rr

(4.84)

68

2 224 4

2 8 2 24

24 2 2

12 2 2

2 2 24 2 2 2 2

1 14 2

1 1 12880 ln 1 5040 ln 1

1440 ln

1 1 12880 ln 1 1 2340 ln ln 1

1 14320 ln ln 1 1800 ln ln 1 120

BrB r BrB rB

BrB Br B rr

BrB r b Br B r bB

3

4 22 2 2 2

1 12 2 4

2 2 22 2 4

1 2 4

2 2 22 2

1 12 2

2

1

ln

1 1 1ln 1 240 ln ln 1 1080 ln 1

1 12340 ln 1 4320 ln ln 1

1 12880 ln ln 1 2880 ln ln 1

480

r

bB r BrBr

B m Br BrB rr r

mB Br r bB Br r

bB B

3 2 32

12 2

2 3 2 2 32 2 2 2

1 12 2

4 2 42 2 2

1 12 2

2 42 2

1 2

1 1ln ln 1 1 2880 ln 1

1 11440 ln 1 840 ln ln 1

1 11440 ln 1 240 ln 1

1600 ln ln 1

r r Br bB Brr

bB Br bB Br rr

bB Br bB Brr r

bB Br r

42

1 2

2 24 2 2

12 4 2 2

2 2 2 32 2 2

1 12 2

2

1 12 2 2 2

11240 ln ln 13

28 17 9 8360 ln ln 11 360 ln ln 1

5 4360 ln ln 4 120 ln ln

36 24 16 1212 11 14

B bBr r

bB Br r bB Br rr

bB Br r B bBr rr

Br br r

2 5 23

1 1

3 5 2 4 42 2 2 2 2

1 1 1 1 1

7 2 2 6 22 2 2 2 2 2

1 1 1 1 1 1

3 4 4 2 32 4 2

1 1 2

ln ln 3 2 ln ln

30 ln ln 8 1 120 ln ln 2 1

3 ln ln 12 30 ln ln 11 120 60

350 ln ln 720 ln ln 5

Br r Br Br r

b Br r Br Br b Br r Br

b Br r Br Br b Br r Br Br

b Br r bB Br r

.

(4.85)

69


Here we have investigated the flow and heat transfer in an incompressible flow of third

grade fluid in a pressure type die. Analyses for velocity field and temperature distribution

have been established in each case. The results have analyzed on various emerging

parameters related to wire coating process and the melt polymer. Numerical values are

given in Tables 4.1-4.6 for different values of perturbation parameter 0 and Brinkman

number Br , which illustrate that for small values of perturbation parameter 0 , there is

insignificant improvement in the solution of both velocity and temperature distribution.

But as the perturbation parameter increases, the error in different order solutions

increases. Perturbation Method gives series form solutions. For small perturbation

parameter approaches to zero, higher-order terms in the series become successively

smaller the solution is obtained by truncating the series. From Tables 4.1 and 4.4, it can

be noted that the results are still valid for “small values” of the perturbation parameter.

In Figs. 4.1–4.11, we have displayed the velocity and temperature profiles for different

values of various parameters. In, Fig. 4.1 the velocity profiles are presented for various

values of perturbation parameters 0 0.1, 0.2, 0.3 and 0.4 keeping the Brinkman

number 5Br and 2 in case of constant viscosity. Here it is observed that with

increase of perturbation parameter, the non-Newtonian behavior of the fluid also

increases. The velocity profile for values of Br equal to 1, 5, 10 and 15 is presented in

Fig. 4.2. As the value of Br is increased, the temperature distribution increases in the

annular gap for fixed values of 0 and in case of constant viscosity. Fig. 4.3 explains

the effect of viscosity parameter m on the velocity profile when the values of 0 and Br

are assumed to be 0.1 and 10 respectively. It is to be noted that when the viscosity

parameter m increases, as a consequence the non-Newtonian effect and the velocity

distribution increases for the case of constant viscosity. Fig. 4.4 admits the fact that

temperature distribution in Reynolds model viscosity case increases while increasing the

Brinkman number for fixed values of 0 0.1 , 2 and the viscosity parameter 10m .

Fig. 4.5 shows that the non-Newtonian effect reduces with decreasing non-Newtonian

parameter 0 for fixed values of Brinkman number Br , and the viscosity parameter

70

m (Reynolds model). Fig. 4.6, 4.8 and 4.10 show the velocity profiles along the radial

distance in a die for the case of Vogel’s model for different viscosity parameter 1 ,

Brinkman number Br and the perturbation parameter 0 respectively. Here it can be

seen that the velocity profiles of the polymer oscillates in the die region and also show

that with increase of these parameters the velocity of the melt polymer also increase with

oscillating behavior. Figs. 4.7, 4.9 and 4.11 present the temperature distribution for

Vogel’s model for different values of viscosity parameter 1 , Brinkman number Br and

the perturbation parameter 0 respectively, keeping the other parameters fixed. Here it

can be noticed that the temperature distribution increases as the viscosity parameter 1 ,

Brinkman number and the perturbation parameter 0 increase correspondingly.


02, 20, 0.01Br .


02, 20, 0.3Br .

71


02, 0.5, 0.5Br .


02, 20, 0.01Br .


02, 20, 0.3Br .

72


02, 0.5, 0.5Br .

Figure 4.1. Dimensionless velocity profiles in case of constant viscosity when

5,2 Br for different values of perturbation parameter 0 .

Figure 4.2. Dimensionless temperature distribution in case of constant viscosity

when 02, 0.01 for various values Brinkman number Br .

73

Figure 4.3. Dimensionless velocity profiles in case of Reynolds model

when 02, 10, 0.1Br for different values of m .

Figure 4.4. Dimensionless temperature distribution in case of Reynolds’s model when

02, 0.1, 10m for various values of Brinkman number Br .

Figure 4.5. Dimensionless velocity profiles in case of Reynolds model when

10,10,2 mBr , for different perturbation parameter 0 .

74


02, 10, 0.2, 0.05,Br B 20m for different values of 1 .


,2.0,5,2 BBr 0 0.05, 5m for different values of 1 .

Figure 4.8. Dimensionless velocity profiles in case of Vogel’s model when 2,

10,m 1 02, 0.05, 0.2B for various values of Brinkman number Br .

75

Figure 4.9. Dimensionless temperature distribution in case of Vogel’s model when 2,

10,m 1 02, 0.05, 0.2B for various values of Brinkman number Br .


2, 20, 5,Br m 3.0,51 B for different values of perturbation parameter 0 .


2, 20, 5,Br m 3.0,51 B for various values of perturbation parameter 0 .

76

4.4 Conclusion

Fluid flow in a pressure type die is considered and influence of non-Newtonian and

viscosity parameter are investigated on the velocity field and heat transfer. Both Constant

viscosity and temperature dependent viscosity cases were under investigation. Fluid is

considered as third grade fluid. Reynolds and Vogel’s models are accommodated to

performed analysis. It is establish that the flow and temperature fields are affected as

viscosity parameters are varied. Further it is found that as the Brinkman number Br and

the non-Newtonian parameter 0 increases, the magnitude of velocity and temperature of

the polymer increases. Furthermore, it is also found that the viscosity parameter 1

strongly affect the velocity field and temperature distribution of the fluid. In this case, the

velocity and temperature of the melt polymer increases as the viscosity parameter 1

increases and this increase is large as compared to other viscosity parameters.

77

Chapter 5

Wire Coating Analysis with Oldroyd 8-Constant Fluid by

Optimal Homotopy Asymptotic Method

78

This chapter explores an approximate solution of the Navier-Stokes equations for wire

coating in a pressure type die with a bath of Oldroyd 8- constant fluid under constant

pressure gradient in the axial direction. The governing differential equations are solved

analytically by using Optimal Homotopy Asymptotic Method (OHAM). The effects of

dilatant constant , the pseudoplastic constant and the constant pressure gradient on

the solution are studied. Moreover, the influences of flow rate and average velocity are

examined with the parameters , and the constant pressure gradient. For better

understanding, some graphs are sketched and discussed.

5.1 The basic equations and boundary conditions

Fig. 5.1 shows the internal geometry of the die considered here, including much of the

nomenclature. The wire of radius wR is dragged with velocity wU in a pool of an

incompressible Oldroyd 8-constant fluid in an annular die of radius dR as shown in Fig.

5.1. The wire and die are concentric. The coordinate system is chosen at the centre of the

wire in which z is taken in the direction of fluid flow and r is perpendicular to z . Here

we assume that the die is uniform and the flow is steady, laminar and isothermal.

We seek a velocity field of the form

rw,0,0u , rSS . (5.1)

Boundary conditions are:

w ww R U and 0dw R . (5.2)

The constitutive equation of Oldroyd 8-constant fluids are defined as [104]

1 1 1 1 1 0 1 1 1

2 2

0 1 2 1 2 2 1 2 1

1 1 1

2 2 2

1.

2

tr tr

tr

S S A S SA S A SA I

A A A A I

(5.3)

Here the constants 210 ,, are respectively zero shear viscosity, relaxation and

retardation time. The other five constants 21210 ,,,, are associated with nonlinear

terms.

The upper contra-variant convected derivative designed by over S and 1A is defined

as follows [102]

79

uSSuS

S

T

tD

D, (5.4)

uAAuA

A

111

1T

tD

D, (5.5)

where TuuA 1 and SuS

ttD

D. (5.6)

Substituting the expression given in Eq. (5.1) into Eqs. (5.3) – (5.6), we obtain nonzero

components of extra stress S as

2

1 1 1 0 2 1 1rr rz

dw dwS S

dr dr

, (5.7)

dr

dw

dr

dwS

dr

dwSS

dr

dwSS zzzzrrrrrz 0

0

011122

1

, (5.8)

2

2220111

dr

dwS

dr

dwS rzzz , (5.9)

2

201

dr

dwS

dr

dwS rz . (5.10)

Solving Eqs. (5.7) – (5.10), we obtain the explicit expressions for the stress components

as

2

1 1 1 0 2 1 1 ,rr rz

dw dwS S

dr dr

(5.11)

2

1 0 2 ,rz

dw dwS S

dr dr

(5.12)

2

1 1 1 0 2 2 2 ,zz rz

dw dwS S

dr dr

(5.13)

2

2

0

1

1

dr

dw

dr

dw

dr

dw

S rz

, (5.14)

where 221220212

3

,

80

111110

2

12

3

.

The constant is known as the dilatant constant while the constant is called the

pseudoplastic constant.

As indicated in Eq. (5.1), that the velocity field u and the stress S as functions of r only,

so the continuity Eq. (1.5) is satisfied identically and the dynamic Eq. (1.6) reduces to

rrrSdr

d

rr

p 1

, (5.15)

0,p

(5.16)

1

.rz

p drS

z r dr

(5.17)

From Eq. (5.16), we have zrpp , .

Substituting the nonzero shear stress given in Eq. (5.14) into Eq. (5.17), we obtain the

differential equation of velocity field as follows:

,023

4

2

25

2

22

2

24

2

223

2

2

dr

dw

z

pr

dr

dw

z

pr

dr

dw

dr

wd

dr

dwr

dr

wd

dr

dwr

dr

wd

dr

dwr

dr

dw

z

pr

dr

dw

dr

wdr

(5.18)

Figure 5.1. Wire coating die.

81

Figure 5.2. Wire coating process in a pressure type die.

We scale length with the radius of the uncoated wire, wR , velocity with the mean velocity,

wU at the die exit, the pressure is scaled with a viscous scale, w wU R . In addition, the

parameters and are scaled with a square of the ratio wU and wR , i.e., viscous scale,

2 2

w wU R . Thus, the dimensionless group that arise are the following

2

2

2

2

,,,w

w

w

w

ww R

U

R

U

U

ww

R

rr

,

ww RU

pp

, (6.19)

so that in non-dimensional form, after dropping the asterisks, and under the assumption

that the pressure gradient in the axial direction is constant, i.e.,

z

p, Eqs. (5.2) and

(5.18) become

,023

24

2

5

2

22

2

24

2

223

2

2

dr

dwr

dr

dwr

dr

dw

dr

wd

dr

dwr

dr

wd

dr

dwr

dr

wd

dr

dwr

dr

dwr

dr

dw

dr

wdr

(5.20)

subject to the following physical conditions of no slip on boundaries

11 w , 0w where 1w

d

R

R . (5.21)

82


Here, Eq. (5.20) is written in the form of Eq. (1.18) by taking

dr

dw

dr

wdrwL

2

2

, rrg ,

3 2 42 2

2 2

2 5 4 222

23 2 .

dw dw d w dw d wN w r r

dr dr dr dr dr

dw d w dw dw dwr r r

dr dr dr dr dr

We then construct a homotopy Rpr 1,0:, that satisfies Eq. (1.20). Now

substitute Eqs. (1.22) and (1.24) in Eq. (5.20) and equating the like powers of p to obtain

Zeroth-order problem

0: 0

2

0

2

0 rdr

dw

dr

wdrp , (5.22)

subject to the boundary conditions

0,11 00 ww , (5.23)

First-order problem

32 22

1 0 0 0 0 01 11 12 2 2

2 4 2 52 2 2

0 0 0 0 0 0 01 1 1 12 2 2

4

2 0 01 1

:

3

2

d w dw d w dw dwd w dwp r r C r C


dw d w dw d w dw d w dwC r rC rC C


dw dwrC rC

dr dr

2

11 0,r C

(5.24)


0,01 11 ww , (5.25)

Second-order problem

83

22 2 22 0 02 2 1 1 1 1

1 2 22 2 2 2

2 3 4 5

20 0 0 02 2 2 2

2

20 01 11

:

2

4 3 4

d w dwd w dw d w dw d w dwp r r C r C r rC


dw dw dw dwrC C C C

dr dr dr dr

dw dwdw dwrC r

dr dr dr dr

3

0 11

4 2 22 2

0 0 0 0 011 2 22 2

5 3

dw dwC

dr dr

dw dw d w dw d wdwC rC rC

dr dr dr dr dr dr

4 32 2 2

0 0 0 0 0 01 12 1 12 2 2

3 22 2

0 0 01 11 12 2

2 42 2

0 01 11 12 2

6 2

4 3

0,

dw d w dw d w dw d wdw dwrC rC rC


dw d w dwdw d wC rC

dr dr dr dr dr

dw dwd w d wrC rC

dr dr dr dr

(5.26)


0,01 22 ww . (5.27)

The solutions for 0 1,w w and 2w are as follows

rrrw ln1312

2

110 , (5.28)

rrrrr

rw ln1

1918

6

17

4

16

2

151421 , (5.29)

rrrrrrrrr

rw ln 111

1918

10

17

8

16

6

15

4

14

2

131221141062 ,

(5.30)

where 181716151413121110191817161514131211 ,,,,,,,,,,,,,,,,, and

19 are constants containing the auxiliary constants 1C and 2C are given in appendix A.

These constants are to be determined such that to minimize the solution error. There are

many methods such as Galerkin’s method, least squares method, collocation method, and

Ritz method which can be used to determine the optimal values of miiC ,...,3,2,1 .

Here, the method of least squares has been applied to locate the optimal values of

auxiliary constants as [70-76]. For detail analysis to obtain these constant, the readers are

referred to chapter 1 (Section 1.7.3).

84

The second-order approximate solution is given by

rwrwrwrw 210 . (5.31)

Substituting Eqs. (5.28) – (5.30) into Eq. (5.31), we obtain

2

10 11 14 12 11 14 13 12 16 146 4 2

4 6 8 10 6

17 15 18 16 17 18 13 19 18

1 1 1( ) ( ) ( )

( ) ( ) ln ( ) ln .

w r rr r r

r r r r r r r

(5.32)


This chapter presents the results obtained by OHAM [72-76] on the problem of wire

coating in an annular die with a bath of Oldroyd 8-constant fluid. The results for velocity

field, flow rate, average velocity and shear stress are obtained. The results of velocity

field and shear stress are expressed graphically in Figs. 5.3-5.9 while numerical values of

the average velocity and flow rate are entered in Tables 5.1-5.4. Fig. 5.3 shows that as the

order of approximation increases, the effect of the nonlinear terms increases in the

solution and as a result the error is reduced and the solution takes a steady state. It is clear

from Fig. 5.4 that increasing the dilatant parameter decreases the velocity profile. Also

it can be seen from Fig. 5.5 that the speed of fluid gradually increases as the value of the

pseudoplastic parameter increases. Fig. 5.6 gives the variation of velocity for different

values of constant pressure gradient 0.5, 1, 1.5 and 2 and for 4.0

and 1 . Here, it can be observed that the velocity distribution increases as the pressure

gradient increases and this increase is comparatively large for high values of pressure

gradients. Here, it is concluded that the pressure gradient plays an important role in fluid

flow. Figs. 5.7 and 5.8 show the profile of shear stress for different values of

pseudoplastic parameter 7.0 and5.0,3.0,1.0 and the dilatant parameter

7.0 and5.0,3.0,1.0 respectively. In this case, it is noticed that the shear stress

decreases with increase of and increases with increase of . It is observed from Fig.

