ANALYTICAL APPROACH TO UNIDIRECTIONAL FLOW OF NON ...

ANALYTICAL APPROACH TO UNIDIRECTIONAL FLOW OF

NON-NEWTONIAN FLUIDS OF DIFFERENTIAL TYPE

MOHAMMED ABDULHAMEED

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Doctor of Philosophy of Science

Faculty of Science Technology and Human Development

Universiti Tun Hussein Onn Malaysia

JULY 2015

v

ABSTRACT

This thesis is regarding the development of mathematical models and analytical

techniques for non-Newtonian fluids of differential types on a vertical plate, horizontal

channel, vertical channel, capillary tube and horizontal cylinder. For a vertical

plate, a mathematical model of the unsteady flow of second-grade fluid generated

by an oscillating wall with transpiration, and the problem of magnetohydrodynamic

(MHD) flow of third-grade fluid in a porous medium, have been developed. General

solutions for the second-grade fluid are derived using Laplace transform, perturbation

and variable separation techniques, while for the third-grade fluid are derived using

symmetry reduction and new modified homotopy perturbation method (HPM). For a

horizontal channel, a new analytical algorithm to solve transient flow of third-grade

fluid generated by an oscillating upper wall has been proposed. A new approach of the

optimal homotopy asymptotic method (OHAM) have been proposed to solve steady

mixed convection flows of fourth-grade fluid in a vertical channel. The accuracy of

the approximate solution is achieved through the residual function. For a capillary

tube, two flow problems of the second-grade fluid were developed. Firstly, oscillating

flow and heat transfer driven by a sinusoidal pressure waveform, and secondly, free

convection flow driven due to the reactive nature of the viscoelastic fluid. The solutions

for the first problem were derived using Bessel transform technique while for the

second problem by using a new modified homotopy perturbation transform method.

For a horizontal cylinder, an unsteady third-grade fluid in a wire coating process

inside a cylindrical die is developed. A special case of the problem is obtained

for magnetohydrodynamic flow with heat transfer for second-grade fluid. Both of

these two problems are solved using a new modified homotopy perturbation transform

method. Data, graph and solutions obtained are shown and were found in good

agreement with previous studies.

vi

ABSTRAK

Tesis ini adalah mengenai pembangunan model matematik dan teknik analisis untuk

cecair bukan Newtonian daripada pengkamiran jenis di atas pinggan menegak,

mendatar saluran, saluran menegak, tiub kapilari dan silinder mendatar. Untuk plat

menegak, satu model matematik bagi aliran tak mantap bendalir kedua-gred dihasilkan

oleh dinding berayun dengan transpirasi, dan masalah magnetohydrodynamic (MHD)

Aliran bendalir ketiga gred di medium berliang, telah dibangunkan. Penyelesaian

am bagi cecair kedua-gred yang diperolehi dengan menggunakan jelmaan Laplace,

usikan dan pemisahan pembolehubah teknik, manakala bagi cecair ketiga-gred yang

diperolehi dengan menggunakan pengurangan simetri dan baru diubah suai kaedah

usikan homotopi (HPM). Untuk mendatar saluran, algoritma analisis baru untuk

menyelesaikan aliran fana cecair gred ketiga yang dihasilkan oleh dinding atas

berayun telah dicadangkan. Pendekatan baru kaedah asimptot homotopi optimum

(OHAM) telah dicadangkan untuk menyelesaikan mantap aliran olakan campuran

cecair keempat gred dalam saluran menegak. Ketepatan penyelesaian hampir dicapai

melalui fungsi baki. Untuk tiub kapilari, dua masalah aliran bendalir kedua-gred telah

dibangunkan. Pertama, didorong aliran dan haba berayun pemindahan oleh tekanan

gelombang sinus, dan kedua, perolakan percuma mengalir didorong kerana sifat

reaktif cecair likat kenyal itu. Penyelesaian untuk masalah pertama telah diperolehi

dengan menggunakan Bessel mengubah teknik manakala bagi masalah kedua dengan

menggunakan baru diubahsuai pengusikan homotopi mengubah kaedah. Untuk

mendatar silinder, bendalir ketiga gred tak mantap dalam proses salutan wayar di

dalam sebuah silinder die dibangunkan. Satu kes khas masalah diperolehi untuk aliran

magnetohydrodynamic dengan haba memindahkan cecair untuk kedua-gred. Kedua-

dua masalah adalah diselesaikan menggunakan homotopi baru diubahsuai pengusikan

mengubah kaedah. Data, graf dan penyelesaian yang diperolehi ditunjukkan dan tidak

terdapat dalam perjanjian yang baik dengan kajian sebelum ini.

vii

TABLE OF CONTENTS

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES xiii

LIST OF FIGURES xv

LIST OF ABBREVIATIONS xxi

NOMENCLATURE xxii

CHAPTER 1 INTRODUCTION 1

1.1 Research background 1

1.2 Problem statement 3

1.3 Objectives of the study 4

1.4 Scope of the study 5

1.5 Significance of findings 5

1.6 Research methodology 6

1.6.1 Problem formulation 6

1.6.2 Analytical computation 7

1.6.3 Result verifications 7

1.7 Thesis organization 7

CHAPTER 2 LITERATURE RIVIEW 1

2.1 Introduction 11

viii

2.2 Unsteady flow due to an oscillating and impulsive

for an infinite plate 11

2.3 Unsteady MHD flow through a porous for an

infinite plate 13

2.4 Couette and Poiseuille flow in a horizontal channel 15

2.5 Free and mixed convention flow near an infinite

wall and channel 16

2.6 Unsteady flow in a capillary tube 17

2.6.1 Convective heat transfer with oscillating

pressure waveform 17

2.6.2 Free-convection due to reactive fluid nature 20

2.7 Unsteady flow past an inner cylinder 22

CHAPTER 3 ANALYTICAL APPROACHES AND

CONSTITUTIVE MODELS 24

3.1 Introduction 24

3.2 Analytical methods 24

3.2.1 Integral transforms 25

3.2 .2 Laplace transformation 25

3.2 .3 Bessel transform (finite Hankel transform) 26

3.3 Approximate Analytical Methods 27

3.3 .1 Perturbation method 27

3.3 .2 Homotopy perturbation method (HPM) 28

3.3 .3 Modified Adomian polynomials:

