MODELLING OF NON-NEWTONIAN FLUID PROBLEMS AND THEIR SOLUTIONS
Transcript of 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
MODELLING OF NON-NEWTONIAN FLUID
PROBLEMS AND THEIR SOLUTIONS
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
CIIT/SP09-PMT-011/ISB
Fall, 2011
iii
MODELLING OF NON-NEWTONIAN FLUID
PROBLEMS AND THEIR SOLUTIONS
___________________________________________
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
MODELLING OF NON-NEWTONIAN FLUID
PROBLEMS AND THEIR SOLUTIONS
By
Rehan Ali Shah
CIIT/SP09-PMT-011/ISB
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: ______________________________________
Prof. Dr. Tahira Haroon
Department of Mathematics, CIIT, Islamabad
HoD: ______________________________________________
Dr. Moiz-ud-Din Khan
Department of Mathematics, CIIT, Islamabad
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
CIIT/SP09-PMT-011/ISB
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:
_____________________
Prof. Dr. Tahira Haroon
Department of Mathematics,
CIIT, Islamabad
Head of Department:
_____________________________
Dr. Moiz-ud-Din Khan
Associate Professor,
Department of Mathematics, CIIT, Islamabad
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Dedicated
To
My parents, wife, children
And
Prof. Dr. A.M. Siddiqui
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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.
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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
CIIT/SP09-PMT-011/ISB
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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.
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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.
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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
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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
3.2 Results and discussion 46
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
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4.2.2 Temperature dependent viscosity 61
4.2.2a Reynolds model 61
4.2.2b Vogel’s model 65
4.3 Results and discussion 69
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
5.2 Solution of the problem 82
5.3 Results and discussion 84
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
6.3 Results and discussion 102
6.4 Conclusion 111
7. Heat Transfer Analysis of a Third Grade Fluid in Post-treatment
Analysis of Wire Coating 112
7.1 Formulation of the problem 113
7.2 Solution of the problem 117
7.2.1 Perturbation solution 117
7.2.2 Solution by optimal homotopy asymptotic method 120
7.3 Results and discussion 122
7.4 Conclusion 125
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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
8.2 Results and discussion 134
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
9.1 Formulation of the problem 142
9.2 Solution of the problem 146
9.2.1 Perturbation solution 147
9.2.2 Solution by modified homotopy perturbation method 150
9.3 Results and discussion 153
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
values of 2.5, 2.X 36
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
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Figure 3.2. Dimensionless velocity profiles for different values of 2
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
Figure 3.5. Dimensionless shear stress profiles for different values of 2
cDe at fixed
values of 0.5, 0.2, 2.X 48
Figure 3.6. Dimensionless normal stress profiles for different values of 2
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
values of 0.2, 10.X 51
Figure 3.12. Thickness of coated wire versus for different values of 2
cDe at fixed
values of 0.2, 10.X 51
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
Figure 3.15. Dimensionless temperature distributions for different values of Brinkman
xix
number at fixed values of 20.5, 2, 10, 10.cX De 52
Figure 3.16. Dimensionless temperature distributions for different values of 2
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
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 . 74
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 . 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
Figure 4.11. Dimensionless temperature distribution in case of Vogel’s model when
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
Figure 8.7. Radius of coated wire cR is plotted against n for different values of by
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
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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
Figure 8.13. The non-dimensional function G for different values of n when
2,10,6.0,5.0 0 SUJ . 139
Figure 8.14. The non-dimensional function G for different values of non-dimensional
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
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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
Table 4.3. Shows velocity distribution at various order of approximations when
02, 0.5, 0.5Br . 71
Table 4.4. Shows temperature distribution at various order of approximations when
02, 20, 0.01Br . 71
Table 4.5. Shows temperature distribution at various order of approximations when
02, 20, 0.3Br . 71
Table 4.6. Shows temperature distribution at various order of approximations when
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
Table 5.2. Shows variation of volume flow rate and average velocity for different values
of when ,5.0,2.0 0.5 . 88
Table 5.3. Shows variation of volume flow rate and average velocity for different values
of when 3,4.0 , 5.0 . 89
Table 5.4. Shows variation of volume flow rate and average velocity for different values
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
Table 5.6. Shows variation of the auxiliary constants 1C and 2C for different values
of when 1.5, 2 , 0.8 . 90
Table 5.7. Shows variation of the auxiliary constants 1C and 2C for different values
of pressure gradient when 0.8, 0.1, 2 . 90
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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
Table 6.8. Error between OHAM and exact solution up to third order of approximation at
different values of time level (example 6.1). 108
Table 6.9. Error between OHAM and exact solution up to third order of approximation at
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
Table 6.11. Error between OHAM and exact solution up to third order of approximation
at 2t (example 6.3). 110
xxv
Table 6.12. Error between OHAM and exact solution up to third order of approximation
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.