5.9 that the shear stress increases with the increase of viscosity coefficient 0 . Tables

5.1-5.4 give the computed values of average velocity and the volume flow rate for

different values of , , and respectively. Table 5.1 gives the computed values for

different values of when 2.0 and 0.5 . Here, it is observed that with

85

increase of , the average velocity and volume flow rate both increase; similar

phenomena can be seen in Table 5.2 with increase of . Table 5.3 shows that these

physical phenomena decrease with the increase of parameter . From Table 5.4, it is

clear that with increase of constant pressure gradient , the computed values also

increase. Tables 5.5-5.7 are given for the values of constants 1C and 2C for different sets

of values of the physical parameters , and , which emphasized that these constants

depend upon the values of the physical parameters.

Figure 5.3. Dimensionless velocity profiles at different order of approximations using

OHAM when ,4.0,2.0 10.5, 0.002154869,C 2 0.0005341298C .

Figure 5.4. Dimensionless velocity profiles for different values of dilatant parameter

when 4.0 , 0.5 .

86

Figure 5.5. Dimensionless velocity profiles for different values of viscoelastic parameter

when 5.0 , 0.5 .

Figure 5.6. Dimensionless velocity profiles for different values of pressure gradient

when ,4.0 1 .

Figure 5.7. Profiles of shear stress for different values of parameter when ,2.0

2.00 , 5.0 .

87

Figure 5.8. Profiles of shear stress for different values of viscosity parameter 0 when

4.0,2.0 , 0.5 .

Figure 5.9. Profiles of shear stress for various values of the the parameter

when 2.0,25.00 5.0 .

88


of when ,3,2.0 0.5 .

Volume flow rate Average velocity

0 11.9378 0.47499

0.1 12.1721 0.48431

0.2 12.4141 0.49394

0.3 12.6639 0.50388

0.4 12.9213 0.51412

0.5 13.1866 0.52468

0.6 13.4595 0.53554

0.7 13.7402 0.54671

0.8 14.0286 0.55818

0.9 14.3248 0.56997

1 14.6287 0.58206


of when ,5.0,2.0 0.5 .


2 3.99037 0. 413392

2.2 5.07004 0. 420272

2.4 6.33704 0.42377

2.6 7.83165 0.432794

2.8 9.59663 0.446594

3.0 11.6765 0.464592

3.2 14.1165 0.486301

3.4 16.9616 0.511274

3.6 20.2553 0.539086

3.8 24.0379 0.569307

4 28.3451 0.601501

89


of when 3,4.0 , 0.5 .


of pressure gradient when 0.5, 0.2 and 3 .


0 13.4286 0.534306

0.2 12.6212 0. 514124

0.4 12.4141 0.493942

0.6 11.9069 0.473760

0.8 11.3997 0.453578

1 10.8924 0. 433396

1.2 10.3852 0. 413214

1.4 9.87796 0.393032

1.6 9.37073 0.372850

1.8 8.86350 0.352668

2 8.35627 0.332485


0 8.66613 0.344814

-0.1 9.27270 0.368949

-0.2 9.88806 0.393434

-0.3 10.5012 0.417829

-0.4 11.1010 0.441696

-0.5 11.6765 0.464592

-0.6 12.2162 0.486068

-0.7 12.7088 0.505666

-0.8 13.1424 0.522919

-0.9 13.5050 0.537346

-1 13.7841 0.548451

90


of when 0.5, 2, 2 .

1C 2C

0 0 0.26010913

0.2 -0.3107991 -0.0689695

0.5 0.3107992 -1.3121701

1.0 -0.255793 -0.0275308

2.0 -0.2562137 -0.0356102


of when 1.5, 2 , 0.8 .


of pressure gradient when 0.8, 0.1, 2 .

1C 2C

0 0 -0.575463

0.5 -0.443776 -0.0441636

1.0 -0.602959 -0.0266326

1.5 0.602959 -2.43847

2.0 -0.541396 -0.0347823

1C 2C

0 0.541396 -2.200374

-0.5 -0.298383 0.0558253

-1.0 -0.541396 -0.0347823

-1.2 -0.355398 -0.0522426

-1.5 0.355398 -1.473830

91

5.4 Conclusion

A theory for wire coating by withdraw from a bath of an Oldroyd 8-constant fluid has

been suggested. The motivation is to determine the effect of the dilatants parameter ,

pseudoplastic parameter , pressure gradient and coefficient of viscosity 0 on the

flow characteristics. The ordinary differential equation is solved for velocity field by

OHAM described by Marinca et al. [72-76]. The present results show that the velocity

profiles at a given point of r decrease with increase of dilatant parameter . It is also

found that the convexity of the velocity profile is more expanded for small values of

pressure gradients. Further, it is also found that the parameter affect the velocity of

fluid. Moreover, it has been concluded that the average velocity and flow rate increase

with increase of and decrease with increase of and . However, it needs to be

mentioned here that the theoretical or experimental data is not available for comparison.

92

Chapter 6

Solution of Differential Equations Arising in Wire Coating

Analysis of Unsteady Second Grade Fluid

93

This chapter describes the flow of unsteady second grade fluid in wire coating analysis.

Two different problems have been discussed (i) when the wire is translated and (ii) the

wire is translated as well as oscillated in a die. The study on the flow of incompressible

fluid formed by the oscillation of a boundary is not only of theoretical importance but it

also occurs in various problems such as acoustic streaming around an oscillating body.

Since the flow is incompressible, it is immaterial whether the wire oscillates or the fluid

oscillates. Exact solution for the velocity field is obtained in first problem and the

Optimal Homotopy Asymptotic Method (OHAM) is applied for obtaining the solution of

the second problem.

6.1 Problem formulation when the wire is translating only

Consider the flow of an incompressible second grade fluid under constant pressure

gradient in a circular die. Here, the wire of radius wR is translating in the axial direction

with velocity wU in a stationary die of radius dR , where the wire and die are concentric

as shown in Fig. 6.1. The coordinate system is chosen at the centre of the wire, in which

the axial direction is taken in the direction in which the fluid is flowing due to the

translation of wire and r is taken to be perpendicular to z .

Figure 6.1. Geometry of coating die.

94

For the problem under consideration, we shall seek the velocity field and pressure

distribution as

trw ,,0,0u , trpp , . (6.1)

The constitutive equation for second grade viscoelastic fluids given by Rivlin and

Ericksen [115], is

12211 AAAIT p , (6.2)

in which p is the pressure, I the identity tensor, the coefficient of viscosity of the

fluid, 21, the normal stress moduli and 21, AA are the line kinematic tensors defined

by [115]

TuuA 1 , (6.3)

.111

2 AuuAA

A T

Dt

D (6.4)

Under the consideration of velocity field given in Eq. (6.1), the continuity equation (1.5)

is satisfied identically.

Substituting Eqs. (6.2) – (6.4) into the balance of momentum given in Eq. (1.6), one

obtains in the absence of body forces as

2 22 2

1 22 2

2 10 4

p w w w w w w

r r r r r rr r

, (6.5)

(6.6)

r

w

rr

w

tr

w

rr

w

t

w 112

2

12

2

. (6.7)

Eq. (6.7) can be solved exactly with the appropriate boundary conditions to obtain the

velocity field. This can be then substituted into Eq. (6.5) to find the pressure distribution

function.

Boundary conditions

At wRr , wUw , 0 t ,

at dRr , 0w , 0 t . (6.8)

0 ,p

95

Initial condition

0w at 0t , r1 . (6.9)

Let’s introduce the following non-dimensional variables and parameters

010 02 2

1

, , , ,w w w

tr wr w t

R U R R

. (6.10)

So, Eqs. (6.7) and (6.8) after dropping the asterisks take the following form

r

w

rr

w

tr

w

rr

w

t

w 112

2

02

2

, (6.11)

with the boundary conditions

At 1r , 1w , 0 t where 1w

d

R

R ,

at r , 0w , 0 t . (6.12)

Initial condition

,0w at 0t , r1 . (6.13)

6.1.1 Solution of the problem

We shall assume that the exact solution of Eq. (6.11) with the non-homogenous boundary

conditions (6.12) consist of steady state solution rV , that satisfy the non-homogenous

boundary conditions plus a solution trF , , i.e.,

trFrVtrw ,, . (6.14)

If we allow t , we obtain steady state solution.

Substituting Eq. (6.14) according to the demand of Eq. (6.11), one obtains

2 2 2

02 2 2

1 1 1F d V dV F F F F

t dr r dr r r r t r r r

. (6.15)

For steady state solution rV , we assume

01

2

2

rd

Vd

rrd

Vd, (6.16)

with boundary conditions

At 1r , 1V ,

at r , 0V . (6.17)

96

The solution of Eq. (6.16) corresponding to the boundary conditions (6.17) is as follows

1

rrV . (6.18)

Under the consideration of Eq. (6.16), Eq. (6.15) takes the form

2 2

02 2

1 1F F F F F

t r r r t r r r

, (6.19)

with the boundary and initial conditions

At 1r , 0F ,

at r , 0F , (6.20)

rVrF 0, . (6.21)

Now we shall seek the solution of Eq. (6.19) with respect to Eqs. (6.20) and (6.21) in the

form of separation of variables as

tTrRtrF 00, . (6.22)

Use Eq. (6.22) in Eq. (6.19), after separating the variables, we have

0 0 0

0 0 0 0 0 0

r R R T

rR R r R T

, (6.23)

where the prime denotes differentiation.

Thus, we have a system of uncoupled ordinary differential equations as

,0

0

T

T (6.24)

and ,01

10

0

00

RRr

R

(6.25)

Eq. (6.25) is the Bessel equation with the boundary conditions

00 R , 010 R , (6.26)

Now, we shall seek the domain of used in both the Eqs. (6.24) and (6.25), for this we

discuss some cases of for Eq. (6.26).

Case (a) If 0 , then we have

CtT 0 , (6.27)

where C is the constant of integration, which means that the velocity field is steady and

so there is a contradiction. Hence 0 is impossible.

97

Case (b) If 0 , we obtain

teCtT 0 . (6.28)

This situation is so contradictory because as t , we arrive at the steady state solution.

Case (c) If 0 , one obtains

teCtT2

0

, (6.29)

where 2 , which implies that as t , we can recover the steady state solution.

So the only possibility for is that it must be negative.

The solution of Eq. (6.19) is given by

rJeCtrF nn

n

n

t

0

2

1

,

, (6.30)

where the coefficients nC are given by

0

1

2

0

1

n

n

n

V r J r rdr

C

J r rdr

. (6.31)

So from Eq. (6.14), the velocity field is

rJeCr

trw nn

n

n

t

0

2

11,

. (6.32)

The expression given in Eq. (6. 32) represents the velocity field in which 0 nJ r are

the eigen functions and n for 1,2,3,...n are the eigen values such that

1 2 3 ... are the positive zeros of 0J with the corresponding eigenfunctions

rJ n0 . These eigen values form a complete orthogonal set.

It is quite straight forward currently to calculate the frictional force, i.e., drag, exerted per

unit length of the die with the support of Eqs. (6.32) and (6.5).

6.2 Problem formulation when the wire is translating as well as

oscillating

In this section, the geometry of the problem is the same as in case (i) except that now the

wire is translating as well as oscillating. Therefore, the governing partial differential

equations are (6.5) – (6.7) with the following initial and boundary conditions

98

At wRr , 1 cosww U a t , 0 t ,

and at dRr , 0w , 0 t . (6.33)

0w at 0t , dw RrR , (6.34)

where wU is the speed, a is amplitude and is frequency of oscillation of wire.

Eq. (6.7) represents the flow due to pressure gradient. After leaving the die, there is only

drag flow. Hence, we consider

r

w

rr

w

tr

w

rr

w

t

w 112

2

112

2

, (6.35)

where 111 ,

We are now interested to solve the above initial-boundary value problem with the help of

OHAM. The pressure distribution function can then be obtained from Eq. (6.5). For

solution, radius of the die is taken 1, i.e., 1dR and the radius of the wire

.10, wR

6.2.1 Solution of the problem

To apply, the optimal homotopy asymptotic method [70] to solve Eq. (6.35) subject to

boundary conditions given in Eq. (6.33), we write

r

w

rr

wwL

12

2

,

r

w

rr

w

tt

wwN

112

2

11

, , 0g r t . (6.36)

After this, we construct a homotopy Rpr 1,0:, that satisfies Eq. (6.21). For

this, substitute Eqs. (6.23) and (6.24) in Eq. (6.21) and equating the identical powers of

p to obtain


20 0 0

2

1: 0

w wp

r r r

, (6.37)


taUtwtw w cos1,,0,1 00 , (6.38)

99

First-order problem

21 0 0 0 01 1 1 11 1

12

2 2 2

0 0 01 11 12 2 2

1 1:

0,

w w w ww w C Cp C

r r r t r r r r r t r

w w wC C

r r t r

(6.39)


0,,0,1 11 twtw , (6.40)


22 0 02 2 1 2 11 2 1 1

2 12

2 22 2

0 01 1 11 1 1 1 11 2 11 22 2 2 2

2

111 1 2

1 1:

0,

w ww w w C C w wp C C

r r r t t r r r t r r r

w wC w C w w wC C C

r r r t r r r r t r

wC

t r

(6.41)


0,,0,1 22 twtw , (6.42)

where

1

and

11

11 .

Zeroth order problem given by Eqs. (6.37) and (6.38) gives the following solution

ln

lncos10

rtaUw w . (6.43)

If Eq. (6.43) is substituted into Eq. (6.39) and solving subject to the boundary conditions

given in Eq. (6.40) gives the first order solution as below

2 2

1 1 1 11 1 1

1sin sin ln sin ln sin

8w ww CU a t CU a r t r t r CU a r t .

(6.44)

Similarly, with the help of zeroth and first-order the second order solution obtains from

Eqs. (6.40) and (6.41) solution is as follows

2 2

2 12 13 14 15

2 2 4 4

16 17 18 19 11 12

ln ln sin

ln ln ln cos .

w r r r r t

r r r r r r r t

(6.45)

Finally, the second order approximate solution is

100

11 12

13 14 15 16

17 18 19 11

12

ln 1 1 21 cos sin sin1 1ln 8 8

1 1 2ln sin ln sin sin18 8

2 2sin ln sin ln sin cos

2 2 4ln cos cos ln cos cos

4 ln cos

w

rw U a t C U a t C U a r t

w w

r t r C U a r t tw

r t r t r r t t

r t r t r r t r t

r r

,t

(6.46)

where 11 12 13 14 15 16 17 18 19 11, , , , , , , , , and 12 are constants involving the

auxiliary constants 21, CC are given in appendix B.

To investigate the stability and convergence of OHAM, we make an effort to solve some

linear and non-linear partial differential equations with known exact solution.

Example 6.1

2

2, 0 1,

w wr

t t

(6.47)


11,,0 , 0, ,r t t

tw r e w t e w e . (6.48)

The exact solution of Eq. (6.47) with the corresponding boundary conditions given in Eq.

(6.48) is as follows

, r tw r t e . (6.49)

Here, we have

2

2, , ,

w wL w r t N w r t

r t

. (6.50)

Following the procedure of OHAM, we obtain the solution to the given problem up to

third order approximation with

1

2

3

0.9724175167,

0.00186254006,

0.0018848878.

C

C

C

The absolute error is presented in the form of numerical data in Table 6.7 and Table 6.8.

101

Example 6.2

2

2

2, 0 1

r tw w ww e r

t r r

, (6.51)


1,0 , 0, , 1,r t tw r e w t e w t e . (6.52)

Having the exact solution

, r tw r t e . (6.53)

Here, we have

2

2

2, , , , ,

r tw w wL w r t N w r t w h r t e

r t r

. (6.54)

Handling the problem with OHAM as discussed earlier, we obtain the third order

approximate solution with

1

2

3

0.555334661,

0.321611989,

0.43579432289.

C

C

C

The absolute error of example 6.2 is presented in the form of numerical data in Table 6.9

and Table 6.10.

Example 6.3

2

2

1, 0 1

2

w w wr

t r t r

, (6.55)


2 2,0 , 0, , 1, 1w r r w t t w t t . (6.56)

The exact solution to the problem is as below

2,w r t r t . (6.57)

In this case, we have

2

2, , , 2 .

w w wL w r t N w r t

r t t r

(6.58)

102

Applying OHAM as discussed in previous section, we obtain the third order approximate

solution with

1

2

3

0.8698907118,

1.523985080,

1.1597421393.