He polynomials 29

3.4 Optimal homotopy asymptotic method (OHAM) 33

3.5 Homotopy perturbation transform method (HPTM) 35

3.6 New modified analytical techniques 36

3.6 .1 New modified optimal homotopy

asymptotic method 36

3.6 .2 New modified HPM for steady-state

problems 40

3.6.3 New modified HPM for unsteady

flow problems 41

ix

3.6.4 New modified HPM for unsteady couple

equations 44

3.7 Differential type fluids 46

3.7 .1 Second-Grade Fluid 47

3.7 .2 Third-grade fluid 48

3.7 .3 Fourth-grade fluid 49

3.8 The basic flow equations 50

3.8 .1 Continuity equation 50

3.8 .2 Momentum equation 51

3.8 .3 Energy equation 52

3.9 Dimensionless numbers 52

3.9 .1 Grashof number 52

3.9 .2 Reynolds number 53

3.9 .3 Prandtl number 53

3.9 .4 Eckert number 53

3.9 .5 Brinkman number 54

3.9 .6 Skin friction coefficient 54

3.9 .7 Nusselt number 54

CHAPTER 4 THE UNSTEADY FLOW OF A SECOND-GRADE

FLUID GENERATED BY AN OSCILLATING

WALL WITH TRANSPIRATION 55

4.1 Introduction 55

4.2 Formulation of the problem 56

4.3 Solution of the problem 60

4.3 .1 Start-up phase solution 60

4.3 .2 Periodic solution 63

4.3 .3 Impulsive start 65

4.4 Results and discussion 67

4.5 Conclusions 80

x

CHAPTER 5 THE UNSTEADY MHD FLOW OF A THIRD- GRADE

FLUID GENERATED BY AN OSCILLATING WALL

WITH TRANSPIRATION IN A POROUS SPACE 82

5.1 Introduction 82



5.3 .1 Start-up phase solution 87

5.3 .2 Impulsive start 90

5.3 .3 Steady-state solution 90


5.5 Conclusions 105

CHAPTER 6 ANALYTICAL SOLUTION OF UNSTEADY FLOW

OF A THIRD-GRADE FLUID IN A HORIZONTAL

CHANNEL WITH OSCILLATING MOTION ON

THE UPPER TRANSPIRATION WALL 107

6.1 Introduction 107



6.3 .1 Solution by using a new modified

HPM method 109

6.3 .2 Solution by using HPM 112

6.3.3 Solution by using OHAM 115


6.5 Conclusions 123

CHAPTER 7 THE STEADY HEAT TRANSFER FLOW OF A FOURTH-

GRADE FLUID IN A DARCY FORCHHEIMER

POROUS MEDIA 125




7.3 .1 Solutions for fourth-grade fluid 0 129

xi

7.3 .2 Solutions for a third-grade fluid 0 136

7.4 Nusselt number and skin friction 137


7.6 Conclusion 145

CHAPTER 8 THE UNSTEADY FLOW OF A SECOND-GRADE

FLUID IN A CAPILLARY TUBE 146


8.2 Problem formulation due to sinusoidal pressure effect 146


8.3 .1 Velocity profile 149

8.3 .2 Temperature distributions 152

8.4 Problem formulation due free-convection flow 159

8.4 .1 Solution of the problem 160

8.4 .2 Nusselt number and wall shear stress 165


8.6 Conclusion 172

CHAPTER 9 THE UNSTEADY FLOW OF A THIRD-GRADE

FLUID ARISING IN THE WIRE COATING PROCESS

INSIDE A CYLINDRICAL ROLL DIE 175


9.2 Formulation of the problem when 0 175


9.3 .1 Solution for a third-grade fluid 0 178

9.3 .2 Solution for a second-grade fluid 0 181

9.3 .3 Formulation of the problem for MHD flow

second-grade fluid with heat transfer 183


9.4 .1 Solutions for steady-state velocity and

temperature fields 186

9.4 .2 Solutions for transient velocity and

temperature fields 188

xii


9.6 Conclusion 206

CHAPTER 10 CONCLUSIONS AND FUTURE RESEARCH 209


10.2 Summary of research 209

10.3 Suggestions for future research 211

REFERENCES 213

APPENDIX A 227

APPENDIX B 232

xiii

LIST OF TABLES

6.1 Numerical results show the comparison of new

algorithm results with the results obtained from HPM

and OHAM for the case of small value of time t

(t = 0.1) when α = 0.5, β = 0.5, ξ = −0.5,

ω = 0.5, C1 = 0.15419514, C2 = −5.54965331 and

C2 = 4.10010429 116

6.2 Numerical results show the comparison of new

algorithm results with the results obtained from HPM

and OHAM for the case of large value of time t

(t = 2) when α = 0.5, β = 0.5, ξ = −0.5 ω =

0.5, C1 = 0.15419514, C2 = −5.54965331 and

C2 = 4.10010429 117

7.1 The results of OHAM and HAM methods for fluid

velocity when γ = 0, ξ = 0, β = 0.5, Pr = Gr =

Re = Br = 1, P = Q = 0, C1 = −0.2843403486

and C2 = −0.8023839208 139

7.2 The results of OHAM and HAM methods for fluid

temperature when γ = 0, ξ = 0, β = 0.5, Pr =

Gr = Re = Br = 1, P = Q = 0, C1 =

−0.40714571 and C2 = −0.35572542 140

8.1 First few eigenvalues of β2cn 157

8.2 Wall friction (−τ) when Pr = 10 and λ = 0.3 166

8.3 Wall friction (−τ) when Pr = 10 and α = 0.4 166

8.4 Nusselt number (Nu) when α = 0.4 and t = 0.2 167

xiv

8.5 Nusselt number (Nu) when α = 0.4 and Pr = 10 167

9.1 Numerical results show the comparison of present

results with related published work for small value

of time t (t = 0.2) and β = 0, ζ = 3, α = 0.5 195

9.2 Numerical results show the comparison of present

results with related published work for large value

of time t (t = 2) and β = 0, ζ = 3, α = 0.5 196

9.3 Variation of frictional force on the total wire with

change in parameters α1, α2 and ζ for M = 0.5, t = 2 201

9.4 Variation of wall shear stress at the surface of the die

with change in parameters α1, M and ζ for t = 2 201

9.5 Variation of wall shear stress at the surface of the

wire with change in parameters M, t and ζ for α1 = 0.5 202

xv

LIST OF FIGURES

1.1 The outline of the thesis 9

4.1 The physical model of transpiration wall 56

4.2 Definition sketch for impulsive and oscillating problem 57

4.3 Velocity profiles versus y for various values of

transpiration parameter ξ in the case of small value

of time t = 2, when α = 0.4, ω = 0.5, are fixed (a)

Cosine oscillation and (b) Sine oscillation 69


transpiration parameter ξ in the case of large value

of time t = 4, when α = 0.4, ω = 0.5, are fixed (a)


4.5 Velocity profiles versus y with injection for various

values of second-grade parameter α in the case of

small value of time t = 0.2, when ω = 0.5, ξ =

−0.9, are fixed (a) Cosine oscillation and (b) Sine

oscillation 72

4.6 Velocity profiles versus y with injection with

injection for various values of second-grade

parameter α in the case of large value of time

t = 0.6, when ω = 0.5, ξ = −0.9, are fixed (a)


4.7 Velocity profiles versus y with suction for various


small value of time t = 0.2, when ω = 0.5, ξ = 0.3,

are fixed (a) Cosine oscillation and (b) Sine oscillation 74

xvi

4.8 Velocity profiles versus y with suction for various


large value of time t = 5, when ω = 0.5, ξ = 0.3,


4.9 Profiles of starting and steady solutions versus y for

various values of the transpiration parameter ξ when

t = 3, ω = 0.5, α = 0.2, are fixed (a) Cosine

oscillation and (b) Sine oscillation 76

4.10 Profiles of starting and steady solutions versus t for


y = 1.2, ω = 0.5, α = 0.1, are fixed (a) Cosine


4.11 Profiles of starting and steady solutions versus t for


y = 1.2, ω = 1, α = 0.1, are fixed (a) Cosine


4.12 Velocity profile versus y for various values of

transpiration parameter ξ with an impulsive start for

small value of time t = 1, when α = 0.4, ω = 0.5,

are fixed 79

4.13 Velocity profile versus y for various values of

transpiration parameter ξ with an impulsive start for

large value of time t = 2, when α = 0.4, ω = 0.5,

are fixed 79

5.1 The physical model of transpiration wall in a porous

space 83

5.2 Comparison of velocity profiles with Aziz and Aziz

(2012) for the transient profiles with no oscillation

(ω = 0) when β∗∗ = 2.5, β∗ = 1.5, M∗ = 1 ϕ∗ =

0.5, ξ = 0.5, t = 1, ν∗ = 0.5 and α∗ = 0.5 are fixed 94

xvii

5.3 Frequency series of the flow velocity with various

values of transpiration parameter ξ, when β∗ = 1,

β∗∗ = 1, y = 0, ϕ∗ = 0.5, M∗ = 1, t = 1, ν∗ = 0.5

and α∗ = 0.5 are fixed (a) Cosine oscillation and (b)