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.
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.
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]
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.
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
Figure 2.3. Dimensionless velocity profiles for different values of 2
cDe at fixed values of
0.5, 2.X
Figure 2.4. Dimensionless shear stress profiles for different values of 2
cDe at fixed
values of 1.5, 2.X
36
Figure 2.5. Dimensionless normal stress profiles for different values of 2
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
cDe at fixed values of 0.65.X
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
Figure 2.9. Dimensionless temperature distributions for different values of Brinkman
number at fixed values of 20.5, 10, 2.cX De
Figure 2.10. Dimensionless temperature distributions for different values of 2
cDe at
fixed values of 0.2, 4, 2.X Br
38
Figure 2.11. Dimensionless temperature distributions for different values of X at fixed
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
3.2 Results and discussion
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
Figure 3.2. Dimensionless velocity profiles for different values of 2
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
Figure 3.5. Dimensionless shear stress profiles for different values of 2
cDe at fixed
values of 0.5, 0.2, 2.X
49
Figure 3.6. Dimensionless normal stress profiles for different values of 2
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
cDe at fixed values of 2.5.X
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
Figure 3.15. Dimensionless temperature distributions for different values of Brinkman
number at fixed values of 20.5, 2, 10, 10.cX De
Figure 3.16. Dimensionless temperature distributions for different values of 2
cDe at
fixed values of 1.2, 10, 2, 2.X Br
53
Figure 3.17. Dimensionless temperature distributions for different values of X at fixed
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 dw dw dwd w dw dw dw d wr r r
dr dr dr dr dr dr dr dr dr
dw d dwdw dw d d wm m mr mr mr
dr dr dr dr dr dr dr
(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 dwd d dw dw dwr rBr rBr rBr
dr dr dr dr dr dr dr
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 dw dw dwd w dw dw dw d wr r r
dr dr dr dr dr dr dr dr dr
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 dwd d dw dw dwr rBr rBr rBr
dr dr dr dr dr dr dr
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
4.3 Results and discussion
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.
Table 4.1. Shows velocity distribution at various order of approximations when
02, 20, 0.01Br .
Table 4.2. Shows velocity distribution at various order of approximations when
02, 20, 0.3Br .
71
Table 4.3. Shows velocity distribution at various order of approximations when
02, 0.5, 0.5Br .
Table 4.4. Shows temperature distribution at various order of approximations when
02, 20, 0.01Br .
Table 4.5. Shows temperature distribution at various order of approximations when
02, 20, 0.3Br .
72
Table 4.6. Shows temperature distribution at various order of approximations when
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
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 .
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 .
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 .
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 .
Figure 4.11. Dimensionless temperature distribution in case of Vogel’s model when
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
5.2 Solution of the problem
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
dr dr dr dr dr dr dr
dw d w dw d w dw d w dwC r rC rC C
dr dr dr dr dr dr dr
dw dwrC rC
dr dr
2
11 0,r C
(5.24)
subject to the boundary conditions
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
dr dr dr dr dr dr dr dr
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
dr dr dr dr dr dr dr dr
dw d w dwdw d wC rC
dr dr dr dr dr
dw dwd w d wrC rC
dr dr dr dr
(5.26)
subject to the boundary conditions
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)
5.3 Results and discussion
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
Table 5.1. Shows variation of volume flow rate and average velocity for different values
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
Table 5.2. Shows variation of volume flow rate and average velocity for different values
of when ,5.0,2.0 0.5 .