C

C

C

The absolute error of example 6.3 can be observed from the numerical data in Table 6.11

and Table 6.12.


In this study, we have model of an unsteady second grade fluid flow in a circular die with

translating as well as with oscillating boundary in the form of partial differential

equations. The problems are solved analytically. The obtained results for the problem

with oscillating boundary verify that OHAM is convergent as the order increases.

Examples 6.1-6.3 are solved up to third order approximations, whose numerical values

are computed in Tables 6.7, 6.9 and 6.11, which show that as we increase the order of this

technique, the accuracy of the solution also increases. Here, it can also be seen that in

each of example, the absolute error in every higher order approximation is smaller than

lower order approximations, which confirms the convergence of OHAM. As the fluid

flow is due to the oscillation and translation of the wire, so the velocity of the fluid will

be high at the surface of the wire as compared to annular gap and will be decrease for the

fluid away from the surface of wire, this phenomena can be observed from Tables 6.1,

6.2, 6.4, 6.5 and Figs. 6.3 and 6.5. Tables 6.3 and 6.6 illustrate the solution of different

order problems at time 10t and 5t respectively for different parameters which show

that the effect of nonlinearity in the problem is less effective because the absolute errors

between different order problems are very less. Figs. 6.2 and 6.4 are plotted for velocity

field verses r and t . Here it can be seen that the velocity of fluid flow decreases as the

distance from the centre of the metal wire increases. The velocity profile in Fig. 6.4

shows oscillatory behavior and the amplitude of oscillations decreases as the distance

from the center of the metal wire increases. It is evident from Tables 6.7-6.12 that

OHAM can be applied for large time domain and the accuracy remains almost consistent.

103

Table 6.1. Shows velocity distribution for different values of time level t when

0.2, 11 0.02, 0.2, 2, 0.5, 0.01,wU a 1 0.5924838150,C

2 0.0902455892C .

Velocity distribution

r 1t 10t 20t 30t

0.2 2.0196 2.01081 1.99168 1.98020

0.3 1.51081 1.50426 1.48995 1.48133

0.4 1.14982 1.14484 1.13396 1.12738

0.5 0.869806 0.866054 0.85782 0.852834

0.6 0.641019 0.63826 0.632192 0.628511

0.7 0.447581 0.445658 0.441421 0.438847

0.8 0.280017 0.278815 0.276165 0.274553

0.9 0.132214 0.131647 0.130396 0.129634

1.0 0 0 0 0

Table 6.2. Shows velocity distribution of fluid flow at different time level t when

,01.0,5.0,2,2.0,02.0,2.0 11 aUw 1 0.5924838150,C

2 0.09024558924C .


t 0.2r 0.22r 0.24r 0.26r

0 2.02 1.90038 1.79117 1.69071

1 2.0196 1.9 1.79082 1.69038

2 2.01842 1.8989 1.78978 1.6894

3 2.01651 1.8971 1.78808 1.6878

4 2.01393 1.89468 1.7858 1.68565

5 2.01081 1.89174 1.78303 1.68304

6 2.00725 1.88839 1.77988 1.68006

7 2.0034 1.88477 1.77647 1.67684

8 1.99942 1.88102 1.77294 1.67351

9 1.99546 1.8773 1.76942 1.67019

10 1.99168 1.87374 1.76607 1.66703

104

Table 6.3. Shows velocity distribution of fluid flow at various order of approximations at

10t when ,01.0,5.0,2,2.0,02.0,2.0 11 aUw 1 0.5924838150,C

2 0.09024558924C .


r Zeroth order First order Second order

0.2 2.00848 2.00848 2.00848

0.3 1.50249 1.50247 1.50245

0.4 1.14348 1.14345 1.14343

0.5 0.865007 0.864975 0.864956

0.6 0.63748 0.637452 0.637434

0.7 0.445109 0.445086 0.445073

0.8 0.27847 0.278454 0.278445

0.9 0.131484 0.131476 0.131471

1.0 0 0 0

Table 6.4. Shows velocity distribution of fluid flow at different values of time by using

0.2, 11 0.02, 0.2, 2, 0.8, 0.5,wU a 1 0.329680663,C 2 0.3060088.C


r 1t 5t 10t 15t

0.2 2.98007 2.5403 1.58385 1.01001

0.3 2.22989 1.90284 1.18755 0.755984

0.4 1.69744 1.44974 0.9055 0.575615

0.5 1.28429 1.09767 0.68605 0.435604

0.6 0.946615 0.80953 0.50623 0.321125

0.7 0.661032 0.565557 0.35381 0.224274

0.8 0.413589 0.353968 0.221506 0.140335

0.9 0.195292 0.167173 0.104633 0.0662685

1.0 0 0 0 0

105

Table 6.5. Shows velocity distribution at different at different time level t when 0.2,

,02.0,2.0 11 2, 0.8, 0.5,wU a ,3296806629.01 C 306008832.02 C .


t 0.2r 0.22r 0.24r 0.26r

0 3.0 2.82234 2.66015 2.51095

2 2.92106 2.74839 2.59074 2.44569

4 2.69671 2.53759 2.39229 2.25859

6 2.36236 2.22322 2.09614 1.9792

8 1.9708 1.85491 1.74905 1.65161

10 1.58385 1.4908 1.4058 1.32756

12 1.26261 1.18839 1.12059 1.05819

14 1.05778 0.995411 0.938457 0.886046

16 1.00171 0.942339 0.888144 0.838293

18 1.10324 1.03755 0.9776 0.922476

20 1.34636 1.26601 1.1927 1.1253

Table 6.6. Shows velocity distribution at various order of approximation at 5t when

,5.0,8.0,2,2.0,02.0,2.0 11 aUw 1 0.3296806,C

2 0.30600883.C


r Zeroth order First order Second order

0.2 1.58385 1.58385 1.58385

0.3 1.18483 1.18581 1.18755

0.4 0.901725 0.903092 0.9055

0.5 0.682128 0.683551 0.68605

0.6 0.502705 0.503985 0.50623

0.7 0.351005 0.352024 0.35381

0.8 0.219596 0.220289 0.221506

0.9 0.103686 0.104029 0.104633

1.0 0 0 0

106

Figure 6.2. Velocity profiles for ,01.0,5.0,2,2.0,02.0,2.0 11 aUw

1 20.5924838150, 0.09024558924C C .

Figure 6.3. Velocity profiles at different position of r when ,02.0,2.0 11

,2.0 ,01.0,5.0,2 aUw 1 0.5924838150,C 2 0.09024558924C .

107

Figure 6.4. Velocity distribution of fluid flow with passage of time t when

,5.0,8.0,2,2.0,02.0,2.0 11 aUw 1 0.3296806629,C

2 0.306008832C .

Figure 6.5. Velocity distribution of fluid flow at different time levels when ,2.0

,5.0,8.0,2,2.0,02.011 aUw 1 0.3296806629,C 2 0.306008832C .

108


2t (example 6.1).

Absolute error

r Zeroth order First order Second order Third order

0.0 0 0 0 0

0.1 0.0814154 0.570537×10-2

0.249482×10-5

7.56031×10-9

0.2 0.149321 0.110026×10-1

0.481734 ×10-4

1.52796×10-8

0.3 0.202296 0.152718×10-1

0.666759×10-4

2.31512×10-8

0.4 0.23877 0.180423×10-1

0.780313×10-4

3.11826×10-8

0.5 0.257007 0.190087×10-1

0.808035×10-4

3.85911×10-8

0.6 0.25509 0.180482×10-1

0.747926×10-4

4.34906×10-8

0.7 0.2309 0.152395×10-1

0.611×10-4

4.32616×10-8

0.8 0.182092 0.108845×10-1

0.41991×10-4

3.56143×10-8

0.9 0.106079 0.553118×10-2

0.205352×10-4

2.01492×10-8

1.0 0 0 0 0


different values of time level (example 6.1).

Absolute error

r 1t 3t 5t

0.0 0 0 0

0.1 6.25207×10-9

6.84085×10-9

1.02053×10-9

0.2 1.26356×10-8

1.38256×10-8

2.06253×10-8

0.3 1.91451×10-8

2.09481×10-8

3.12508×10-8

0.4 2.57868×10-8

2.82152×10-8

4.20921×10-8

0.5 3.19133×10-8

3.49187×10-8

5.20926×10-8

0.6 3.59649×10-8

3.93519×10-8

5.87061×10-8

0.7 3.57756×10-8

3.91447×10-8

5.8397×10-8

0.8 2.94516×10-8

3.22251×10-8

4.80743×10-8

0.9 1.66625×10-8

1.82317×10-8

2.71985×10-8

1.0 0 0 0

109


0.5t (example 6.2).

Absolute error


0.0 0 0 0 0

0.1 0.673272×10-2

0.284105×10-2

0.107105×10-3

0.36837×10-4

0.2 0.123482×10-1

0.491567×10-2

0.238502×10-3

0.70689×10-4

0.3 0.16729×10-1

0.62991×10-2

0.37333×10-3

0.10249×10-3

0.4 0.197453×10-1

0.705434×10-2

0.492424×10-3

0.13098×10-3

0.5 0.212534×10-1

0.723054×10-2

0.578245×10-3

0.15322×10-3

0.6 0.210949×10-1

0.68619×10-2

0.614799×10-3

0.16496×10-3

0.7 0.190945×10-1

0.596588×10-2

0.587582×10-3

0.16111×10-3

0.8 0.150583×10-1

0.454153×10-2

0.483547×10-3

0.13609×10-3

0.9 0.877234×10-2

0.256752×10-2

0.29108×10-3

0.84219×10-4

1.0 0 0 0 0


at different values of time level (example 6.2).

Absolute error

r 0.2t 0.7t 1.2t

0.0 0 0 0

0.1 0.36837×10-4

0.37959×10-4

0.445452×10-4

0.2 0.70689×10-4

0.72843×10-4

0.854817×10-4

0.3 0.102489×10-3

0.10561×10-3

0.123934×10-3

0.4 0.130985×10-3

0.134974×10-3

0.158394×10-3

0.5 0.15322×10-3

0.157886×10-3

0.185281×10-3

0.6 0.16496×10-3

0.169984×10-3

0.199478×10-3

0.7 0.16111×10-3

0.166016×10-3

0.194822×10-3

0.8 0.136094×10-3

0.140239×10-3

0.164572×10-3

0.9 0.842191×10-4

0.867839×10-4

0.101842×10-3

1.0 0 0 0

110


at 2t (example 6.2).

Absolute error


0.0 0 0 0 0

0.1 0.285×10-1

0.134087×10-3

0.111993×10-4

0.402434×10-5

0.2 0.48×10-1

0.102312×10-2

0.199196×10-4

0.805369×10-5

0.3 0.595×10-1

0.308721×10-2

0.245956×10-4

0.115625×10-4

0.4 0.64×10-1

0.55984×10-2

0.245932× 10-4

0.139915×10-4

0.5 0.625×10-1

0.803744×10-2

0.201174×10-4

0.148591×10-4

0.6 0.56×10-1

0.98477×10-2

0.122106×10-4

0.138719×10-4

0.7 0.455×10-1

0.10462×10-1

0.280969×10-5

0.110492×10-4

0.8 0.32×10-1

0.933614×10-2

0.517164×10-5

0.687801×10-5

0.9 0.165×10-1

0.59802×10-2

0.777698×10-5

0.250961×10-5

1.0 0 0 0 0


at different values of time level (example 6.1).

Absolute error

r 0.5t 1.5t 2.5t 3t

0.0 0 0 0 0

0.1 0.208495×10-4

0.197658×10-4

0.186455×10-5

0.17491×10-5

0.2 0.376054×10-4

0.35773×10-4

0.338665×10-5

0.318911×10-5

0.3 0.479629×10-4

0.459034×10-4

0.437308×10-5

0.414529×10-5

0.4 0.517735×10-4

0.49929×10-4

0.479772×10-5

0.458895×10-5

0.5 0.503948×10-4

0.491237×10-4

0.476926×10-5

0.461118×10-5

0.6 0.459324×10-4

0.452381×10-4

0.443965×10-5

0.434159×10-5

0.7 0.394581×10-4

0.392185×10-4

0.338482×10-5

0.384126×10-5

0.8 0.303416×10-4

0.303955×10-4

0.30385×10-5

0.31313×10-5

0.9 0.169306×10-4

0.171267×10-4

0.172969×10-5

0.174421×10-5

1.0 0 0 0 0

111

6.4 Conclusion

Here, the solutions for velocity field corresponding to the motion of a second grade fluid

in a straight annular die, are derived. Two different problems are discussed in case (i) and

(ii). For the problem (i) the solution presents as a sum of steady-state and transient

solution, describes the motion of the fluid for different times. For large values of time,

when the transient’s part vanish, the initial solution reduces to the steady state solution. In

case (ii) an approximate solution was founded by OHAM. Here, the velocity variations

subsequent to the fluid flow with the cosine oscillations of wire have been established. It

is found that the fluid velocity reduces with the passage of time passing through a single

point. It is also found that at the same time the fluid velocity decreases along the radial

direction. Furthermore, it is concluded that the oscillation profile of the fluid reduces

away from the wire and becomes zero at the wall of die. This review would serve as a

major reference for researchers in this area, such that duplication of efforts would be

minimized.

112

Chapter 7

Heat Transfer Analysis of a Third Grade Fluid in

Post-treatment Analysis of Wire Coating

113

In this chapter, we extended the work of Kasajima and Ito [17] and investigate the flow

of a thermodynamically compatible third grade fluid in case of post-treatment of wire

coating. The expressions for velocity field and temperature distribution are derived by

using the well known traditional Perturbation Method (PM) and the Optimal Homotopy

Asymptotic Method (OHAM). Also the volume flow rate, thickness of the coated wire

and force on the total wire have been derived explicitly. Moreover, the effect of emerging

parameters on velocity and temperature distribution is investigated with the help of

several graphs.


In wire coating, the quality of the material and wire drawing velocity are important

within the die, after leaving the die temperature and the shape of the transverse sectioning

is also very important. Consider the flow of the polymer extrudate is given in Fig. 7.1,

denoted by the solid line. To analyze the flow behavior of a polymer used in wire coating,

it is convenient to divide the flow transversely into many short sections as shown in

broken lines in Fig. 7.1 with the assumption that each section has almost the same shape,

we analyze only one section because each section can be assumed to be approximately of

the shape shown in Fig. 7.2 and readily analyzable.

Consider the wire of radius wR and temperature 1 is dragged in the z direction through

an incompressible third grade polymer (II) with a velocity 1V and the gas (III)

surrounding the polymer (II) at the surface of coated wire of radius 0R is at

temperature 2 and flowing with a velocity 2V .

Consider the cylindrical coordinates zr ,, such that r is perpendicular to the direction

of flow.

Assuming that the flow is steady, laminar, unidirectional and axisymetric:

We seek the velocity field of the form

rw,0,0u , rSS , r . (7.1)

The following are some more assumptions which are made during the formulation of the

problem:

i The flow is incompressible due to the high viscosity of the polymer.

ii. Polymer II holds the third order fluid model for shear rate.

114

iii. In Fig. 7.2, the metal wire I, the polymer II and gas III are in contact with each

other and consider no slippage occurs along the contacting surfaces of the wire,

polymer and the gas.

Boundary conditions are:

1Vw , 1 at wr R ,

2Vw , 2 at 0r R , (7.2)

For third grade fluid, the extra stress tensor S is defined as [56-58, 114]

123122123112211 AAAAAAAAAAS tr , (7.3)

in which is the coefficient of viscosity of the fluid, 32121 ,,,, are the material

constants and 321 ,, AAA are the line kinematic tensors defined by [56]

LLA T

1 , (7.4)

11 1 , 2,3T n

n n n

Dn

Dt

AA A L LA , (7.5)

where the superscript T denotes the transpose of the matrix.

Figure 7.1. Polymer extrudate in wire coating.

115

Figure 7.2. Drag flow in wire coating.