Sine oscillation 95


values of the magnetic field parameter M∗, when

β∗ = 1.5, β∗∗ = 1, y = 0, ϕ∗ = 0.5, ξ = 1,

t = 1, ν∗ = 0.5 and α∗ = 0.5 are fixed at (a) Cosine



values of the distances from the plate y, when β∗ =

1.5, β∗∗ = 0.5, M∗ = 1 ϕ∗ = 0.5, ξ = −1,

t = 1, ν∗ = 0.5 and α∗ = 0.5 are fixed (a) Cosine



values of the material parameters β∗, when β∗∗ = 1,

y = 0, M∗ = 1 ϕ∗ = 0.5, ξ = 0.5, ν∗ = 0.5, t = 1


Sine oscillation 98


values of the material parameters β∗∗, when β∗ = 1,

y = 0, M∗ = 1 ϕ∗ = 0.5, ξ = 0.5, ν∗ = 0.5, t = 1


Sine oscillation 100


values of the porosity of the porous medium

parameter ϕ∗, when β∗ = 1.5, β∗∗ = 1, y = 0,

M∗ = 0.5, ξ = 1, t = 1, ν∗ = 0.5 and α∗ = 0.5


xviii

5.9 Velocity profiles versus y of the starting and steady-

state flow velocity with various values of the time

t, when ω = 0.5, k0 = 1, β∗∗ = 2.5, β∗ = 1.5,

M∗ = 0.5, ϕ∗ = 1, ξ = −0.5, ν∗ = 0.5 and α∗ = 0.5

are fixed 102

5.10 Velocity profiles versus y of the starting and steady-

state flow velocity with various values of the time

t, when ω = 0.5, k0 = 1, β∗∗ = 2.5, β∗ = 1.5,

M∗ = 0.5, ϕ∗ = 1, ξ = 0, ν∗ = 0.5 and α∗ = 0.5 are

fixed 103

5.11 Velocity profiles versus y for various values of the

wall velocity parameters k0, when β∗∗ = 2.5, β∗ =

1.5, M∗ = 1 ϕ∗ = 0.5, ξ = 0.5, t = 1, ν∗ = 0.5 and

α∗ = 0.5 are fixed 104

6.1 The physical model of transpiration horizontal channel 108

6.2 Comparison of the present results with HPM and

OHAM for small value of time t = 0.1 118

6.3 Comparison of the present results with HPM and

OHAM for large value of time t = 2 119

6.4 Velocity profiles versus y for various values of ω for

small time t = 1 and α = 0.5, ξ = −0.7, and β =

0.5 with (a) Cosine oscillation and (b) Sine oscillation 120

6.5 Velocity profiles versus y for various values of β for

small value of time t = 1 and α = 0.5, and ξ = −0.7

with (a) Cosine oscillation and (b) Sine oscillation 121

6.6 Effect of blowing on wall stress τω when ω = 0.5,

α = 0.5, t = 0.2 and β = 0.5 are fixed with (a)


7.1 The physical model of transpiration vertical channel

in a porous space 126

7.2 Effects of Darcian parameter P on wall shear stress τω 138

xix

7.3 Effects of Forchheimer parameter Q on wall shear

stress τω 141

7.4 Effects of Darcian parameter P on Nusselt number Nu 141

7.5 Effects of Forchheimer parameter Q on Nusselt

number Nu 142


fourth-grade parameter γ on velocity u 142

7.7 Effects of Darcian parameter P on velocity u 143

7.8 Temperature profiles versus y with zero transpiration

(ξ = 0) for different values of the constant pressure

gradient A 143

7.9 Temperature profiles versus y with suction (ξ = 2)

for different values of the constant pressure gradient A 144

8.1 The physical model of capillary tube 147

8.2 Temperature profiles versus r for various values of

λ when α = 0.4, ϵ = 0.01, Pr = 10, t = 0.2 and

Gr = Ec = Re = 1 167


Pr when α = 0.4, ϵ = 0.01, λ = 0.2, t = 0.2 and

Gr = Ec = Re = 1 168

8.4 Velocity profiles versus r for various values of λ

when ϵ = 0.01, Pr = 10, α = 0.4 and Gr = Ec =

Re = 1. at (a) t = 0.2 and (b) t = 6 169

8.5 Velocity profiles versus r for various values of Gr

when ϵ = 0.01, Pr = 10, α = 0.4, λ = 0.1 and

Ec = Re = 1. at (a) t = 0.2 and (b) t = 6 170

8.6 Velocity profiles versus r for various values of α

when ϵ = 0.01, Pr = 10, λ = 0.1 and Ec = Gr =

Re = 1. at (a) t = 0.2 and (b) t = 6 171

9.1 The physical model configuration 176

xx

9.2 Comparison of velocity profiles with related

published paper for the case of small value of time t

when t = 0.2 and β = 0, ζ = 3, α = 0.5 are fixed 197

9.3 Comparison of velocity profiles with related

published paper for the case of large value of time t

when t = 2 and β = 0, ζ = 3, α = 0.5 are fixed 198

9.4 Velocity profiles versus r for various values of third-

grade parameter β for the case of small value of time

t when t = 0.2, ζ = 3 and α = 0.5 are fixed 198

9.5 Velocity profiles versus r for various values of third-

grade parameter β for the case of large value of time

t when t = 2, ζ = 3 and α = 0.5 are fixed 199

9.6 Velocity profiles versus r for various values of

distance ζ parameter for the case of small value of

time t when t = 0.2, β = 0.5 and α = 0.5 are fixed 199

9.7 Velocity profiles versus r for various values of

distance ζ parameter for the case of large value of

time t when t = 2, β = 0.5 and α = 0.5 are fixed 200

9.8 Velocity profiles versus r for various values of time

parameter t when β = 0.5, α = 0.5 and ζ = 3 are fixed 200


Pr when M = 0.5, Br = 1, α = 0.5 and t = 2 202

9.10 Temperature profiles versus r for various values of t

when M = 0.5, Br = 1, α = 0.5 and Pr = 10 203


Br when when M = 0.5, Pr = 10 and α = 0.5 203


M when α = 0.5, Pr = 10, Br = 1 and t = 2 204

9.13 Nusselt number versus t for various values of Pr

when r = 1, M = 0.5, α = 0.5, Br = 1 and ζ = 3 204

9.14 Nusselt number versus t for various values of αwhen

r = 1, M = 0.5, Pr = 20, Br = 1 and ζ = 3 205

xxi

LIST OF ABBREVIATIONS

HPM - Homotopy perturbation method

OHAM - Optimal homotopy asymptotic method

ADM - Adomian decomposition method

HPTM - Homotopy perturbation transform method

MHD - Magnetohydrodynamic flow

HAM - Homotopy analysis method

xxii

NOMENCLATURE

Roman Letters

a - Lower limit of range integration

an - Constant defined by equation (8.24)

az - Unit vector along z−axis

A - Constant pressure gradient

Ae - Rate constant

Ai (i = 1, 2, 3, 4) - Revlin-Ericksen tensors

A(r) - Function defined by equation (8.24)