Volume flow rate Average velocity
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
Table 5.3. Shows variation of volume flow rate and average velocity for different values
of when 3,4.0 , 0.5 .
Table 5.4. Shows variation of volume flow rate and average velocity for different values
of pressure gradient when 0.5, 0.2 and 3 .
Volume flow rate Average velocity
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
Volume flow rate Average velocity
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
Table 5.5. Shows variation of the auxiliary constants 1C and 2C for different values
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
Table 5.6. Shows variation of the auxiliary constants 1C and 2C for different values
of when 1.5, 2 , 0.8 .
Table 5.7. Shows variation of the auxiliary constants 1C and 2C for different values
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
Zeroth-order problem
20 0 0
2
1: 0
w wp
r r r
, (6.37)
subject to the boundary conditions
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)
subject to the boundary conditions
0,,0,1 11 twtw , (6.40)
Second-order problem
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)
subject to the boundary conditions
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)
with the boundary and initial conditions
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)
with the boundary and initial conditions
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)
with the boundary and initial conditions
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.
6.3 Results and discussion
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 .
Velocity distribution
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 .
Velocity distribution
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
Velocity distribution
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 .
Velocity distribution
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
Velocity distribution
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
Table 6.7. Error between OHAM and exact solution up to third order of approximation at
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
Table 6.8. Error between OHAM and exact solution up to third order of approximation at
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
Table 6.9. Error between OHAM and exact solution up to third order of approximation at
0.5t (example 6.2).
Absolute error
r Zeroth order First order Second order Third order
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
Table 6.10. Error between OHAM and exact solution up to third order of approximation
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
Table 6.11. Error between OHAM and exact solution up to third order of approximation
at 2t (example 6.2).
Absolute error
r Zeroth order First order Second order Third order
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
Table 6.12. Error between OHAM and exact solution up to third order of approximation
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.
7.1 Formulation of the problem
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)
Eq. (7.11) represents the flow due to pressure gradient. After leaving the die, there is only
drag flow. Hence, we consider
116
02
3
32
dr
dwr
dr
d
rdr
dwr
dr
d , (7.12)
and the energy Eq. (1.11) becomes
.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 Solution of the problem
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
dr dr dr dr dr dr dr dr dr
, (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
Zeroth-order problem with boundary conditions
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)
First-order problem with boundary conditions
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)
Second-order problem with boundary conditions
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
Zeroth-order problem with boundary conditions
0: 0
2
0
2
0 dr
dw
dr
wdrp , (7.52)
110 w , .0 Uw (7.53)
First-order problem with boundary conditions
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
dr dr dr dr dr dr dr
dw d wr C
dr dr
(7.54)
011 w , .01 w (7.55)
Second-order problem with boundary conditions
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
dr dr dr dr dr dr dr
dw dw d w dw d wdw dwC r C r C
dr dr dr dr dr dr dr
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.
7.3 Results and discussion
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.
Eq. (8.10) represents the flow due to pressure gradient. After leaving the die, there is only
drag flow. Hence, we consider
00
n
dr
dwr
dr
d , (8.11)
and the energy Eq. (1.11) becomes
.
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
S U r U n rG r r S r
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
S U r U n rG r r S r
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
8.2 Results and discussion
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.
Figure 8.6. Radius of coated wire cR is plotted against n for different values of by
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.
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 .
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
Figure 8.11. The non-dimensional function G for different values of n when
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
Figure 8.13. The non-dimensional function G for different values of n when
2,10,6.0,5.0 0 SUJ .
Figure 8.14. The non-dimensional function G for different values of non-dimensional
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.
9.1 Formulation of the problem
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 .
9.2 Solution of the problem
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
Zeroth-order problem
0: 00
0 uLwLp , (9.61)
subject to the boundary conditions
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)
subject to the boundary conditions
011 w , .01 w (9.64)
Second-order problem
2
2 0 1 22: 3 e
dw dw Kp L w R
dr dr r
, (9.65)
subject to the boundary conditions
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)
9.3 Results and discussion
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 .
Velocity distribution
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
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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 .