Using the velocity field, the continuity Eq. (1.5) is satisfied identically and the nonzero

components of Eq. (7.3) become

2

212

dr

dwSrr , (7.6)

2

2

dr

dwS zz , (7.7)

3

322

dr

dw

dr

dwS zr . (7.8)

Substituting the velocity field and Eqs. (7.6) – (7.8) in the equation of balance of

momentum (1.6) in the absence of body force take the form

2

21

12

dr

dwr

dr

d

rr

p , (7.9)

0

p, (7.10)

3

322dr

dwr

dr

d

dr

dwr

dr

d

rz

p

. (7.11)



116

02

3

32

dr

dwr

dr

d

rdr

dwr

dr

d , (7.12)


.021

4

32

2

2

2

dr

dw

dr

dw

dr

d

rdr

dk (7.13)

The average velocity is

0

2 20

2

w

R

ave

w R

w r w r drR R

. (7.14)

At some control surface downstream, the volume flow rate of coating is

2 21 c wQ V R R . ` (7.15)

where cR is the radius of the coated wire. On the other hand at the cross-section within

the die, the volume flow rate is

0

2

w

R

R

Q rw r dr . (7.16)

The thickness of the coated wire can be obtained from Eqs. (7.15) and (7.16) as

0

1

2

2

1

2.

w

R

c w

R

R R r w r drV

(7.17)

The force on the wire is derived by obtaining the shear stress at the surface of wire. This

is given by

3

02 .w

w

rz r R

r R

dw dwS

dr dr

(7.18)

The force on the surface of the total wire is

2w

w w rz r RF R LS

. (7.19)

Introduce the dimensionless parameters

1

1 2 1

, , ,w

r wr w

R V

0 2

0 2 3

1

, 1, 1w

R VU

R V . (7.20)

The system of Eqs. (7.2), (7.12) – (7.19) after dropping the asterisks, take the following

form

117

032

32

2

2

02

2

dr

dw

dr

dw

dr

wdr

dr

dw

dr

wdr , (7.21)

11 w and Uw , (7.22)

021

4

0

2

2

2

dr

dwBr

dr

dwBr

dr

d

rdr

d , (7.23)

01 and 1 . (7.24)

2 20

1 12

ave w

ave

w

w R Rw r w r dr

R V

, (7.25)

2

1 12 w

QQ r w r dr

R V

, (7.26)

1

2

1

1 2 ,cc

w

RR r w r dr

R

(7.27)

3

011 1 1

2 ,rz wrz r

r r

S R dw dwS

V dr dr

(7.28)

3

0

11

2 ,2

ww

r

F dw dwF

LV dr dr

(7.29)

where the Brinkman number

2

010 2

2 1

2

1

, .w

VBr

k R

V

(7.30)

The traditional perturbation method is used to solve momentum and energy equation with

the corresponding boundary conditions given in Eq. (7.22) and (7.24).


7.2.1 Perturbation solution

The approximate solution of Eq. (7.21) subject to the boundary conditions given in Eq.

(7.22) can be obtained by selecting 0 as perturbation parameter.

The velocity field is chosen as

2

0 0 0 1 0 2, ...w r w r w r w r . (7.31)

118

Substituting the above series in Eqs. (7.21) and (7.22), and comparing the coefficient

of 0 1 2

0 0 0, , , we obtain the zeroth, first and second order problems with the

corresponding boundary conditions in the following form:

Zeroth-order problem with boundary conditions

2

0 0 00 2

: 0d w dw

rdr dr

, (7.32)

110 w , .0 Uw (7.33)

First-order problem with boundary conditions

2 3221 0 0 01 1

0 2 2: 6 2 0

d w dw dwd w dwr r

dr dr dr dr dr

, (7.34)

011 w , .01 w (7.35)

Second-order problem with boundary conditions

2 2 22 22 0 0 0 02 2 1 1 1

0 2 2 2: 6 6 12 0

dw dw dw d wd w dw d w dw dwr r r


, (7.36)

012 w , .02 w (7.37)

Now we solve the above sequence of problems and construct the series form solution.

Zeroth-order solution

1ln

ln10 U

rrw

, (7.38)

which is the Newtonian solution.

First-order solution

22

3

1

11

11

ln

ln

ln

1

r

rUrw

. (7.39)

Second-order solution

22244

5

2

1111

ln

111

11

ln

ln

ln

13

rr

rUrw

. (7.40)

Next, we find the approximate solution for temperature profile, for which we write

2

0 0 0 1 0 2, ...r r r r . (7.41)

119

Substituting Eqs. (7.31) and (7.41) into the energy equation and their corresponding

boundary conditions after collecting the same power of , yields different order problems


220 0 0 0

0 2

1: 0

d d dwBr

dr r dr dr

, (7.42)

01,1 00 , (7.43)

with the solution

rUBr

rr ln1ln2

ln2

ln 2

20 . (7.44)


421 0 01 1 1

0 2

1: 2 2 0

dw dwd d dwBr Br

dr r dr dr dr dr

, (7.45)

01,0 11 , (7.46)

with the solution

lnln2lnln2

11ln

111

ln2

1 2

22

4

51 rrrr

UBrr . (7.47)


3 222 0 02 2 1 2 1

0 2

1: 8 2 0,

dw dwd d dw dw dwBr Br Br

dr r dr dr dr dr dr dr

(7.48)

01,0 22 , (7.49)

with the solution

2 2 2

2 22 2

2

4 2 2 2

2 2 4 4

61

4 ln

ln 1 ln 114 1 8 1

ln ln

ln 1 ln 1 ln 112 1 17 1 11 1

ln ln ln

1 1 1 1 18 1 1 12ln 1 3 1 .

ln

Br Ur

r r

r r r

rr r

(7.50)

120

7.2.2 Solution by optimal homotopy asymptotic method

In Eq. (7.21), we have

dr

dw

dr

wdrwL

2

2

, 0rg and .32

32

2

2

0

dr

dw

dr

dw

dr

wdrwN (7.51)

We construct a homotopy Rpr 1,0:, that satisfies Eq. (1.20). Now substitute

Eqs. (1.22) and (1.24) in Eq. (7.21) and collecting like powers of p , we obtain various

order problems as


0: 0

2

0

2

0 dr

dw

dr

wdrp , (7.52)

110 w , .0 Uw (7.53)


3 2 221 0 0 0 0 01 1

1 0 1 12 2 2

2 2

0 00 1 2

: 2

6 0

dw dw dw d w d wd w dwp r C C r rC


dw d wr C

dr dr

(7.54)

011 w , .01 w (7.55)


3 222 0 0 02 2 1 1

2 0 2 1 22 2

2 2 2 2

0 0 0 0 01 10 1 0 2 0 12 2

22 2 2

01 1 11 0 12 2 2

: 2

6 6 12

6 0,

dw dw d wd w dw dw dwp r C C C rC


dw dw d w dw d wdw dwC r C r C


dwd w d w d wr rC r C

dr dr dr dr

(7.56)

012 w , .02 w (7.57)

Solving the above set of problems in conjunction associate to the corresponding boundary

conditions, we obtain

0

ln1 1

ln

rw r U

, (7.58)

121

3

1 0

1 3 2 2

1 3 1 ln 11 1

ln2 ln

U C rw r

r

, (7.59)

23 2

20

2 06 2 4 2

2

204 2 20 12 2 4 2

2

0

1 9 1ln 1 1 11 9 1 1 2 1

ln ln2 ln

9 1 1 1 1 1ln 1 1 9 1 1 2 ln 1

ln

3 1ln

ln

U Urw r U

Ur U r C

r r r

Ur

2

2

02 4 2

1 1 11 3 1 1 1 ,

lnU

(7.60)

where 1C and 2C are auxiliary constants which are optimally determined by the method

of least square.

Collecting the results, we write the velocity field obtained by the optimal homotopy

asymptotic method

0 1 1 2 1 2, , , ....w r w r w r C w r C C . (7.61)

Finally, incorporating the expressions for 0 1,w w and 2w from Eqs. (7.58) – (7.60), we

obtain the optimal homotopy asymptotic solution up to second order as

1410 11 15 12 16 13 174 2

11 lnw r r

r r

. (7.62)

The volume flow rate in dimensionless form is obtained from Eqs. (7.26) and (7.62) as

2

14 12 16 13 17 18 13 152

1 11 1 ln 2 ln .

2Q

(7.63)

The thickness of the coated wire is obtained with the help of Eqs. (7.26) and (7.27) as

1

22

14 12 16 13 19 13 152

11 1 1 ln 2 ln ,cR

(7.64)

Similarly, the force on the surface of the total wire is obtained by using Eq. (7.62) in Eq.

(7.29) as

122

14 11 15 13 17

2 2 2

0 13 14 17 11 11 13 14 17 14 13 17 13 17

2

2 9 4 6 2 .

wF

(7.65)

Finally, we solve the energy Eq. (7.23) with respect to the boundary conditions given in

Eq. (7.24). Therefore substituting the solution w r from Eq. (7.62) into the energy Eq.

(7.13) and integrating twice with respect to r , we obtain the solution for the temperature

field as

0 10 13 15 16 1711 12 1418 1918 16 14 12 10 8 6 4 2

lnr rr r r r r r r r r

, (7.66)

where 0 10 11 12 13 14 15 16 17 18 19, , , , , , , , , , and 10 11 12 13 14 15 16, , , , , , ,

17 18 19, , are constants which involve the auxiliary constants 1C and 2C are given in

appendix C.


In this section, we discuss the feature of some results concerning the melt polymer flow.

For this reason, Figs. 7.3-7.8 are prepared. Fig. 7.3 is plotted for comparison of PM and

OHAM, and both the results achieved approximately the same accuracy as the non-

Newtonian parameter 00 . The consequence of different emerging parameters is

discussed on the velocity and temperature profiles. Fig. 7.4 shows that the velocity at any

point in the domain decreases with the increase in the value of velocity ratioU . Fig. 7.5

depicts that the velocity decreases with the increase in the values of non-Newtonian

parameter 0 . Moreover, as the velocity ratio U approaches to zero, the velocity profile

comes close to a linear distribution. Fig. 7.6 gives the comparison of temperature

distribution obtained from PM and OHAM and both the results are in good agreement to

each other for small values of dimensionless parameter 0 . It can be seen from Fig. 7.7

that the temperature raises with the increase in the values of Brinkman number. Fig. 7.8

represents the temperature distribution. It shows that with the increase of velocity ratio,

the temperature distribution decreases.

123

Figure 7.3. Comparison of dimensionless velocity profiles using PM and OHAM when

0 1 20.6, 0.01, 0.001357286, 0.0027125721U C C .

Figure 7.4. Dimensionless velocity profiles for different values of the velocities ratioU

when .01.00

Figure 7.5. Dimensionless velocity profiles for different values of the dimensionless

parameter 0 when .2.0U

124

Figure 7.6. Comparison of dimensionless temperature distribution using PM and OHAM

when 0 1 20.7, 0.01, 10, 0.001473286, 0.0002569261U Br C C .

Figure 7.7. Dimensionless temperature distribution for different values of Brinkman

number Br when .7.0U

Figure 7.8. Dimensionless temperature distribution for different values of the velocity

ratioU when Brinkman .10Br

125

7.4 Conclusion

This study has successfully established an analysis of the post-treatment problem of wire

coating, a processing problem of industrial relevance. The polymer considered is to be

satisfies the third grade fluid model. The equations of momentum and heat transfer are

solved by PM and OHAM. Explicit expressions for the distribution of velocity and

temperature are obtained. Moreover, the volume flow rate, thickness of coated wire and

the total force on the surface of coated wire are also derived. The effects of emerging

parameters are examined on the fluid velocity and temperature distribution. It is found

that PM and OHAM are in good agreement for small values of 0 ( 00 ). It is also

found that the magnitude of velocity decreases with the increase of non-Newtonian

parameter 0 . Furthermore, it is found that with increasing the dimensionless velocity

ratio U and the Brinkman number Br causes increase in the temperature profiles.

126

Chapter 8

Exact Solutions of a Power Law Fluid Model in Post-

treatment Analysis of Wire Coating with Linearly Varying

Boundary Temperature

127

In this chapter, analysis of post-treatment of wire coating is presented. Coating material

satisfies power law fluid model. Exact solutions for the velocity field, flow rate and

average velocity are obtained. Moreover, the heat transfer results are presented for

different cases of linearly varying temperature on the boundaries. The variations of

velocity, volume flow rate, radius of coated wire, shear rate and the force on the total

wire are presented graphically and discussed.

8.1 Formulation and solution of the problem

In this section, we work under the same geometry and with the same assumptions as

discussed in chapter 7. The discrepancy is that the z -axis of the coordinate system is

chosen in the opposite direction of fluid flow due to drag of wire as shown in Fig. 8.1.

Moreover, we use power law fluid model instead of third grade fluid and consider that the

temperature is not constant on the boundaries but linearly varying with boundary.

Figure 8.1. Drag flow in wire coating.

We seek the velocity field of the form

rw,0,0u , rSS , (8.1)

then the boundary conditions for the problem become

1Vw at ,wr R

2Vw at 0r R , (8.2)

128

For power law fluid model, extra stress tensor S is defined as

1AS , (8.3)

where

1

2

0

: 1;

2 2

n

T

L L , (8.4)

where : is the scalar invariant, the coefficient of viscosity of the fluid , T in

superscript denotes the transpose of the matrix L , 0 the consistency index and n is

the power law index. The parameter n , further divide liquids into pseudoplastic 1n ,

dilatant 1n and Newtonian liquid for 1n . Therefore, the variation of n from unity

shows the measure of variation from Newtonian behavior.

In the flow through the tube, the scalar invariant is

2

2:

r

w. (8.5)

Substituting Eq. (8.5) into Eq. (8.4), one obtains

1

0

nw

r

. (8.6)

Using the velocity field given in Eq. (8.1) the continuity Eq. (1.5) is satisfied identically

and the nonzero components of Eq. (8.3) with the help of Eq. (8.6) become

n

zrdr

dwS

0 . (8.7)

Substituting the velocity field and Eq. (8.7) in the momentum Eq. (1.6) neglecting the

body force takes the following forms

0

r

p, (8.8)

0

p, (8.9)

n

dr

dwr

dr

d

z

p0 . (8.10)

129

If the z - axis is chosen correspond to the direction of increasing pressure, polymer (II)

moves in the negative direction of the z - axis and the shear rate dr

dw0 becomes

positive for all value of .r Therefore the absolute value can be discarded.



00

n

dr

dwr

dr

d , (8.11)


.

1

0

2

n

pdr

dwk

tD

Dc (8.12)

For linearly varying temperature [116], consider

rgzzr , , (8.13)

where is the temperature gradient.

Substituting Eq. (8.13) into Eq. (8.12), we have

.1

1

02

2

n

pdr

dwg

dr

d

rdr

dkwc (8.14)

The force on the wire is computed by determining the shear stress at the wire surface.

This is given by

0 .w

w

n

rz r R

r R

dwS

dr

(8.15)

Introduce the dimensionless parameters

1

, ,w

r wr w

R V

1

0 2

110 10 1

1

, , 1,

np w

nnw

nw

c R R VgG S U

R VVV

kR

(8.16)

Eqs. (8.2), (8.11), (8.14) and (7.15) – (7.19) after dropping the “ ” take the following

form

0

n

dr

dwr

dr

d, (8.17)

130

11 w and Uw , (8.18)

wrSdr

dwr

dr

dG

dr

Gdr

n

1

2

2

, (8.19)

2 20

1 12

ave w

ave

w

w R Rw r w r dr

R V

, (8.20)

20 1 1

2

QQ r w r dr

R V

, (8.21)

1

2

1

1 2 ,cc

w

RR r w r dr

R

(8.22)

10 1 1 1

,

nnrz w

rz nr

r r

S R dwS

drV

(8.23)

1

1 1

,2

nnw w

w n

r

F R dwF

drLV

(8.24)

The solution of Eq. (8.17) corresponding to the boundary conditions given in Eq. (8.18) is

1

1

11 1,

1

n

n

n

n

rw r U

for 1n (8.25)

For 1n , the velocity field can be obtained from Eq. (8.17).

ln

1 1,ln

rw r U

(8.26)

For 1n , the average velocity is obtained from Eqs. (8.20) and (8.25) as follows

11 1

2

2

1 21 1 1 1 1 1 ,

2 3 1 1

n n

n nave

nw U

n

(8.27)

and for 1n , the average velocity is obtained from Eq. (8.20) and (8.26) as given by

1

2

2

1 1 11 1 1 1 .

2 2lnavew U

(8.28)

Similarly for 1n , the shear rate can be obtained from Eq. (8.25) as

1

11

1 1,n

n

n

dw n Ur

dr n

(8.29)

and for 1n , the shear rate is obtained from Eq. (8.26) as

131

1

11

1 1.n

n

n

dw n Ur

dr n

(8.30)

The thickness of the coated wire for 1n is obtained from Eqs (8.22) and (8.25) as

1

211 1

2 2

2

1 22 1 1 1 1 1 1 ,

2 3 1 1

n n

n nc

nR U

n

(8.31)

Similarly, the thickness of the coated wire for 1n is obtained from Eqs. (8.22) and

(8.26) as given by

1

21

2 2

2

1 1 12 1 1 1 1

2 2lncR U

. (8.32)

In a similar manner, the force on the surface of the total wire for 1n in the die is

1

1

1 1

n

w n

n

n UF

n

, (8.33)

and the force on the surface of the total wire for 1n in the die is

1

ln

n

w

UF

. (8.34)

In dimensionless form, the volume flow rate and the average velocity for n is or is not

equal to 1 are the same as given in Eqs. (8.27) and (8.28) respectively.