A1(r) - Function defined by equation (8.87)

b - Upper limit of range integration

bn - Constant defined by equation (8.25)

B0 - Applied magnetic field

B1 - Amplitude of the oscillatory pressure gradient

B1(r) - Function defined by equation (8.88)

Bs - Amplitude of the steady pressure gradient

B - Boundary differential operator

B(r) - Function defined by equation (8.25)

B(y, t) - Source term arising form integration

Br - Brinkman number

c - Constant wave speed

cn - Constant defined by equation (8.26)

cp - Specific heat capacity at constant pressure

C(r, t) - Function defined by equation (8.26)

C1(r, t) - Function defined by equation (8.89)

C2(r, t) - Function defined by equation (8.90)

Ci (i = 1, 2, ...,m) - Auxiliary constants

CF - Skin friction coefficient

xxiii

C0 - Initial concentration of the reactant speciesd

dt- Material time derivative

D(r, t) - Function defined by equation (8.27)

D2(r, t) - Function defined by equation (8.92)

D - Second order differential operator

E2(r, t) - Function defined by equation (8.92)

E(r, t) - Function defined by equation (8.29)

Ec - Eckert number

EA - Activation energy

1F1 - Confluent hypergeometric function (Kummer function)

g - Acceleration due to gravity

g1, g2 - Known analytical function

g3, g4 - Source terms

G1 - Differential operator for velocity field

G2 - Differential operator for temperature field

Gr - Grashof number

h (q) , h1 (q, y) , h2 (q, y) - Auxiliary functions

h(r, t) - Function defined by equation (8.84)

h - Reference length between two point

Hn - He polynomials

i - Imaginary units

I - Identity tensor

Jn - Bessel function of fist kind and order n

J×B - Magnetic body force

k - Thermal conductivity

k0 - Constant wall velocity

K - Permeability of the porous medium

l - Length of die

L - Linear operator

L - ∇V

M - Magnetic parameter

N - Nonlinear operator

Nu - Nusselt number

xxiv

p - Pressure gradient

p∗ - Modified pressure gradient

P - Darcian parameter

Pr - Prandtl number

q - Embedding parameter

Q - Forchheimer parameter

Qr - Heat reaction parameter

r - Transverse distance to flow direction

R1 - Radius of wire

R2 - Radius of die

R - Universal gas constant

R - Darcy’s resistance due to porous medium

R - Linear differential operator of less order than D

Rm - Residual error function

Re - Reynolds number

S1(r, t) - Function defined by equation (8.85)

S2(r, t) - Function defined by equation (8.86)

S (x, s) - Kernel of the transform

t - Time

T - Cauchy stress tensor

u0 - Initial approximation for velocity

us - Steady velocity

ut - Transient velocity

u - Velocity field

U0 - Characteristic velocity

U1 - Dimensional fluid velocity of wire

U2 - Dimensional fluid velocity of die

V - Velocity vector

V0 - Amplitude of wall oscillations

Vw - Dimensional transpiration velocity

W - Solution of Kummer’s equation

xxv

W1 - First particular solution of Kummer equation

W2 - Second particular solution of Kummer equation

X1 - Time translation

X2 - Space translation

X3 - Third prolongation operator

X1 − cX2 - Wave-front type travelling solutions

X, Y - Banach spaces

x, y - Perpendicular distances

z - axial position

Greek Letters

α1, α2, α - Viscoelastic parameters of second-grade fluid

β1, β2, β3, β - Viscoelastic parameters of third-grade fluid

βc - Separation constant

βT - Volumetric coefficient of thermal expansion

γ0 - Amplitude of the pressure

γ1, γ2..., γ6, γ7, γ - Viscoelastic parameters of fourth-grade fluid

γc - Euler-Mascheroni constant

δ (t) - Dirac delta function

ϵ - Activation energy parameter

ε - Perturbation parameter

ζ - Radii ratio

θ0 - Bulk fluid temperature

θ - Temperature field

θs - Steady temperature

θt - Transient temperature

θw - Wall temperature

λ - Non-dimensional reactant consumption parameter

µ - Dynamic viscosity

ν - Kinematic viscosity

ξ - Non-dimensional transpiration velocity

xxvi

ρ - Fluid density

σ - Electrical conductivity

ς - Constant that controls the amplitude of the pressure

fluctuation

τ - Dummy variable

τw - Wall shear stress

ϕ - Porosity of the porous medium

ω - Frequency of the oscillation

ρ - Fluid density

φ - Homotopy mapping

ψ (z) - Logarithmic derivative of Γ (z)

Γ (z) - Gamma function

Γ - Boundary of domain Ω,

ℓn - Eigenvalue of the Bessel function of first kind of order

zero

Ω - Domain

ℑ - Integral transform operator

κ - Thermal conductivity of the fluid

ψ (z) - Logarithmic derivative of Γ (z)∂

∂n- Differentiation along the normal drawn outwards from

Ω.

ϱ - Grouping parameter of convenience

Φn - Sequence of function

ȷ - Indexing set

H - Convex homotopy

∇ -∂

∂xi+

∂

∂yj

Superscripts

∗ - Dimensional conditions′ - Derivative

xxvii

Subscripts

s - Steady state

t - Transient state

m - Mean value

w - Wall

∗ - Material parameter variables

CHAPTER 1

INTRODUCTION

1.1 Research background

The theoretical investigation of flow of non-Newtonian fluids continues to receive

special status in the literature due to growing importance of such fluid in modern

technology and industries. The flows of non-Newtonian fluids occur in a lot of

numbers of applications, with important examples in industrial processes which

involve synthetic fibbers, extrusion of molten plastic, flows of polymer solutions,

bioengineering and heat exchange efficiency. Simple examples of non-Newtonian

fluids are slurries, pastes, polymer solutions, multigrade engine oils, toothpaste, liquid

soaps and peanut. The main distinguishing feature of many non-Newtonian fluids

is that they exhibit both viscous and elastic properties, shear stress depends only on

the rate of shear and the relation between shear stress and shear rate depends on time

(Schowalter (1978)). These fluids exhibit numerous strange features, for example shear

thinning or thickening and display of elastic effects. Hence, the classical Navier-Stokes

equations are not appropriate in describing their rheological behaviour.

Because of the complex diversity in the physical structure of non-Newtonian

fluids, there is no single constitutive equation in the literature to describe all the flow

properties of non-Newtonian fluids. For this reason, various rheological models have

been proposed to portray their non-Newtonian flow behaviour (Schowalter (1978),

Rivlin and Ericksen (1955) and Bird et al. (1987)). These rheological models are

classify under the following type: rate type, differential type and integral type. Among

2

these types, the fluids of differential type have received special attention as well

as much controversy, see (Dunn and Rajagopal (1995)). Due to ability of fluid of

differential type to describe several non-standard features such as shear thinning,

shear thickening and normal stresses, has been the subject of many investigations

covering various facets, for example, thermodynamical aspects (Fosdick and Yu

(1996), Makinde (2007), Makinde (2009) and Coulaud (2014)), the existence and

uniqueness of solutions (Galdi et al. (1995), Passerini and Patria (2000) and Vajravelu

et al. (2002)) and some basic flow situations ((Hayat et al. (2010), Okoya (2011),

Danish et al. (2012), Baoku et al. (2013) and Zhao et al. (2014)). Also the study

of such flows with application of magnetic field through a porous medium has become

an active area of research due to its applications in several technological processes.

Among these processes are petroleum exploration/recovery, cooling of electronic

equipment, catalyst and chromatography. Few recent studies (Ali et al. (2012), Aziz

and Aziz (2012), Baoku et al. (2013), Ellahi (2013) and Hatami et al. (2014)) may

be mentioned in this direction. Similarly, the study of flows with heat transfer in free

convection and mixed convection occurs in many industrial and engineering process

and natural phenomena. Few recent studies of the topics has been discussed by

(Hayat et al. (2008), Ziabakhsh and Domairry (2009), Sajid et al. (2010) and Olajuwon

(2011)).

The governing equations of motions for such fluids are strongly non-linear and

higher than the Navier-Stokes equations for Newtonian fluids. Hence progress was

limited in the previous time, in the recent times analytical solutions become available

to more problems of particular interest. Recent studies in finding the analytical

investigation can be found in (Islam et al. (2011), Fakhar et al. (2011), Jha et al.

(2011a), Aziz et al. (2013), Shah et al. (2013), Sivaraj and Rushi K (2013) and Farooq

et al. (2014)). Although the analytical solutions are limited to particular combinations

of simple geometry and boundary conditions, they provide a great insight on more

complex flow situations.

This thesis is regarding the development of mathematical models and analytical

techniques to study non-Newtonian fluids of differential types: namely, (i) second-

grade fluid for: (a) an infinite flat plate, (b) horizontal cylinder and (c) capillary tube;

3

(ii) third-grade for: (a) an infinite flat plate, (b) horizontal channel and (c) horizontal

cylinder; (iii) fourth-grade fluid in a vertical channel.

The problem statements, objectives and scope of this study are stated in the

next three sections. The significance of findings of the study is stated in Section

1.5, followed by a section on the research methodology. The thesis organization is

described in Section 1.7.

1.2 Problem statement

The previous models had considered flow about second-grade, third-grade and fourth-

grade fluids with or without convective heat transfer, zero transpiration rates and

stagnant wall velocity. However, higher order models will be derived by considering

fluid transpiration and oscillating velocity wall velocity. These models will be

significant in many industrial processes such as in acoustic streaming around an

oscillating body and manufacturing techniques (de Almeida Cruz and Ferreira Lins

(2010)).

The models of non-Newtonian fluid of differential type, form a very

complicated system, and these kinds of models have limited number of analytical

solutions. Several methods have been proposed in order to handle this. For

example, Adomian decomposition method (ADM), homotopy analysis method

(HAM), homotopy perturbation method (HPM) and optimal homotopy asymptotic

method (OHAM) have been employed by various researchers to solve several

basic flow problems of third grade and fourth-grade fluid, and the approximate

solutions were found for the velocity profiles. However, modification of homotopy

perturbation method and optimal homotopy asymptotic techniques to handle unsteady

and couple equations are not considered yet. A newly modified homotopy perturbation

method and optimal homotopy asymptotic method for unsteady and couple non-linear

equations are needed in order to improve the existing methods and give an insight to

describe complex problems.

4

The recent theoretical studies of free, forced and mixed convection flows of

non-Newtonian fluids in channels and cylinder contains zero transpiration rate, absence

of pressure gradient, non-porous space and steady problem. These can be extended

in four directions which are: (i) by considering higher order viscoelastic fluid, (ii) to

consider MHD effects (iii) to take into account the porous medium, (iv) by considering

suction/injection effects (v) by considering unsteady problem.

The recent theoretical studies of oscillating laminar flow round capillary tube

driven by oscillating pressure waveform are considering the fluid are Newtonian. For

viscoelastic fluids, no analysis have showed the viscoelastic effects in convective heat

transfer with sinusoidal pressure gradient in a capillary tube to apply to bioengineering

field..

1.3 Objectives of the Study

To investigate theoretically the problems of unidirectional flow of non-Newtonain

fluids of differential type by solving the mathematical models for the following

problems:

(i) Unsteady flow of second-grade fluid for an infinite vertical plate with

transpiration generated by impulsive and oscillating boundary;

(ii) Unsteady MHD flow of third-grade fluid for an infinite vertical plate

with transpiration generated by impulsive and oscillating boundary

embedded in a porous medium;

(iii) Unsteady flow of third-grade fluid in a channel with transpiration

generated by impulsive and oscillating motion on the upper wall;

(iv) Steady mixed convection flows of fourth-grade fluid in a Channel

with transpiration in a Darcy-Forchheimer medium;

(v) Unsteady flow of second-grade fluid in a capillary tube generated

by sinusoidal pressure gradient and without sinusoidal pressure

gradient;

(vi) Unsteady flow of third-grade arising in the wire coating process

inside a cylindrical die.

5

1.4 Scope of the Study

The scope of the study refers to the problems involving unsteady and steady

unidirectional flow of non-Newtonian fluid of differential type. The problems are

limited to no-slip conditions and the induced motion on the fluid is either by impulsive

or oscillation wall boundary. The study contains analytical solutions describing

the behaviour of unsteady and incompressible second-grade, third-grade and fourth-

grade fluids. Effect of heat transfer is considered. The fluid is assumed to be

electrically conducting, passing through a porous medium, with wall transpiration and

in the presence of magnetic field. The analytical solution technique used is Laplace

transform, Bessel transform, perturbation technique, symmetric reduction approach,

extended version of variable separation method combined with similarity arguments,

new modified homotopy perturbation method and new modified optimal homotopy

asymptotic method.

1.5 Significance of findings

The outcomes from this research work will be significant in the following ways:

(i) Newly improved analytical methods could be applied to solved more

general problems for other non-Newtonian fluids flow models;

(ii) It is hoped that newly extended mathematical models may bring out

more research for other fluids flow problems;

(iii) The obtained analytical solutions will be very useful for assessing

the accuracy of approximate numerical and theoretical procedures,

as well as experimental practices;

(iv) The applications of wall transpiration near an infinite plate wall

could be found in boundary layer control with important examples

in manufacturing techniques, aeronautical systems, chemical and

mechanical engineering processes;

6

(v) Heat transfer in free and mixed convection in vertical channels

occurs in many industrial processes and natural phenomena. These

applications could be found in the design of cooling systems for

electronic devices and in the field of solar energy collection;

(vi) Heat transfer in transpiration walls can be used actually in many

of engineering applications such as solid matrix heat exchanger,

electronic cooling, heat pipes chemical reaction, solar collector,

design of thrust bearings, drag reduction and thermal recovery of oil;

(vii) Application of MHD flow through a porous medium could be found

in petroleum technology to study the movement of natural gas,

in chemical engineering for filtration and purification process, as

well as in agricultural engineering to study the underground water.