Keeping in view, the importance of temperature, we are looking for the temperature

distribution through different cases.

Case 1. Constant temperature of the wire and linearly varying temperature at the surface

of the coated wire.

Consider the temperature of the wire is 0 and it is z on the surface of the coated wire,

so from Eq. (8.13), we have

01,1 gzz , zgzz , . (8.35)

After transformation, we obtain

132

HG 1 , 0G , (8.36)

where 0

1

0 11

n

n

w

zH

V

kR

.

For 1n , substituting the velocity field from Eq. (8.25) into Eq. (8.19) and solving the

resulting equation corresponding to the boundary conditions given in Eq. (8.36), we

obtain the expression for temperature distribution in form of G as

2 3 1 3 1

2 2001 1

12 1 1

1

1 ln 1 ln1 1 1 1 1

2 ln 3 1 ln1 1

1 ln ln1 1 1

1 ln ln1

n n

n nn n

n n

n

n n

n nn

n

S U r U n rG r r S r

n

U n r rr H

n

.

(8.37)

Now for 1n , substituting the velocity field from Eq. (8.26) into Eq. (8.19) and solving

the resulting equation, one obtains

2 2 2 20 01 ln 1 ln1 1 1 ln 1 .

4 ln ln 4 ln ln

S SU r U rG r r r r H

(8.38)

Case 2. Linearly varying temperature of the wire and constant temperature at the surface

of the coated wire.

In this case, consider the temperature at the surface of wire is 1 and z on the surface

of continuum.

Under the above consideration, Eq. (8.13) gives

zgzz 1,1 , 0, gzz . (8.39)

After transformation of the boundary conditions given in Eq. (8.39) for the non-

dimensional temperature distribution G takes the following form

01 G , JG , (8.40)

where 1

1

0 11

n

n

w

zJ

V

kR

.

133

For 1n , we now substitute the velocity field from Eq. (8.25) into Eq. (8.19) and solved

corresponding to the boundary conditions given in Eq. (8.40), we have

2 3 1 3 1

2 2001 1

12 1 1

1

1 ln 1 ln1 1 1 1 1

2 ln 3 1 ln1 1

1 ln ln1 1 .

1 ln ln1

n n

n nn n

n n

n

n n

n nn

n


n

U n r rr J

n

(8.41)

Next, for 1n , substituting w r

in Eq. (8.19) and solved corresponding to the boundary

conditions (8.40), we have

2 2 2 20 01 ln 1 ln1 1 1 ln .

4 ln ln 4 ln ln

S SU r U rG r r r r J

(8.42)

Case 3. Linearly varying temperature with same temperature gradient on both of the wire

and on the surface of the coated wire.

Consider the temperatures at the surface of wire and on the surface of continuum are z .

From Eq. (8.13), we have

zgzz 1,1 , zrgzz , . (8.43)

After simplification according to the demand of our problem, we obtain

01 G , 0G , (8.44)

For 1n , using Eq. (8.26) in Eq. (8.19) and solving the resulting equation, we have

2 3 1 3 1

2 2001 1

12 1 1

1

1 ln 1 ln1 1 1 1 1

2 ln 3 1 ln1 1

1 ln1 1 .

1 ln1

n n

n nn n

n n

n

n n

n nn

n


n

U n rr

n

(8.45)

For 1n , substituting w r

from Eq. (8.25) into Eq. (8.19) and solving along with the

boundary conditions given in Eq. (8.44), we obtain the non-dimensional temperature as

2 2 2 20 01 ln 11 1 1 ln .

4 ln ln 4 ln

S SU r UG r r r r

(8.46)

134


In this study, results have been evaluated on the basis of equations derived in the

theoretical analysis. The figures in the following section demonstrate the way in which

the fluid velocity, average velocity, flow rate, thickness of coated wire, force of polymer

on the surface of wire and temperature vary during the post-treatment process of wire

coating. For linearly varying wall temperature, we have discussed three cases. One can

see the behavior of the physical quantities such as velocity function, non-dimensional

function of temperature profile and the differential form of these functions from Figs.

8.2-8.14. Fig. 8.2 illustrates the well known effect of power law index n on the velocity

profile; i.e., for pseudoplastic, the profile becomes progressively flatter and for dilatant

fluids, the profile becomes progressively linear. Figs. 8.3 and 8.4 show the fluid velocity

and variation of velocity profiles for different velocity ratios

7.0and6.0,5.0,4.0,3.0,2.0U . Here, it can be seen that with the increase of velocity

ratio U , the fluid velocity and its variation increase almost with the same ratio and gives

flatten profiles. Figs. 8.5-8.8 are plotted to investigate the force on surface of wire,

thickness of coated wire and volume flow rate respectively. Here it is observed that the

force on the wire increases as the velocity ratio increases. Also it can be seen that the

thickness of coated wire increases in case of dilatant fluids and decreases in pseudoplastic

fluids. Figs. 8.9 and 8.10 present the temperature profiles for the case when temperature

of the wire is constant and varying linearly on the surface of the coated wire respectively.

Fig. 8.9 shows that dimensionless temperature reduces as the non-Newtonian parameter

0S increases and this increase in velocity is relatively low at the centre of domain as

compared to the fluid velocity near the boundary. Fig. 8.10 admits that the temperature is

approximately linearly distributed in the fluid and it is found that it increases with the

increase of parameter H . Figs. 8.11 and 8.12 present the non-dimensional temperature

profiles for the case when temperature of the wire is varying linearly and constant on the

surface of the coated wire respectively. Fig. 8.11 is plotted for temperature distribution

against r for different values of power law index 1.6 and4.1,2.1,8.0,6.0,4.0n . It is

observed that as we increase n in the domain 10 n , the temperature distribution

increases and decreases when we increase n in the domain 1n . Also the temperature

135

profiles overshoots near the surface of coated wire due to the linearly varying

temperature at the surface of coated wire. Fig. 8.12 depicts that when we increase the

non-dimensional parameter H , as a result the temperature distribution reduces. Figs.

8.13 and 8.14 give the temperature profiles for the power law index n and the non-

dimensional parameter 0S for the case when temperature of the wire and the surface of

coated wire are varying linearly at the same temperature gradient. It is observed in both

of these graphs that the maximum temperature rises at the centre of the boundaries and

tends to zero at the boundaries.

Figure 8.2. The velocity profiles for different values of n when .5.0,2 U

Figure 8.3. Dimensionless velocity profiles for different values of velocity ratio U

when ,1.0n .2

136

Figure 8.4. The shear rate for different values of velocity ratio U when ,1.0n .2

Figure 8.5. Force wF is plotted against U for different values of velocity ratio n when

2.


taking 1.2.U

137

Figure 8.7. Radius of coated wire cR is plotted against n for different values of

by taking 1.2.U

Figure 8.8. Volume flow rate is is plotted against U for different values of power law

index n when 2.


parameter 0S when 2,05.0,5.0,5.0 HUn .

138

Figure 8.10. The non-dimensional function G for different values of H taking

15.0,5.0,5.0 0 SUn and .2


2,5,6.0,10 0 SUJ .

Figure 8.12. The non-dimensional function G for different values of H when

2,5.0,25,6.0,4 0 nSUJ .

139


2,10,6.0,5.0 0 SUJ .


parameter 0S when 2,4.0,3.0,2 nUJ .

8.3 Conclusion

In the present study, the post-treatment of the wire coating analysis is carried out for

power law model fluid. The velocity field, volume flow rate, average velocity, thickness

of coated wire, force of polymer on the surface of wire and shear rate have been derived

exactly for n is or is not equal to 1. In post-treatment problem, the temperature is

extremely important for cooling the wire. Therefore, regarding the importance of

temperature, we have discussed three cases for linearly varying temperature. Expressions

for temperature distributions in non-dimensional form are obtained for 1n and 1n .

140

The interpretations of the results are carried out under the influence of non-dimensional

parameters. It is concluded that the force on the wire increases as the velocity ratio

increases. It is also concluded that the thickness of the coated wire increases in case of

dilatant fluids and decreases in pseudoplastic fluids. Further, it is found that the non-

Newtonian parameter reduces the fluid velocity. Moreover, it is observed that the

force on the coated wire increases as the velocity ratio increases and it decreases while

increases . It can be seen that for 1n , the thickness of coated wire increases. It is

also established that as we increase n , lies in the range 10 n , the distribution of

temperature increased and it decreases when we increase n in the domain 1n .

Moreover, the peak temperature rise in the centre of the die depends on the dimensionless

number S and the power law index n .

141

Chapter 9

Heat Transfer by Laminar Flow of an Elastico-Viscous

Fluid in Post-treatment Analysis of Wire Coating with

Linearly Varying Temperature along the Coated Wire

142

In this chapter, the flow of an elastico-viscous fluid in post-treatment of wire coating

analysis with linearly varying temperature is studied. The constitutive equation of motion

and energy equation are solved by using Perturbation Method (PM) and Modified

Homotopy Perturbation Method (MHPM) for velocity, temperature and pressure

distribution. The theoretical analysis of volume flow rate, thickness of coated wire and

force on the total wire are also derived. Moreover, the flow phenomenon is studied under

the influence of elastic number ,eR the cross-viscous number ,c velocity ratio U and

the dimensionless number S . Solutions are also presented graphically.


The geometry and assumptions for the problem under investigation are the same as in

previous problems discussed in chapters 7 and 8. However, here we assume that the melt

polymer obeys an elastico-viscous liquid model and consider that the temperature is

linearly varying along the coated wire. The velocity field and the boundary conditions are

given in Eqs. (8.1) and (8.2) respectively.

The basic equations governing the flow of an incompressible elastico-viscous fluid with

thermal effects are

Continuity equation 0, iiv , (9.1)

Momentum equation ,,i i

j jj

iv i iv v T ft

, (9.2)

Energy equation

jjjjp kv

tc ,, , (9.3)

where iv is the velocity vector, the constant density, if the body force, i

j the

Cauchy stress tensor, the fluid temperature, k the thermal conductivity, pc the

specific heat and is the dissipation function defined as

j

i

i

j d , (9.4)

in which j

id is the rate of strain tensor defined by

, ,2 .i j i j j id v v (9.5)

143

According to Noll [61] the rheological equation of state for an elastico-viscous fluid

model is given by

i i i

j j jT p S , (9.6)

where p is the isotropic mean pressure, i

j the Kronecker’s delta and the extra stress

tensor i

jS is defined as [60]:

j

i

c

i

j

i

j

i

je

i

j dddpSS 42 , (9.7)

where e is the elastic parameter having the dimension of time, the coefficient of

viscosity, c the coefficient of cross viscosity and i

jS is the rate of stress tensor

according to Truesdell [117] is given by

k

k

i

j

i

k

k

j

k

j

i

k

ki

kj

i

ji

j vSvSvSvSt

SS ,,,,

. (9.8)

Using the velocity field given in Eq. (8.1) the continuity Eq. (9.1) is satisfied identically.

With the help of Eqs. (9.1), (9.4) and (9.7) the nonzero components of extra stress tensor

S and the dissipation function are

,

2

dr

dwS crr (9.9)

,

3

dr

dw

dr

dwS cezr (9.10)

,22

4

2

2

dr

dw

dr

dwS ceeczz (9.11)

.

42

dr

dw

dr

dwce (9.12)

The equation of momentum (9.2) in the absence of body force gives

rrrr S

rr

S

r

p 1

, (9.13)

0

p, (9.14)

rzrz S

rr

S

z

p 1

. (9.15)

144

Since the flow takes place due to uniform motion of the wire in the z direction, we have

0

z

p. (9.16)

From Eqs. (9.14) and (9.16), we have rpp only.

Also from Eq. (9.15), we get

0rzrSdr

d, (9.17)

which on integration gives

r

CS rz

11 , where 11C is a constant of integration. (9.18)

From Eqs. (9.9) and (9.13), we have

2

dr

dwr

dr

d

rdr

dp c , (9.19)

and from Eqs. (9.10) and (9.18), we get

r

C

dr

dw

dr

dwce

11

3

. (9.20)

Assuming the temperature of the boundaries are linearly varying with z according to the

expressions

0,w wR z z F R , 0 0 1,R z z F R . (9.21)

We set temperature as [116]

rFzzr , , (9.22)

where is the temperature gradient.

Using Eqs. (9.12) and (9.22), the energy Eq. (9.3) reduces to

.01

42

2

2

wc

dr

dw

dr

dw

dr

dF

rdr

Fdk pce (9.23)

145

Introducing the dimensionless variables

2

0 12

2 2

1 1 1

0110 1 02

0

, , , 1, , , ,

, , when

e ce

w w w

d dcc

w w w

R VVr w pr w U R P

R V R V R V

F r F RCK

R R F R F R

0

0

0

d dF r F R

when 01 ,

where

0 d 0

0 1 0 0 1 02 2

1 1

, for and , forwd

F R F R kk

V V

,

where eR is the elastic number, P the dimensionless pressure and c the cross viscous

number.

The set of Eqs. (9.19) and (9.20) after dropping the asterisks, determining the radial

pressure distribution, velocity w , the temperature d and the corresponding boundary

conditions reduce to

2

dr

dwr

dr

d

rdr

dP c , (9.24)

3

e

dw dw KR

dr dr r

, (9.25)

01

2

2

wSdr

dw

r

K

dr

d

rdr

d dd

, (9.26)

0 1 0

1 0

1 1, ,

1 0, for ,

and

1 0, 0 for .

d d d

d d

w w U

(9.27)

The non-dimensional form of average velocity, volume flow rate, thickness of coated

wire, shear stress at the surface of coated wire and the force on the total wire given in

Eqs. (7.15) – (7.19) become

146

2 20

21 1

2

ave w

ave

w

w R Rw r w r dr

R V

, (9.28)

2

1 12 w

QQ r w r dr

R V

, (9.29)

1

2

1

1 2 ,cc

w

RR r w r dr

R

(9.30)

3

11 11 1

,rz wrz er

rr r

S R dw dw KS R

V dr dr r

(9.31)

3

1 11

.2

ww e

rr

F dw dw KF R

LV dr dr r

(9.32)

We use the suitable form of 0

d in the cases where 1 is or is not equal to 0 .


The solution of Eq. (9.25) can be obtained in a closed form by using the following

approach.

Writing dr

dwq ,

Eq. (9.25) can be re-written in the form

3e

Kq R q

r , (9.33)

Differentiating Eq. (9.33) with respect to ,w we have

2

2 21 3 1e e

dq qR q R q

dw K , (9.34)

which yields

2 2

2

21

1 1

e e

ee

R q KR qdw K dq dq

q R q R q

, (9.35)

After integration equation (9.35) yields

2

12

1log

1

e

e

R q Kw K K

q R q

, (9.36)

where 1K is constant of integration.

147

Removal of q from Eqs. (9.33) and (9.36) gives the solution of velocity field. Using the

boundary conditions given in Eq. (9.27) the unknown constants K and 1K can be

evaluated.

But this appearance of result is very complicated to be tackled in physical problem.

Therefore, we make an attempt to obtain an approximate solution of Eq. (9.25) by

perturbation method.

9.2.1 Perturbation solution

To apply perturbation method (PM) we assume that the elastic number eR is a small

perturbation parameter and expand , ew r R and the constant K in series of the form as

follows:

2

0 1 2, ~ ...e e ew r R w r R w r R w r , (9.37)

2

0 1 2~ ...e eK K R K R K . (9.38)

Substitute Eqs. (9.37) and (9.38) into Eq. (9.25) and equating the coefficient of same

powers of , we get the following equations of various orders

0 0 0: 0e

dw KR

dr r , (9.39)

3

1 01 1: 0e

dwdw KR

dr dr r

, (9.40)

2

2 02 1 2: 3 0e

dwdw dw KR

dr dr dr r

, (9.41)

and so on. Also the boundary conditions for velocity field in Eq. (9.27) reduces to

0 1 2

0 1 2

at 1 1, ... 0,

and

at ... 0.

r w w w

r w w w

(9.42)

We solve the above sequence of problems with the corresponding boundary conditions

given in Eq. (9.42) and build the series solution.

Zeroth-order solution

1ln

ln10 U

rw

, (9.43)

148

ln

10

UK . (9.44)

First-order solution

Substituting the zeroth-order solution from Eq. (9.43) into Eq. (9.40) and integrating

yields

22

3

1

11

11

ln

ln

ln

1

2

1

r

rUw

, (9.45)

24

3

1

11

ln2

1

UK . (9.46)

Second-order solution

Substituting Eqs. (9.43) and (9.45) into Eq. (9.41), and integrating twice by using the

conditions given in Eq. (9.42) yields the second order solution

4224

2

22

5

2

11

11

11

ln

111

ln

ln11

ln

ln

ln

1

4

3

rr

rrUw

,

(9.47)

4

2

26

5

2

11

11

ln

1

ln4

13

UK . (9.48)

Similarly, the higher order perturbation solution can be obtained. But for small elastic

number, we carry on only up to the terms containing square of . Hence from Eqs. (9.37)

and (9.38), we have

2 2 3

0 1 2e ew w R w R w O , (9.49)

2 2 3

0 1 2e e eK K R K R K O R , (9.50)

where the superscript “ )2( ” represent the approximation up to the second order.