Other areas of applications include the design high and low voltage

of electrical appliances, MHD power generation, plasma studies,

nuclear reactor using liquid metal coolant and geothermal energy

extraction;

(viii) Wire coating is used for the purpose of generating high and low

voltage, for the protection of humans, and for the processing of

signals such as in cable and telephone wires. Wire coating is an

important chemical process in which different types of polymer are

used.

1.6 Research methodology

The thesis undertakes the following research methodology:

1.6 .1 Problem formulation

The governing flow and heat transfer equations for the new models outlined in the

objectives are modelled mathematically using the fully developed flow conditions

7

and then expressed in dimensional form. The dimensional governing equations are

transformed into non-dimensional equations using non-dimensional variables.

1.6 .2 Analytical computation

Non-dimensional governing equations for the problems are solved using Laplace

transformation, Bessel transformation, perturbation techniques, translational type of

symmetry reduction method, new modified homotopy perturbation transform method

and new modified optimal homotopy asymptotic method.

Graphical representations of the solutions of the problems are developed using

MATHEMATICA 5.2, MAPLE 17 and MATHCAD 15.

1.6 .3 Result verifications

The obtained solutions (results) will be validated by comparing the limiting cases of

the present work with the results of the related published papers/articles. The graphical

results will be analyzed and discussed over various physical parameters.

1.7 Thesis organization

In this thesis an overview of the thesis is given in Chapter 1, which include the

research background, the problem statement, research objectives and scope, significant

of findings and research methodology. The structure of the remaining part of the

thesis and its relation to the objectives listed in Section 1.3 are given in Figure 1.1.

Chapter 2 is regarding a literature review for the research problem. Various works

by different researchers regarding unsteady and steady flow with heat in vertical

plate, vertical channel, horizontal channel, horizontal cylinder and capillary tube are

8

Chapter 1

Chapter 4

Chapter 2

Chapter 3

Chapter 5

Chapter 6

Chapter 8

Chapter 7

Chapter 10

Objective (i)

Chapter 9

Objective (ii)

Objective (iii)

Objective (iv)

Objective (v)

Objective (vi)

Figure 1.1: The outline of the thesis

9

reviewed. Chapter 3 present analytical approaches and constitutive models. This is

the review of different analytical approaches used to solve nonlinear problems. In this

chapter, new modified analytical algorithms using the idea of HPM, OHAM and He

polynomials are proposed. Constitutive models of differential type fluids, basic flow

equations and dimensionless numbers are given.

In Chapter 4, an exact solution for unsteady second-grade fluid over flat plates

with impulsive, oscillating motions and wall transpiration have been derived. The exact

solutions have been derived using the Laplace transform, the perturbation technique

and extension of the variable separable technique together with similarity arguments.

Limiting cases have been considered for zero transpiration rates and Newtonian fluid.

The solution agreed favorably with the existing results of (Asghar et al. (2006)) and

(de Almeida Cruz and Ferreira Lins (2010)).

In Chapter 5, the problem of unsteady MHD flow of third-grade over an

infinite flat wall with impulsive, oscillating motions, wall transpiration and porous

space have been derived. Under the flow assumptions, the governing nonlinear partial

differential equation has been transformed into steady-state and transient nonlinear

equations. The reduced equation for the transient flow has been solved using symmetry

reduction approach while the nonlinear steady-state equation has been solved using a

new modified homotopy perturbation transform technique. The accuracy of the new

modified HPM solutions for the velocity field has been achieved by comparing with the

exact solutions for the unsteady flow equation. Limiting cases have been considered

for cases of non-oscillation motion and zero transpiration rate. The results agreed with

(Aziz et al. (2012)) and (Aziz and Aziz (2012)).

The problem of unsteady flow of a third-grade fluid in a transpiration horizontal

channel with oscillating motion on the upper wall is presented in Chapter 6. We applied

a new modified homotopy perturbation method for unsteady problems, which is also

solved using HPM and OHAM. Comparison between results obtained from HPM,

OHAM and (Siddiqui et al. (2008)) solutions reveals that the proposed algorithm is

highly accurate.

10

Chapter 7 describes the modelling of mixed convection flows of fourth-grade

fluids in a vertical channel with transpiration in a Darcy-Forchheimer medium. Explicit

analytical expressions for the velocity field and the temperature distribution have been

derived using a new modified OHAM. The results are validated with Ziabakhsh and

Domairry (2009) and found to be in good agreement.

Chapter 8 considers two problems involving the unsteady flow of second-grade

fluid in a capillary tube. Modelling of two flow problems where considered; one with

sinusoidal pressure waveform and heat transfer and the other one is free-convection

flow due to reactive nature of the fluid. Exact solution of the first problem was derived

using Bessel transform technique while the second problem was solved using a new

modified HPM method. The results agreed favorable with (Jha et al. (2011a)) and (Yin

and Ma (2013)).

Chapter 9 concentrates on the mathematical modelling of two problems arising

in the wire coating process inside a cylindrical die. Firstly, the mathematical modelling

of unsteady third-grade fluid without magnetic field and heat transfer is developed.

Secondly, the mathematical modelling of unsteady flow of second-grade fluid with

magnetic effect and heat transfer is developed. Both the problems were solved

analytically using a new modified HPM method. The results are validated with (Shah

et al. (2013)) and found to be in good agreement.

Finally, the conclusion and the outline of possible extensions to this work are

outlined in Chapter 10.

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

This chapter will be categorized into six sections according to the arrangements of

the objective in Chapter 1 Section 1.3. Sections 2.2 presents the general review on

unsteady flow due to an oscillating or impulsive motion of an infinite plate. Sections

2.3 presents the review of unsteady MHD flow through a porous medium for an infinite

plate. Section 2.4 contains the literature on Couette and Poiseuille flows for a channel

flow. Sections 2.5 discusses the literature of the free and mixed convection near an

infinite wall and channel. Sections 2.6 presents the literature on unsteady flow in a

capillary tube while Sections 2.7 outlines the literature on unsteady flow past an inner

cylinder.

2.2 Unsteady flow due to an oscillating and impulsive for an infinite plate

The study of the fluid flow over an oscillating and impulsive plate is not only of

fundamental theoretical interest but also occurs in many applied problems such as

in acoustic streaming around an oscillating body and unsteady boundary layer with

fluctuations. This problem is also named as Stokes or Rayleigh problem in the

literature by (Schlichting and Gersten (2000)). In the first problem of Stokes, the wall

is initially at rest and a transient flow is induced to the fluid by a sudden impulsive

motion. In the second problem of Stokes, the motion is generated by an oscillating

12

plate. For longer time period, the transient motion vanished and the fluid velocity at

any point is just a harmonic oscillation with the same wall frequency.

Some important and fundamental studies have been conducted for flow of a

viscous fluid which is described by the Navier-Stokes equation. The first closed form

of the steady velocity for an incompressible fluid past an infinite oscillating plate was

presented by (Stokes (1851)). Panton (1968) presented the first closed form expression

for the transient solution due to the oscillating plate, where the transient part contains

exponential and error functions. A serious deficiency in (Panton (1968)) solutions was

that he could not express the solutions as a sum of steady-state and transient solutions.

Erdogan (2000) criticized this by applying the Laplace transform to solve the same

problem and obtained the exact solutions for transient motion due to the cosine and sine

oscillations of the plate. He found that for large times t, the transient solutions tend

to the steady-state solutions. However, the results presented by (Erdogan (2000)) are

not directly presented as a sum of steady-state and transient solutions. (Fetecau et al.