Hence, we have

3 522

2 2

2

2 2 4 2 2 4

3ln 1 ln 1 1 11 1 1 1

ln 2 ln ln 4 ln

ln 1 ln 1 1 1 1 11 1 1 1 1 ,

ln lnln

e eR Rr U r Uw w U

r

r r

r r

(9.51)

149

3 5 22

2

4 62 2 4

1 3 11 1 1 1 11 1 1

ln 2 lnln 4 ln

eeU R URU

K K

. (9.52)

We point out that if we set 0eR in Eq. (9.51), we recover the Newtonian solution.

Substituting the expressions for w and K from Eqs. (9.51) and (9.52) into the energy Eq.

(9.26) and integrating twice after using the boundary conditions for the temperature from

Eq. (9.27), we get the temperature distribution function for 01 .

23 52 2 2 22

2

222 2 2 2

2 2 4 2

ln 1 ln1 1 1 1 3 1ln 1 1

2 4 ln 2 ln 4 ln 4 2 4 ln

ln 1 ln 1 ln1 1 1 11 1 1 1

4 4 ln ln 4 2 4ln

d

e e

r rr r U U r r US r R R

r r rr r r r

4

2 223 5 2

2

2 2 2

2

04 2 2 4

1

ln ln1 1 1 1 1 3 1 1 ln 11 1

ln 2 ln 2ln 2 4 ln 2 ln 2ln

1 1 1 1 1 11 1

2 ln 4 1 2

e e

d

r

r rU U U rK R R

r

rS

r r

2

23 52 22

2

22 2

2 2 4 2

1ln 1 1

ln 4

ln1 1 1 1 1 3 11 ln 1 1

2 ln ln 4 2 4 4 ln

1 1 1 1 1 1ln 1 1 1 ln 1 1 1 1

4 4 ln lnln

e e

U

U UR R

2 2 3 22 2

4 2 2

5 2 2

2

2 4 2 4

ln ln1 1 1 1 1 1 1 ln 11 1 1

4 2 4 ln 2 2 ln 2 2

3 1 1 1 ln 1 1 1 1 11 1 1 1

4 ln 2 2 2ln 4

e

e

U UK R

UR

3 5 2

2

22 2 4

3 5

2

2 2

1 1

2 4ln

1 1 1 1 1 3 1 1 1 1 11 1 1

2 ln 4ln 4 4 ln 4ln4 ln

1 1 1 1 3 1 1 11 1

4ln 4 ln 4 ln 2ln

e e

e e

US

U UR R

U UK R R

1.

4

(9.53)

Similarly, the temperature distribution for 01 can be obtained by putting 00 d in

Eq. (9.53).

150

The result for velocity and temperature distribution obtained by Mishra [118] can be

recovered when we transform our problem to the original parameters and take the non-

dimensional parameter S and the non-dimensional velocity ratio U both equal to zero.

9.2.2 Solution by modified homotopy perturbation method (MHPM)

According to modified homotopy perturbation method [68] in Eq. (1.15), we have

dr

dwwL , (9.54)

,

3

dr

dwRwN c (9.55)

and

K

g rr

. (9.56)

We decompose rg into an infinite series as given in Eq. (1.14).

The only feasible choice for choosing the terms of series rg for this problem is to take

,00 rg (9.57)

0 11 ,

K Kg r

r

(9.58)

22 ,

Kg r

r (9.59)

and so on.

Take the initial guess approximation

1ln

ln10 U

ru

, (9.60)

that satisfies the boundary conditions given in Eq. (9.27) for zeroth order velocity field.

Substitute Eqs. (9.54), (9.55) and (9.57) – (9.60) in Eq. (1.15) and comparing the

coefficients of same power of p , we obtain the different order problems as


0: 00

0 uLwLp , (9.61)


110 w , .0 Uw (9.62)

First-order problem

151

3

1 0 0 11 0: e

dw K Kp L w L w R

dr r

, (9.63)


011 w , .01 w (9.64)


2

2 0 1 22: 3 e

dw dw Kp L w R

dr dr r

, (9.65)


012 w , .02 w (9.66)

The solutions of above equations are

00 1ln

ln1 uU

rrw

, (9.67)

3

1 2 2

1 ln 1 11 1

2 ln ln

eR U rw r

r

, (9.68)

5 22

2 2 2 4 2 2

4

3 1 ln 1 ln 1 1 1 11 1 1 1

4 ln ln lnln

11 .

eR U r rw r

r

r

(9.69)

Thus, from Eq. (1.17), the second order approximate solution is

3 52

2 2

2

2 2 4 2 2 4

3ln 1 ln 1 1 11 1 1 1

ln 2 ln ln 4 ln

ln 1 ln 1 1 1 1 11 1 1 1 1 .

ln lnln

e eR Rr U r Uw r U

r

r r

r r

(9.70)

where

3

0 1 4 2

11 11

ln 2 ln

eR UUK K

,

5 22

2 6 2 4

3 1 1 1 11 1

ln4 ln

eR UK

.

In dimensionless form, average velocity and volume flow rate are the same as given in

Eqs. (9.28) and (9.29). Therefore, we make an attempt to determine only the flow rate.

152

The volume flow rate of coating material per unit width is obtained from Eqs. (9.29) and

(9.70) as

2 22

3

2 2 2

52

2 2 2

1 1 1ln

2 ln

2 1 1 1 1 ln2 1 1

2 ln 2ln

3 1 2 1 1 1ln 1

4 ln 2ln

e

e

U

QR U

R U

2

2 2

11

1 31

ln

(9.71)

The thickness of the coated wire

2 2

2

3

2 2 2

2 2 2

52

1 1 1ln

2 ln1

1 1 1 1 ln2 1 1

2 ln 2ln

2 1 1 1ln 1

2ln

3 1

4 ln

e

c

e

U

R U

R

R U

1

2

2

2 2

,

11

1 31

ln

(9.72)

Similarly, the force on the surface of the total wire is

3 5 22

4 2 6 2 4

1 3 11 1 1 1 11 1 1 .

ln 2 lnln 4 ln

eew

U R URUF

(9.73)

Now substituting Eq. (9.70) into the energy Eq. (9.26) and solving the resulting equation

corresponding to boundary conditions given in Eq. (9.27), we obtain the explicit

expression for temperature distribution as

153

23 52 2 2 22

2

222 2 2 2

2 2 4 2

ln 1 ln1 1 1 1 3 1ln 1 1

2 4 ln 2 ln 4 ln 4 2 4 ln

ln 1 ln 1 ln1 1 1 11 1 1 1

4 4 ln ln 4 2 4ln

d

e e

r rr r U U r r US r R R

r r rr r r r

4

2 223 5 2

2

2 2 2

2

04 2 2 4

1

ln ln1 1 1 1 1 3 1 1 ln 11 1

ln 2 ln 2ln 2 4 ln 2 ln 2ln

1 1 1 1 1 11 1

2 ln 4 1 2

e e

d

r

r rU U U rK R R

r

rS

r r

2

23 52 22

2

22 2

2 2 4 2

1ln 1 1

ln 4

ln1 1 1 1 1 3 11 ln 1 1

2 ln ln 4 2 4 4 ln

1 1 1 1 1 1ln 1 1 1 ln 1 1 1 1

4 4 ln lnln

e e

U

U UR R

2 2 3 22 2

4 2 2

5 2 2

2

2 4 2 4

ln ln1 1 1 1 1 1 1 ln 11 1 1

4 2 4 ln 2 2 ln 2 2

3 1 1 1 ln 1 1 1 1 11 1 1 1

4 ln 2 2 2ln 4

e

e

U UK R

UR

3 5 2

2

22 2 4

3 5

2

2 2

1 1

2 4ln

1 1 1 1 1 3 1 1 1 1 11 1 1

2 ln 4ln 4 4 ln 4ln4 ln

1 1 1 1 3 1 1 11 1

4ln 4 ln 4 ln 2ln

e e

e e

US

U UR R

U UK R R

1.

4

(9.74)


In this study, the heat transfer of an elastico-viscous liquid in post-treatment of wire

coating with linearly varying temperature is examined carefully. To obtain an

approximate solution for velocity field, we choose the elastic number ,eR as a small

perturbation parameter. The velocity and heat transfer analysis are investigated under the

effect of cross-viscous number ,c elastic number ,eR velocity ratioU and the

dimensionless number S graphically. For all the graphs, we fixed radii ratio .2 From

Eq. (9.25), it is quite clear that the velocity distribution is effected by the elasticity of the

liquid only when the liquid posses cross- viscosity and vice versa. But Eq. (9.24) shows

154

that cross- viscosity modifies the radial pressure field even in the absence of elasticity of

the liquid. Fig. 9.1 presents the velocity distribution of the polymer for different values of

velocity ratio .U It has been observed that for 1U , the velocity at any point increases

from 1 to , on the other hand, it decreases in this domain for 1U . Also it can be seen

that as the velocity ratio U approaches to 1, the velocity profile becomes to be a linear

distribution. Fig. 9.2 shows that as the elastic number eR increases, the velocity of the

fluid decreases, in other words, we can say that the elasticity of the liquid reduces the

speed of flow. Fig. 9.3 shows the theoretically predicted change in the velocity of

polymer for different elastic number eR . Fig. 9.4 presents the change in the thickness of

coated wire for different elastic number. The thickness of coated wire increases in the

nonlinear form up to the optimum value as the elasticity of the polymer increases. Fig.

9.5 shows temperature profile for the case 01 . In this case, it can be observed that

as the temperature at the surface of coated wire increases, the distributed temperature at

any point of the domain increases. Fig. 9.6 shows the temperature profile in the problem

when 01 for different values of non-dimensional parameter S . In this plot, it is

concluded that for S close to zero, the result approaches to linear profile and become

more parabolic as the value of S increases. Figs. 9.7 and 9.8 gives the profile of

temperature distribution in the case when 01 for different values of radii ratio

U and the non-dimensional parameter S respectively. Here, it has been found that the

temperature decreases with increase of velocity ratio U and the non-dimensional

parameter S correspondingly.

Table 9.1 shows that as the elastic number eR increases gradually, the velocity of the

fluid decreases slowly, in other words, we can say that the elasticity of the liquid reduces

the speed of flow. Table 9.2 gives that the rate of change of velocity for various values of

elastic number eR . Here, it is concluded that as we increase the elastic number eR slowly,

the rate of change of velocity decreases almost at the same rate and decrease vice versa at

every point of the domain. We observe from Table 9.3 that the elasticity of the liquid

decreases the temperature at any point when the walls are at constant temperature. This

conclusion agrees with Jain’s [61] conclusion who has observed that the elasticity of the

liquid increases the temperature.

155

Figure 9.1. Dimensionless velocity profiles for different values of radii ratio U for fixed

value of elastic number 0.2eR .

Figure 9.2. Dimensionless velocity profiles for different values of eR for fixed value of

.5.0U

Figure 9.3. Thickness of coated wire against elastic number Re for different values

when radii ratio 0.5.U

156

Figure 9.4. Force on the surface of coated wire against elastic number Re for different

values of when radii ratio 1.2.U

Figure 9.5. Dimensionless temperature distribution for different values of d

0 when

0.1,eR ,5.0S 4.0U .

Figure 9.6. Dimensionless temperature distribution for different values of S when

0.1,eR 2.1U .

157

Figure 9.7. Dimensionless temperature distribution for different values of U when

,5.00 d 0.3,eR 10S .

Figure 9.8. Dimensionless temperature distribution for different values of non-

dimensional parameter S when ,20 d 0.2, 0.4eR U .

Table 9.1. Shows velocity distribution for different values of elastic number eR

when 2.1U .


r 0.01eR 0.03eR 0.05eR 0.07eR 0.09eR

1.0 1 1 1 1 1

1 2 1.05259 1.05257 1.05254 1.05252 1.05249

1.4 1.09707 1.09704 1.09701 1.09698 1.09695

1.6 1.13560 1.13558 1.13555 1.13553 1.13551

1.8 1.16959 1.16958 1.16957 1.16955 1.16954

2.0 1.2 1.2 1.2 1.2 1.2

158

Table 9.2. Shows shear rate distribution for different values elastic number eR

when 2.1U .

Rate of change of velocity

r 0.01eR 0.03eR 0.05eR 0.07eR 0.09eR

1.0 0.288429 0.288210 0.287994 0.287779 0.287567

1.2 0.240418 0.240357 0.240296 0.240236 0.240175

1.4 0.206105 0.206115 0.206125 0.206135 0.206145

1.6 0.180359 0.180404 0.180449 0.180493 0.180537

1.8 0.160330 0.160396 0.160454 0.160515 0.160575

2.0 0.144304 0.144374 0.144443 0.144512 0.144581

Table 9.3. Shows temperature distribution for different values elastic number eR when

5.0,20,2.1 0 dSU .

Temperature distribution

r 0.1eR 0.5eR 0.8eR 1.2eR 1.6eR

1.0 0 0 0 0 0

1. 2 -1.77982 -1.77977 -1.77973 -1.77968 -1.77960

1.4 -2.50795 -2.50788 -2.50782 -2.50776 -2.50769

1.6 -2.31992 -2.31986 -2.31981 -2.31975 -2.31969

1.8 -1.29927 -1.29924 -1.29921 -1.29917 -1.29914

2.0 0.5 0.5 0.5 0.5 0.5

9.4 Conclusion

In the present work, PM and MHPM are employed to present analytical solution for

laminar polymer flow while the fluid is of an elastico-viscous form. Explicit expressions

are developed for the fluid velocity and temperature distribution. It is established that

unlike the Newtonian fluid model, the acquired result strongly depends upon the behavior

of the non-Newtonian fluid. It is also found that the elasticity of the fluid reduces the

velocity of flow. Further, it is found that if the cross viscous coefficient c disappears; the

effect of the elastic parameter eR also vanishes. But the case is not the same in reverse.

159

Moreover, the influence of non-Newtonian parameter on the radius of coated wire, force

on the total wire and on the temperature distribution are seen. The solution obtained by

the PM requires the presence of a small perturbation parameter in an equation, which is

not so with the solution obtained by the MHPM. In the MHPM, we look for an

asymptotic solution with few terms (usually 2 to 4 terms) and therefore no convergence

theory is needed. Comparison shows that the MHPM can completely overcome the

limitations arising in traditional perturbation methods. The result reveals that its second

order of approximation obtained by the MHPM is valid uniformly even for large

parameter, due to limitations, it is not possible in perturbation solutions.

160

Chapter 10

Conclusions and Future Work Directions

161

In this chapter, we conclude this thesis by summarizing our contributions and discussing

directions for future work.

10.1 Conclusions

The thesis presents the theoretical analysis of wire coating process. Study of wire coating

process under the influence of non-Newtonian fluids such as Phan-Thien and Tanner,

second grade, third grade, elastico-viscous and Oldroyd-8- constant and power law fluid

models are developed using both slip and no-slip conditions for analysis of polymeric

wire coating flows. Both problems within the pressure tooling die and outside the die are

under investigation. In case of post-treatment, the investigation is performed by

considering the slippage which exists at the contact surfaces of wire, polymer and the gas.

Flow is assumed to be the drag flow and the pressure gradient is considered to be

constant in the direction of drag of wire in each study. The analysis concentrate on flow

conditions around the die exit: particularly on shear and strain rates and identification of

the influence of slip. A long-term goal is to improve the understanding of process

performance and hence for this to ultimately impact upon product optimization. Chapter

wise conclusion is given at the end of each chapter in detail. In this section, we briefly

review our main conclusions.

In wire coating analysis, expressions are presented for the radial variation of the axial

velocity and the temperature distribution. In industrial point of view, some results were

also performed for steady shear and pure extensional flows such as for the flow rate,

average velocity, shear stress, normal stress, thickness of coated wire and force on the

total wire.

From the modelling of non-Newtonian fluids in coated process, a number of guiding

results in general may be identified. It was found that under the influence of involved

model parameters, all the above mentioned quantities are affected. Generally, it was also

concluded that the coating thickness is directly controlled by die design, boundary of the

die and the polymer used for coating the wire.

Effects of the slip and elastic parameters on the wire coating operation are also discussed.

It is noticed that the variation in the fluid velocity, temperature distribution, volume flow

rate, thickness of coated wire and the force on the surface of wire is quite interesting. It is

found that the velocity, volume flow rate, thickness of wire and the force on the wire

162

increases with the increase in slip parameter while the temperature distribution decreases

with increase of these parameters. Shear stress and normal stress decrease with increase

of slippage at the boundary of the die only for a limiting value of the boundary slip

coefficient. The thickness of coated wire increases in the nonlinear form up to the

optimum value as the elasticity of the polymer increases. It is observed that the elasticity

of the liquid reduces the speed of flow. It is also found that the elasticity of the liquid

decreases the temperature within the region.

The focus of this work was mainly on the effects associated to develop the understanding

of wire coating process performance and to improve product quality.

10.2 Future work directions

Among the many other possibilities for further studies, we would like to conclude the

thesis by suggesting:

We have used only single layer polymer flow for analysis using various non-

Newtonian fluids. The problems can be done by taking two or more polymer

layers. This problem particularly useful for industry.

We have used coaxial coextrusion process of wire coating. The problems can be

reformulated and can be done using electro-statical deposition process in which

the thermal treatment in an electric field and treatment by a beam of non-

penetrating electrons can be applied during coating operation. This process will

provide strong bonding between the melt polymer and wire.

It would be interesting to extend the problem for analysis of pipe manufacturing

by considering the polymer flow between concentric annulus. This type of

analysis can also be very useful in industry.

163

References

[1] Z. Tadmor and C.G. Gogos, Principle of Polymer Processing, John Wiley and Sons,

New York, (1978).

[2] C.D Han and D. Rao, The rheology of wire coating extrusion, Polym. Eng. Sci.,

18(13) (1978) 1019-1029.

[3] B. Caswell and R.J. Tanner, Wire coating die using finite element methods, Polym.

Eng. Sci., 18(5) (1978) 417-421.

[4] C.L. Tucker, Computer Modeling for Polymer Processing, Hanser, Munich, (1989)

311-317.

[5] S. Basu, A theoretical analysis of non-isothermal flow in wire coating co-extrusion

dies, Polym. Eng. Sci., 21(17) (1981) 1128-1138.

[6] H. Zhang, M.K. Moallemi and S. Kumar, J. Heat Transfer Metals and Container

less Process. Manuf., ASME, (1991) 49-55.

[7] S. Nakazawa and J.F.T. Pittman, Numerical Methods in Industrial Forming

Process, Pineridge, Swansea, UK, (1982) 523-533.

[8] V.S. Mirnov, P.K. Dutta and J.S. Gray, Surface Engineering Volume II:

Engineering Applications, The Roy. Soc. of Chem., 2 (1993) 80-90.

[9] S. Akter and M.S.J. Hashmi, Modeling of the pressure distribution within a

hydrodynamic pressure unit: Effect of change in viscosity during drawing of wire

coating, J. Material Proc. Tech., 77 (1998) 32-36.

[10] S. Akter and M.S.J. Hashmi, Wire drawing and coating using a combined geometry

hydrodynamic unit, J. Material Proc. Tech., 178 (2006) 98-110.

[11] V.M. Hovey, History of Extrusion Equipment for Rubber and Plastics, Wire and

wire Prod., 36 (1961) 192-195.

[12] J.F. Carley, Die design, in processing of thermoplastic material, Reinhold Pub.

Corp., New York, (1959).

[13] J.B. Paton, P.H. Squire, W.H. Darnell, F.M. Cash and J.F. Carley, Proc. of

Thermoplas. Materials., Reinhold Pub. Corp., New York, (1959) 269-299.

164

[14] J.M. McKelvey, Polymer Processing, John Wiley and Sons, New York, (1962).

[15] E.B. Bagley and S.H. Storey, Processing of thermoplastic materials, Wire and wire

Prod., 38 (9) (1963) 1104-1122.

[16] R.T. Fenner and J.G. Williams, Analytical methods of wire coating die design,

Trans. Plast. Inst. London, 35 (1967) 701-706.

[17] R.T. Fenner, Extruder Screw Design, Iliffe, London, (1970).

[18] M. Kasajima and K. Ito, Post-treatment of Polymer Extrudate in Wire Coating,

Appl. Polym. Symp., 20(1973) 221-235.

[19] H.H. Winter, Thermal capacitance and cooling length in the wire coating process,

in SPE 36th

ANTEC, Tech. Papers, 24 (1978) 462-469.

[20] Z. Tadmor and R.B. Bird, Rheological analysis of stabilizing forces in wire coating

dies, Polym. Eng. Sci., 14(2) (1974) 124-136.

[21] J.F. Carley, T. Endo and W. Krantz, Realistic analysis of flow in wire coating dies,

Polym. Eng. Sci., 19(16) (1979) 1178-1187.

[22] S. Middleman, Fundamentals of polymer processing, McGraw-Hill, New York,

(1977) 249-259.

[23] M.S.J. Hashmi, G.R. Symmons and H. Pervinmehr, A novel technique of wire

drawing, J. Mech. Eng. Sci. Inst. Mech. Eng., 24 (1982) 1-4.

[24] G.R. Symmons, M.S.J. Hashmi and H. Pervinmehr, Plasto-hydrodynamic dieless

wire drawing, theoretical treatments and experimental results, Proc. Int. Conf.

Developments in Drawing of Metals, Met. Soc., London, (1992) 54-62.

[25] R. Wagner and E. Mitsoulis, Effect of die design on the analysis of wire coating,

Adv. Polym. Tech., 5 (1985) 305-325.

[26] E. Mitsoulis, Finite element analysis of wire coating, Polym. Eng. Sci., 26(2)

(1986) 171-186.

[27] D.A. White and J.A. Tallmadge, The gravity corrected theory for cylinder

withdrawal, AICHE J. 13 (1967) 745-750.

[28] A.J. Soroka, J.L. Fitzjohn and J.A. Tallmadge, Cylinder Apparatus for Continuous

Withdrawal Studies, Ind. Eng. Chem. Proc. Des. Dev., 10(4) (1971) 592-594.

[29] S.C. Roy and D.K. Dutt, Wire coating by withdrawal from a bath of power law

fluid, Chem. Eng. Sci., 36(12) (1991) 1933-1939.

http://pubs.acs.org/doi/abs/10.1021/i260040a029



165

[30] J.F. Dijksman and E.P.W. Savenije, Flow of Newtonian and non-Newtonian liquids

through annular converging regions, Rheol Acta, 24 (1985) 105-118.

[31] Y.H. Shu, M. Charmchi and S.J. Chen. A theoretical study of non-isothermal

multilayer co-extrusion flow, J. Polym. Eng. Sci., 7 (1987) 255-273.

[32] Y.H. Shu, Theoretical studies of non-isothermal multilayer coextrusion flow, Ph.D.

thesis, University of Lowell, (1988).

[33] Y.H. Shu, S.J. Chen and J.S. Cheng, Non-isothermal multilayer coextrusion flow

in circular and annular dies, J. Reinf. Plast. Comp., 8 (1989) 358-369.

[34] P.B. Kuyl, Coating flow simulation of 24 AVG wire with FEP fluoropolymer, 45th

Int. wire and cable symp. Proc., (1996) 736-742.

[35] P.B. Kuyl, Non-isothermal viscoelastic flow simulation of high speed wire coating

process, proc., ANTEC, 97 (1997) 298-302.

[36] A.J. Hade and A.J. Giacomin, Wire coating under vacuum, J. Eng. Materials and

Tech., 123 (2001) 100-105.

[37] S. Akter and M.S.J. Hashmi, Plasto-hydrodynamic pressure distribution in a tapered

geometry wire coating unit, in: Proceedings of the 14th

Conference of the Irish

manufacturing committee (IMC14), Dublin, (1997) 331-340.

[38] S. Akter and M.S.J. Hashmi, Analysis of polymer flow in a canonical coating unit:

power law approach, Prog. Org. Coat., 37 (1999) 15-22.

[39] M. Sajjid, A.M. Siddiqui and T. Hayat, Wire Coating Analysis using MHD

Oldroyd 8-Constant Fluid, Int. J. Eng. Sci., 45 (2007), 381-392.

[40] A.M. Siddiqui, M. Sajjid and T. Hayat, Wire coating by withdrawal from a bath of

fourth order fluid, J. Phy. Lett. A, 372 (2008) 2665-2670.

[41] A.M. Siddiqui, T. Haroon and H. Khan, Wire coating extrusion in a pressure-type

die in flow of a third grade fluid, Int. J. of Non-Linear Sci. Numeric. Simul., 10 (2)

(2009) 247-257.

[42] W.R. Schowalter, Mechanics on non-Newtonian fluids, Pergamon Press, (1978).

[43] R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of polymeric liquids, Fluid

Mechanics, John Wiley and Sons, Vol. 1 (1987).

[44] D. Tripathi, Mathematical modeling of peristaltic transport of non-Newtonian

fluids, Ph. D thesis, Department of Applied Mathematics Institute of Technology

166

Banaras Hindu University, (2009).

[45] P. N. Garay, Pump Application Desk Book, Prentice Hall, 3rd edition, ISBN

0-88173-231-1, ISBN 978-0-88173-231-3, (1996), p.358, Google books.

[46] M. A. Rao, Rheology of Fluid and Semisolid Foods, Principles and Applications,

Publisher: Springer, ISBN 0-387-70929-0, ISBN 978-0-387-70929, 2nd edition,

(2007), p.8, Google books.

[47] L.L. Schramm, Emulsions, Foams and Suspensions: Fundamentals and

Applications, Publisher: Wiley VCH, (2005), ISBN 3-527-30743-5, ISBN 978-3-

527- 30743-2p.173, Google books.

[48] A.M. Vinogradov and B.A. Kupershmidt, the structure of Hamiltonian mechanics,

London Math. Soc. Lect. Notes Ser., 60, London: Camb. Univ. Press (1981).

[49] E. Schrödinger, Quantisierung als Eigenwertproblem [Quantification of the eigen

value problem], Annalen der Physik Bibcode 1926AnP...385..437S.

DOI:10.1002/andp.19263851302, 80 (13) (1926) 437–490.

[50] G. Batchelor, (2000). Introduction to Fluid Mechanics.

[51] A.H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow,

Volume 1, Ronald Press, (1953).

[52] A.M. Vinogradov and B.A. Kupershmidt, The structure of Hamiltonian mechanics,

London Math. Soc. Lect. Notes Ser., 60, London: Cambridge Univ. Press (1981).

[53] J. W. S. Rayleigh, Theory of Sound, London: Macmillan, ISBN 1-152-06023-6, 2

(1894) 115–118..

[54] P. Dirac, The quantum theory of dispersion, Proceedings of the Royal Society,

London, 114 (1927) 710-728.

[55] Y. Aksoy and M. Pakdemirli, Approximate analytic solutions for flow of a third

grade fluid through a parallel-plate channel filled with a porous medium, Transp.

Porous Med., 83 (2010) 375-395.

[56] A.M Siddiqui, R. Mahmood and Q.K Ghori, Thin film flow of a third grade fluid

on a moving belt by He’s homotopy perturbation method, Int J Nonlinear Sci,

Numer Simul. 7(1) (2006) 7-14.

[57] A.M. Siddiqui, R. Mahmood and Q.K Ghori, Thin film flow of a third grade fluid

on an inclined plane, Chaos, Solitons and Fractals, 35 (2008) 140-147.

http://en.wikipedia.org/wiki/Special:BookSources/0881732311


http://books.google.co.uk/books?id=pww5cxwitHAC&lpg=PP1&dq=0881732311&pg=PA359#v=snippet&q=thixotropic&f=false




http://books.google.co.uk/books?id=qFi61f1NqNIC&lpg=PA173&dq=pseudoplastic&pg=PA173#v=onepage&q=pseudoplastic&f=false

http://en.wikipedia.org/wiki/Bibcode

http://adsabs.harvard.edu/abs/1926AnP...385..437S

http://en.wikipedia.org/wiki/Digital_object_identifier

http://dx.doi.org/10.1002%2Fandp.19263851302

http://diffiety.ac.ru/djvu/structures.djvu

http://en.wikipedia.org/wiki/International_Standard_Book_Number

http://en.wikipedia.org/wiki/Special:BookSources/1-152-06023-6

167

[58] A.M. Siddiqui, A. Zeb, Q.K. Ghori and A.M. Benharbit, homotopy perturbation

method for heat transfer flow of a third grade fluid between parallel plates, Chaos,

Solitons & Fractals, 36 (2008) 182-192.

[59] W. Noll, On the continuity of the solid and fluid states, J. Rat. Mech. Anul., 4

(1955) 3-81.

[60] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy.

Soc., 200 (1950) 523-541.

[61] M.K. Jain, Heat transfer by laminar flow of elastico-viscous liquid between two

parallel walls with heat transfer, Appl. Sci. Res., 11 (1963) 295-307.

[62] P.J. Oliveira and F.T. Pinho, Analytical solution for fully developed channel and

pipe flow of Phan-Thien and Tanner fluids, J. Fluid Mech., 387, (1999) 271-280.

[63] F.T. Pinho and P.J. Oliveira, Axial annular flow of a nonlinear viscoelastic fluid-an

analytical solution, J. Non-Newtonian Fluid Mech., 93 (2000) 325-337.

[64] C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, Annu.

Rev. Fluid Mech., 23 (1991) 159-177.

[65] A.H. Nayfeh, Perturbation Methods, John Wiley and Sons, New York, 1973.

[66] A.H. Nayfeh, Introduction to perturbation techniques, John Wiley and Sons, New

York, (1979).

[67] R.H. Rand and D. Armbruster. Perturbation methods, bifurcation theory and

computer algebraic. Springer, (1987).

[68] Z.M. Odibat, A new modification of the homotopy perturbation for linear and non

linear operators, Appl. Math. Comput., 189 (1) (2007) 746-753.

[69] A.M. Siddiqui, T. Haroon and S. Irum, Torsional flow of third grade fluid using

modified homotopy perturbation method, Comput. Math. Appl., 58 (2009) 2274-

2285.

[70] V. Marinca, N. Herisanu and l. Nemes, A New Analytic Approach to Nonlinear

Vibration of an Electrical Machine, Proc. of the Rom. Acad., 9 (2008) 229-236.

[71] V. Marinca, N. Herisanu and I. Nemes, An optimal homotopy asymptotic method

with application to thin film flow, Centr. Euro. J. of Phy., 6(3) (2008) 648-653.

[72] V. Marinca, N. Herisanu, C. Bota and B. Marinca, An optimal homotopy

asymptotic method applied to steady flow of a fourth-grade fluid past a porous

168

plate, Appl. Math. Lett., 22 (2009) 245-251.

[73] V. Marinca and N. Herisanu, Application of optimal homotopy asymptotic

method for solving nonlinear equations arising in heat transfer, Int. Comm. in Heat

and Mass Trans., 35 (2008) 710-715.

[74] V. Marinca and N. Herisanu, Optimal homotopy perturbation method for strongly

nonlinear differential equations, Non-linear Sci. Lett. A, 1(3) (2010) 273-280.

[75] S. Islam, Rehan Ali Shah and Ishtiaq Ali, Optimal Homotopy Asymptotic Solutions

of Couette and Poiseuille Flows of a Third Grade Fluid with Heat Transfer

Analysis, Int. J. Non-Linear Sci. and Numeric. Simul., 11(6) (2010) 389-400.

[76] J. Ali, S. Islam, G. Zaman, The solution of multipoint boundary value problems by

the optimal homotopy asymptotic method, Comput. Math. Appl., 59 (2010)

2000-2006.

[77] K.U. Haas and F.H. Skewis, The Wire Coating Process: Die Design and Polymer

Flow Characteristics, S.P.E. ANTEC, Tech. Papers, 20 (1974) 8-11.

[78] D.A. White and J.A. Tallmadge, Theory of drag out on flat plates, Chem. Eng.

Sci. 20 (1965) 33-37.

[79] R.P. Spiers, C.V. Subbaraman and W.L. Wilkinson, Free coating of a liquid onto a

vertical surface, Chem. Eng. Sci. 29 (1974) 389-396.

[80] A. J. Soroka and J. A. Tallmadge, A test of the inertial theory of plate

withdrawal, Inst. Chem. Eng. Sci. J. 17 (1971) 505-508.

[81] M. N. Esmail and R.L. Hummel, Nonlinear theory of free coating on to vertical

surface, Chem. Eng. Sci. J. 21(5) (1975) 958-965.

[82] H. Mavridis and A.N. Hrymak, Finite element simulation of stratified multiphase

flows, J. Vlachopoulos, AIChE J., 33 (1987) 410-422.

[83] E. Mitsoulis, Extrudate swell in double-layer flows, J. Rheol., 30(8) (1986) 23-44.

[84] C. Fetecau, The Rayleigh-Stokes problem for heated second grade fluids. Int. J.

Non-Linear Mech., 37 (2002) 1011-1015.