(2008)) reinvestigated the work of (Erdogan (2000)) and presented new exact solutions.

These solutions, unlike those obtained by (Erdogan (2000)), were simpler and directly

presented as a sum of steady-state and transient solutions. It was found that the time

required to attain the steady-state for the cosine oscillations of the boundary is smaller

than that for the sine oscillations. However, the transient parts of these solutions still

involved the unevaluated integrals. Later, Erdogan and Imrak (2009) reinvestigated

the same problem by (Erdogan (2000)) using two different transform methods, and

obtained the starting solutions. These solutions were presented as the sum of the

steady-state and transient solutions again in terms of unevaluated integrals. However,

they noted that the integrand contains an oscillatory function and therefore will give

erroneous results upon numerical integration. In order to obtain correct results, the

transient part is presented in terms of the tabulated functions. de Almeida Cruz and

Ferreira Lins (2010) derived more general form of Stokes’ problems for viscous fluid

when fluid transpiration at the wall is considered.

The exact solutions for the second-grade fluid past an infinite oscillating plate

was first presented by (Rajagopal (1982)), where the steady-state velocities for the fluid

are presented in several states of motion. This solution has a serious drawback because

13

no expression for the starting velocity field. Asghar et al. (2006) applied Laplace

transform and perturbation techniques to treat this problem and obtained the closed-

formed expression for the starting velocity due to oscillation of the wall boundary.

Also, Nazar et al. (2010) solved the same problems and obtained the starting velocity

field due to oscillation of the wall boundary by means of Laplace transform method.

Yao and Liu (2010) studied the effects of the side walls on unsteady flow of a second-

grade fluid over a plane wall.

All the above mentioned studies of Stokes’ problems for a second-grade fluid

are based on the hypothesis that the bounding plate has no transpiration velocity.

A more general solution for the Stokes’ problems for the second-grade fluid can

be derived when the fluid transpiration is considered at the bounding plate. The

formulations must then be modified with the help of a brand new term representing

the momentum introduced into the flow through the transpiration of fluid. Having this

in mind, Chapter 4 of this thesis gives a more general solution of unsteady second-

grade generated by impulsive and oscillating wall with transpiration are derived.

2.3 Unsteady MHD flow through a porous for an infinite plate

The study of MHD flow through a porous medium has become an active area of

research due to its varied applications in science and engineering. Several researchers

have studied it for a variety of fluids in various forms. Awang Kechil and Hashim

(2009) analyzed MHD stagnation flow against a flat plate in porous media by using

Adomian decomposition method (ADM) to obtain an approximate analytical solution

of the problem. Alizadeh-Pahlavan et al. (2009) examined the analytical solution

of MHD upper-convected Maxwell (UCM) fluid over a porous stretching plate by

homotopy analysis method (HAM). Based on the results obtained in the (Alizadeh-

Pahlavan et al. (2009)) it is concluded that in applying HAM to highly nonlinear

equations such as those encountered in the flow of viscoelastic fluids, it might be

advisable to base the analysis on two auxiliary parameters instead of one. It is also

advisable that for each set of working parameters such as the elasticity number, the

14

magnetic number, and the suction/injection number different values should be assigned

to the auxiliary parameters to guarantee the convergence of the series. Raftari and

Yildirim (2010) considered MHD UCM fluid over a porous stretching plate means

of homotopy perturbation method (HPM). Comparison between the HPM and HAM

methods for the studied problem showed a remarkable agreement and revealed that the

HPM method need less work. Furthermore, the first-order approximate solution leads

to a very highly accurate solution.

Abdulhameed et al. (2013) extended (de Almeida Cruz and Ferreira Lins

(2010)) problem to the electrically conducting fluid passing through a porous

space and established new exact solutions using a modified version of the variable

separation technique. Ali et al. (2012) extended the problem by (Nazar et al. (2010))

to the electrically conducting second-grade fluid passing through a porous space

and established new exact solutions for transient oscillation motion using Laplace

transform techniques. Aziz et al. (2012) examined the analytical solutions for unsteady

magnetohydrodynamic flow of a third-grade fluid past a porous plate within a porous

medium due to an arbitrary wall velocity. They obtained results by applying a Lie

symmetry and numerical methods. Later, Aziz and Aziz (2012) extended this problem

to the porous wall and established analytical solution using similar techniques.

To the best of our knowledge, the time-dependent transient

magnetohydrodynamic flow of a third-grade fluid due to an oscillating plate in

a porous space with and without transpiration has not been studied before, and it is the

main aim of Chapter 5 to study these problems. The translational type of symmetry

method is used, such that the transient governing nonlinear partial differential

equations reduced to a nonlinear ordinary differential equations, which further solved

analytically for the time-dependent transient in the form of wave-front type travelling

solution. The nonlinear steady-state equation is solved using a new modified HPM for

steady-state problems.

15

2.4 Couette and Poiseuille flow in a horizontal channel

Couette flow is the flow between two parallel plates, one of which is moving relative to

the other. The flow is driven by virtue of viscous drag force acting. Here the external

pressure gradient is not cooperated. When the pressure is induced by the flow it is

referred to as Poiseuille flow. In Poiseuille flow, the movement of fluid is solely due to

the pressure gradient. Siddiqui et al. (2008) studies the heat transfer analysis of third-

grade fluid between two heated parallel plates for plane Couette flow, plane Poiseuille

flow and Couette-Poiseuille flow. HPM has been used to obtain approximate analytical

solutions. They found that the fluid velocity depends upon the non-Newtonian

parameter except in the case of simple plane Couette flow where it does not depend on

this parameter. Siddiqui et al. (2010) investigated again the problems of plane Couette

flow, plane Poiseuille flow and Couette-Poiseuille flow and presented new analytical

solutions using ADM. Similarly, Islam et al. (2011) solved the same problem using

OHAM and HPM with the application of heat transfer. The results obtained using

OHAM and HPM are compared with the numerical solutions. It is observed that

OHAM gave good accurate results, by comparing with the numerical results. They

concluded that the approach is simple to apply, as it does not require discretization or

perturbation like other numerical and approximate methods. Moreover, the technique

converges quickly to the numerical solution and requires less computational work.

Danish et al. (2012) derived the explicit forms of exact analytical solutions for the

velocity profiles in the cases of Poiseuille and Couette-Poiseuille flow of a third-grade

fluid between two parallel plates. It is found that the Couette flow features is dominate

as the third-grade parameter or absolute value of the pressure gradient are increased.

For a given pressure gradient, the Poiseuille flow effects are more pronounced in the

case of Newtonian fluid.

The studies mentioned above deal with steady flows, but many problems of

practical interest may be unsteady. The problem of unsteady flow of third-grade

fluid in a horizontal channel generated by impulsive and oscillating upper wall with

transpiration is formulated in Chapter 6. A new analytical algorithm is developed,

tested and verified.