[84] C. Fetecau, Decay of a potential vortex in a Maxwell fluid. Int. J. Non- Linear

Mech., 28 (2003) 985-990.

[86] C. Fetecau, Decay of a potential vortex in an Oldroyd-B fluid. Int. J. Non-Linear

Mech., 43 (2005) 340-351.

169

[87] W.C. Tan and M.Y. Xu, The impulsive motion of a flat plate in a generalized

second grade fluid. Mech. Res. Comm., 29 (2002) 3-9.

[88] W.C. Tan and T. Masuoka, Stokes first problem for a second grade fluid in a

porous half space with heated boundary. Int. J. Non-Linear Mech., 40 (2005) 515-

522.

[89] W.C. Tan and T. Masuoka, Stoke first problem for an Oldroyd-B fluid in a porous

half space. Phys. Fluids, 17 (2005) 023101-023107.

[90] C.I. Chen, C.K. Chen and Y.T. Yang, Unsteady unidirectional flow of an Oldroyd-

B fluid in a circular duct with different given volume flow rate. Int. J. Heat and

Mass Trans., 40 (2004) 203-209.

[91] K.R. Rajagopal, A note on unsteady unidirectional flows a non-Newtonian fluid,

Int. J. Non-Linear Mech., 17 (1982) 369-373.

[92] K.R. Rajagopal, On the creeping flow of the second grade fluid, J. Non-Newtonian

Fluid Mech., 15 (1984) 239-246.

[93] K.R. Rajagopal and A.S. Gupta, Flow and stability of a second grade fluid between

two parallel plates rotating about non coincident axes, Int. J. Eng. Sci., 19 (1981)

1401- 1409.

[94] A.M. Siddiqui, T. Hayat and S. Asghar, Some unsteady unidirectional flows of a

non-Newtonian fluid, Int. J. Eng. Sci., 38 (2000) 337-346.

[95] A.M. Siddiqui, T. Hayat and S. Asghar, Some exact solutions of an elastico-

viscous fluid, Appl. Math. Lett., 14 (2001) 571-579.

[96] A.M. Siddiqui, R.P. Burchard and W.H. Schwarz, An undulating surface model for

the motility of bacteria gliding on a layer of non-Newtonian slim, Int. J. Non-

Linear Mech., 36 (2001) 743-761.

[97] A.M. Siddiqui and A. Benharbit, Hodograph transformation methods in an

incompressible second grade fluid, Mech. Res. Comm., 24(4) (1977) 463-472.

[98] A.M. Siddiqui, M. Ahmed, S. Islam and Q.K. Ghori, Homotopy analysis of Couette

and Poiseuille flows for fourth grade fluids. Acta Mech., 180 (2005) 117-132.

[99] C. Truesdell and W. Noll, The nonlinear field theories of mechanics, Handbuch

der physic, Berlin Heidelberg New York Springer Verlag, Vol. 3 (1965).

[100] J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod.

170

Phys., B 20 (10), (2006) 1141-1199.

[101] J.H. He, Homotopy perturbation method for solving boundary value problems.

Phys. Lett. A, 350(1-2) (2006) 87-88.

[102] G.L. Liu, New research directions in singular perturbation theory: artificial

parameter approach and inverse-perturbation technique, in: Conf. of 7th Mod.

Math. and Mech., Shanghai, (1997).

[103] S. J. Liao, An approximate solution technique not depending on small

parameters: a special example, Int. J. Non-Linear Mech., 30 (3) (1995), 371–380.

[104] J.H. He, Homotopy perturbation technique. Comput. Math. Appl. Mech. Eng., 178

(1999) 257-263.

[105] J.H. He, Homotopy perturbation method, a new analytical technique. Appl. Math.

Comput., 135 (2003) 73-79.

[106] J.H. He, Homotopy perturbation technique, Comput. Meth. Appl. Math. Eng., 178

(1999) 257-262.

[107] N. Phan-Thien and R.I. Tanner, A new constitutive equation derived from

network theory, J. Non-Newtonian Fluid Mech. 2, (1977) 353-365.

[108] N. Phan-Thien, A nonlinear network viscoelastic model, J. Rheol. 22, (1978)

259-283.

[109] D.O.A. Cruz and F.T. Pinho, Skewed Poiseuille-Couette flows of PTT fluids in

concentric annuli and channels, Non-Newtonian Fluid Mech., 121 (2004) 1-14.

[110] M.A. Alves, F.T. Pinho and P.J. Oliveira, Study of steady pipe and channel flows

of single-mode Phan-Thien and Tanner fluid, J. Non-Newtonian Fluid Mech.,

101, (2001) 55-76.

[111] G. Karapetsas and J. Tsamopoulos, Steady extrusion of viscoelastic materials

from an annular die, J. Non-Newtonian Fluid Mech., 154, (2008) 136–152.

[112] R.I. Tanner, Engineering Rheology, Clarendon Press, Oxford, (2000).

[113] R. Ellahi, T. Hayat, F.M. Mohamed andS. Asghar, Effects of slip on the non

linear flows of a third grade fluid, Int. J. of Non-linear Anal., 11(2010) 139-146.

[114] R.L. Fosdick and K.R. Rajagopal, Anomalous features in the model of second

order fluids, Arch. Rational Mech. 70 (1979) 145-152.

[115] R.S. Rivlin and J.L. Ericksen, Stress deformation relations for isotropic materials,

171

J. Rat. Mech. Anul., 4 (1955) 323-425.

[116] S.P. Mishra, Heat transfer by laminar flow of an elastico-viscous liquid in a

circular cylinder with linearly varying wall temperature, Appl. Sci .Res.,14

(1965) 182-190.

[117] C. Treusdell, The Simplest Rate Theory of Pure Elasticity, Commun. Pure Appl.

Math., 8 (1955) 123-132.

[118] S.P. Mishra and J.S. Roy, Heat transfer by laminar motion of an elastico-viscous

liquid between two coaxial circular cylinders due to longitudinal motion of the

inner cylinder, Appl. Sci .Res., 35 (1968) 6-16.

List of Publications/Submissions

1. R. A. Shah, S. Islam, A. M. Siddiqui and T. Haroon, Optimal Homotopy

Asymptotic Method Solution of Unsteady Second Grade Fluid in Wire Coating

Analysis, Journal of Korean society of industrial and applied mathematics, 15(3)

(2011) 201-222.

2. R. A. Shah, S. Islam, A. M. Siddiqui and T. Haroon, Wire coating analysis with

Oldroyd 8- constant fluid by optimal homotopy asymptotic method, Journal of

Computer and mathematics with applications, doi:10.1016/j.camwa.2011.11.033.

3. R. A. Shah, S. Islam, A. M. Siddiqui and T. Haroon, Heat transfer by laminar flow

of an elastic‐ viscous fluid in post‐treatment analysis of wire coating with linearly

varying temperature along the coated wire, Journal of Heat and Mass Transfer,

DOI 10.1007/s00231-011-0934-1.

4. R. A. Shah, S. Islam, Manzoor Ellahi, A. M. Siddiqui and T. Haroon, Analytical

Solutions for Heat Transfer Flows of a Third Grade Fluid in Post-treatment of

Wire Coating, International journal of physical sciences, 6(17) (2011) 4213-4223.

5. R. A. Shah, S. Islam, A. M. Siddiqui and T. Haroon, Heat Transfer by Laminar

Flow of a Third Grade Fluid in Wire Coating Analysis with Temperature

172

Dependent and Independent Viscosity, Journal of analysis and mathematical

physics, DOI 10.1007/s13324-011-0011-4.

6. S. Islam, R. A. Shah, A. M. Siddiqui and T. Haroon, Exact solution of a

differential equation arising in wire coating analysis of unsteady second grade

fluid (Accepted in “Mathematical and Computer Modelling”).

7. R. A. Shah, S. Islam, A. M. Siddiqui, T. Haroon, Exact Solutions of a Power Law

Fluid Model in Postt-reatment Analysis of Wire Coating with Linearly Varying

Boundary Temperature, (Submitted “Quarantine Mathematics”).

8. R. A. Shah, S. Islam, A. M. Siddiqui, T. Haroon, Exact solution of non-isothermal

PTT fluid in wire coating analysis, (Submitted “Communication in nonlinear

Science and Numerical Simulation”).

9. R. A. Shah, S. Islam, A. M. Siddiqui, T. Haroon, Wire coating with heat transfer

analysis flow of a viscoelastic PTT fluid with slip conditions, (Submitted

“Physics Letter A”).

Appendix A

4111

412

2

13 4ln4

1

1

24

13

3

13

3

13

4

1312144

1

2

1

2

1C

1

24

131

22

13

3

121

2

13

3

121

24

121

4

1312

1

2

13

3

12113

4

121

5

12113121

2

12

1

3

13113

2

12113

2

121

3

1311121215

4

162

9

4

2489

162

2

1

2

126

2

3

4

1

4

1

CCCCC

CCCCC

CCCCCC

173

1

23

13

2

121

2

13

3

12

11312113

2

12113

2

1211121216

624

2264

1

4

1

CC

CCCCC

1

2

13

3

1213

4

12

3

12

3

1217 262

12 C

1

23

13

5

12189

4

9

16C

2 2 3 3 4

19 12 12 13 1 13 1 12 13 12 2 2

3 2 3 2 3 2

13 1 12 1 1 12 1 12 13 1 13 1 12 132

3 2 3 2 4 4

13 1 12 1 13 13 12 13 1 12 13 1 12

1 1 1 1 1 1

ln 4 4 2 2

1 1 12 6 2

4 4 2

1 1 42 24

2 2 9

C C C

C C C C C C

C C C C

2

1

3 2 2 3 2 2 2 3 2 3 2

12 13 1 12 13 1 12 1 1 12 13 1 12 13 1

2 3 2 2 2 2 2 3 4 2 4

12 13 1 12 13 1 12 13 1 12 1 12 1

4 4 2 2

12 13 13 1 13 1

1 12 6 6 2

ln 2

12 24 6 2

2

8 2

C

C C C C C C

C C C C C

C C

114

4

13109

1C 1

2

14

3

13114

2

13114

2

13112

1

2

3CCC

1

2

19

3

131

2

14

2

1312

119

3

131216

2

13114

2

13

2

1211413

119

2

131141312119

2

1311413121141412

12

4242

2

312

2

312

CC

CCCC

CCCCC

1

2

1714

3

13

1

2

18

3

121

2

9

3

13121

2

1814

3

13

11614

3

13119

2

12116

2

1312

119

3

13117

2

12119

3

121161413

116

4

121181312117

2

131181312

1161312118

2

12116

2

12114

2

13

11413121171171161141817161413

2

9

866

2408

9

1

2

32

26612

12662

3

12

C

CCC

CCC

CCCC

CCCC

CCCC

CCCCC

174

1

2

1913

2

121

2

17

3

131

2

16

2

13121

2

1914

3

12

11913

3

12118

4

13117

3

1312116

2

13

2

12

114

4

121191211613119

2

12117

2

13

1161312119

2

121161312117

2

131161614

1241216

7294872

482222

46126

CCCC

CCCC

CCCCC

CCCCC

1

2

19

3

12

1

2

18

3

131

2

17

2

13121

2

1613

2

12119

4

12

118

3

1312117

2

13

3

1211613

3

1211713

11612118

2

131181312116

2

121171715

2

2

3668

244832

2

9126

C

CCCC

CCCC

CCCCC

1

2

18

2

13121

2

1713

2

121

2

16

3

12118

2

13

3

12

11713

3

12116

4

121181311712

1181312117

2

121181312117

2

121181816

49

48

9

1640

9

320

9

80

3

2

9

8

3

4

9

8128

CCCC

CCCC

CCCCC

2 2 4 3

17 12 18 1 12 18 1 12 18 1 12 17 1 12 13 18 1

3 2 2 2

12 17 1 12 13 18 1

3 39 12 36

2 4

92

2

C C C C C

C C

118

3

121218 4833625

1C

141312112

2

1813

3

12

2

2

16

3

122

2

9

3

13121

2

1916

3

1321714

3

12

216

3

13219

2

131211914

3

12114

2

12

218

2

121191412214

4

132191412216

2

13

21613121191412119

2

12219

4

13218

2

13

21613122181312119

2

12114

2

13113

4

12

11614131171171161141817161419

6

2

3312

1049

12

2

12012

9

86268

2292

3

8

C

CCCC

CCCC

CCCCC

CCCCC

CCCCC

CCCCC

where

175

)9

320

9

8032

129

11836

242

343

6402(ln

1

2

2

17

2

122

2

18

2

13122

2

1913

3

12

21814

2

12217

3

12219

2

1312218

2

13

3

12

214

2

132141221614216

4

12

218

2

121191412217

2

13119141211

CCC

CCCC

CCCC

CCCC

)24212

9

808

2

12

229

32(

ln

2

2

18

4

132

2

1614

3

1221916

2

13

219

2

1312216

2

13

2

122191321714

217

3

12219

2

1321614122161312

2

12

CCC

CCCC

CCCC

)9

143

1298

9

40162(

ln

2

2

18

3

13122

2

1714

3

13216

2

12

219

2

1312217

2

13

3

122191321812

219

2

132161413214

3

132181412

4

13

CCC

CCCC

CCCC

63 2

14 12 14 17 2 13 16 19 2 13 14 18 2 12 19 2

4 3 2 3 213 16 18 2 13 16 2 12 13 17 2 12 18 2

16( 8 24

ln 9

14 96 6 )

2

C C C C

C C C C

Appendix B

wUCa

11

ln1ln 2

2

11

,

wUaCCC 2

2

11126

1 ,

wUaCCC 2

2

11136

1 ,

wUaCCCCCCC

211

2

22

22

1

2

1

2

1114

2

12

1,

wUaCCC 2

2

11156

1 ,

176

wUaCCCCCCC 2

2111

2

1

22

1111

2

1

22

116 324

58

48

1

,

2 2 2 4 2 2 2 2 2

17 1 1 1 11 1 11 1 1

2 2 4 2 4 2 2 2

1 1 1 11 1 2 11 1 2

1 1 1 1( 2 )

48ln 12 12 414

1 13 3 3 9,

48 414 96 48 48

(

) w

C C C C C C

C C C C C C C a U

wUaCCCCCCC 2

2111

2

1

22

1

2

111

22

1

2

118 422848

1 ,

wUaCCCCCCC 2

2111

2

1

22

1

2

111

22

1

2

119 422848

1 ,

2 2 2

11 1

1

64wa C U , 2 2 2

12 1

1

96wa C U .

Appendix C

1ln

110 U

10

3

1011 C

10

3

1012 C

0

3

10

2

1213 1ln

1

C

1011

2

1014 3 C

20

3

101013

2

1011115 31 CCC

20

3

101011

2

10111113

2

1016 313 CCCC

22

101111

2

102

10

20

3

101013

2

10111

2

217 33

311ln

1

CCCC

0171310

4

1418 56.1 Br

01511

3

1419 4 Br

0171310

3

140

2

1511

2

1410 261.0918.3 BrBr

01713101511

2

140

3

15111411 33.577.1 BrBr

177

0

2

171310

2

14

0171310

2

1511140

4

151112

92.1

368.032.0

Br

BrBr

2 3

13 11 15 11 15 10 13 17 0

2

14 11 15 10 13 17 0

0.25

3

Br Br

Br

0

3

17131014

0

2

171310

2

151115111414

88.0

33.1454.0

Br

BrBr

0

3

1713101511

017131014

2

151115 5.025.0

Br

BrBr

0

4

17131014171310151116 5.0 BrBr

0

4

1713100

3

1713101511

0

2

171310140

2

171310

2

1511

0

2

1713101511140

2

171310

2

14

0171310

3

15110171310

2

151114

01713101511

2

140171310

3

14

0

4

1511

3

1511140

2

1511

2

14

01511

3

140

4

141713101511

17131014

2

1511151114

2

1417

5.0

88.033.1

392.1

84.3

33.561.2

32.077.1918.3

458.1

5.025.044.025.0

BrBr

BrBr

BrBr

BrBr

BrBr

BrBr

BrBrBr

BrBrBrBr

178

19

3

14016

2

20

2

1401420

3

14014

3

1914012

2019

2

14012

2

1914010

2

20

2

14010

20

2

19010

2

14820

2

1908

2

20190819146

2

20

2

1906

3

201406

2

19420144

3

201904

4

20022019218

114

1191.3

1161.2

1177.1

1133.5

1132.0

1192.1

1184.3

1125.0

11

113

1144.0

1133.1

1188.0

1125.0

115.0

11

115.0

111

ln

1

Br

BrBrBr

BrBrBr

BrBrBr

BrBrBr

BrBrBr

BrBrBr

where 151119 and 17131020 .

modelling of non-newtonian fluid problems and their solutions

Documents