16

2.5 Free and mixed convention flow near an infinite wall and channel

Theoretical studies on free convection from vertical plate with step discontinuities

in surface temperature continue to receive attentions in the literature due to its

industrial and technological applications. Chandran et al. (2005) investigated the

unsteady natural convection flow of a viscous incompressible fluid near a vertical

plate with ramped wall temperature. They compared the results with that of plate

with constant temperature. Seth and Ansari (2010) investigated magnetohydrodynamic

natural convection flow past an impulsively moving vertical plate with ramped wall

temperature in the presence of thermal diffusion with heat absorption. Makinde and

Olanrewaju (2010) studied the buoyancy effects on thermal boundary layer over a

vertical plate with a convective surface boundary. Seth et al. (2011) studied the

effects of radiation on unsteady magnetohydrodynamic natural convection flow of

a viscous fluid near an impulsively moving vertical flat plate with ramped wall

temperature in porous medium. Patra et al. (2012) investigated the effect of radiation

on natural convection incompressible viscous fluid near a vertical plate with ramped

wall temperature. Mohammed et al. (2012) solved the problem of transient natural

convection flow past an oscillating infinite vertical plate in present of magnetic field

and radiative heat transfer. The governing equations are solved analytically using

Laplace transform technique. The results are expressed in terms of the velocity and

temperature profiles as well as the skin-friction and Nusselt number. In line with these,

a more general solutions could be driven when fluid is considered as non-Newtonian

at the bounding plate. Motivated by this consideration, the unsteady free convection

transient flow of second-grade viscoelastic fluid near a vertical flat plate with ramped

wall temperature under the Boussinesq approximation have been derived.

Heat transfer in mixed convection in vertical channels have been the subject of

many detailed, mostly numerical studies of different flow configurations. Ziabakhsh

and Domairry (2009) obtained an analytical solution of natural convection flow of

a third-grade fluid between two vertical flat plates using homotopy analysis method

(HAM). Effects of the non-Newtonian nature of fluid on the heat transfer where

studied. Sajid et al. (2010) obtained exact solution of the steady fully developed mixed

17

convection flow of a viscoelastic second-grade fluid in a parallel-plate vertical channel.

Effects of pertinent parameters flow parameters have been obtained.

To the best of our knowledge, the steady fully developed mixed convection

flow of fourth-grade fluid in a transpiration channel in a Darcy- Forchheimer medium

has not been studied before and it is the main aim of Chapter 7 to study this problem.

Such a study, however, is needed for the understanding of flow pattern or thermal

characteristics and hence is worthy of additional investigation.

2.6 Unsteady flow in a capillary tube

2.6 .1 Convective heat transfer with oscillating pressure waveform

Theoretical interest about the flow and heat transfer round pipe driven by oscillating

pressure waveform has increased substantially over the past few decades due to its

application in natural systems for example in the circulatory system, respiratory system

in addition to engineering systems such as with reciprocating pumps, IC engines, pulse

combustors, hot wire anemometer inside a changing flow, and in tube bundles where

vortex losing in the leading tube induces flow fluctuations on subsequent tubes. Some

theoretical and fundamental researches have been the subject of many investigations

covering various facets. The very first closed form of the velocity distribution of an

oscillatory pipe flow was presented by (Richardson and Tyler (1929)). They consider

a Newtonian flow in a pipe and demonstrated the way an oscillating flow could cause

another velocity profile close to the wall surface and discovered that the annular effect

occurs near the wall rather than at the center of the pipe as in the case of unidirectional

steady flow.

Since that time, numerous research has been dedicated to oscillatory flow for

heat transfer enhancement. Zhao and Cheng (1995) presented numerical solution for

laminar forced convection of an incompressible periodically reversing flow in a pipe

of finite length at constant wall temperature. The results revealed that annular effects

18

also exist in the temperature profiles near the entrance and the exit of the pipe during

each half cycle at high kinetic Reynolds numbers. The averaged heat transfer rate

is found to increases with both the kinetic Reynolds number and the dimensionless

oscillation amplitude but decreases with the length to diameter ratio. Moschandreou

and Zamir (1997) developed an analytical technique of pulsatile flow in a tube with

constant heat flux at the wall. The results indicate that in a range of moderate values

of the frequency, there is a positive peak in the effect of pulsation whereby the bulk

temperature of the fluid and the Nusselt number increased. However, the effect is

reversed when the frequency is outside this range. The peaks of pulsation are higher

at lower Prandtl numbers. Guo and Sung (1997) presented an analytical investigation

of the Nusselt number in pulsating pipe flow. The analysis proposed many versions

of the Nusselt number and have been tested to clarify the conflicting results in the

heat transfer characteristics for pulsating flow in a pipe. An improved version of the

Nusselt number was obtained which shown to be in close agreement with available

measurements.

Guo et al. (1997) investigated pulsating flow in a circular pipe with partial

filling of porous media. It was found that the porous layer added to the inner surface

of the pipe can enhance the heat transfer from oscillating flow depending on the Darcy

number, an oscillating frequency, and oscillating amplitude. Hemida et al. (2002)

presented a new time average heat transfer coefficient for pulsating flow is defined

such as to produce results that are both useful from the engineering point of view,

and compliant with the energy balance. The current derived average is compared with

intuitive averages used in the literature. New results are numerically obtained for the

thermally developing region with a fully developed velocity profile. Different types

of thermal boundary conditions are considered, including the effect of wall thermal

inertia. The impact of Reynold and Prandtl numbers, as well as pulsation amplitude

and frequency on heat transfer, are investigated. The mechanism by which pulsation

affects the developing region, by creating damped oscillations along the tube length of

the time average Nusselt number, is also demonstrated.

Yu et al. (2004) studied the effects of oscillating laminar heat convection

with heat flux in a circular tube where the flow is considered symmetrical. Their

19

results showed that both the temperature profile, and the Nusselt number fluctuate

periodically about the solution for steady laminar convection, with the fluctuation

amplitude depending on the dimensionless pulsation frequency, ω∗, the amplitude of

the pressure, γ0, and the Prandtl number, Pr. It is also shown that pulsation has no

effect on the time-average Nusselt numbers when ω∗ = 0.1, γ0 = 0.01 and Pr = 0.1.

Wang and Zhang (2005) carried out an analysis of convection heat transfer in pulsing

turbulent flow in velocity oscillating amplitudes and constant wall temperature inside

a pipe. Their results showed that the heat transfer enhancement is principally impacted

by Womersley number and velocity oscillation amplitude. In addition, the heat transfer

enhancement can also be affected by Reynolds number, especially at lower Reynolds

number. However the temperature fluctuations at inlet triggered by velocity oscillation

do not have any effect on the heat transfer enhancement.

Akdag and Ozguc (2009) analyzed experimentally the heat transfer flow with

constant heat flux and oscillating flow inside a vertical annular liquid column. The flow

effects of parameters including the frequency, amplitude, heat flux, Prandtl number and

geometric parameter were analyzed. The final results demonstrated that the the positive

impact of heat transfer with the increasing both frequency and amplitude oscillation

parameters. Mehta and Khandekar (2010) presented a numerical study of the effect

of periodic pulsations on heat transfer in concurrently developing laminar flows. The

outcomes demonstrated the results of oscillation frequency and Reynolds number are

minimal, and also the Prandtl number comes with an adverse impact on heat transfer.

Shailendhra and AnjaliDevi (2011) examined heat transfer within the pulsing flow of

liquid metals having a constant axial temperature gradient superimposed. The result

showed that, under constant axial thermal gradient, sinusoidal oscillation of the fluid

results in the substantial enhancement of heat transfer and it does not matter how the

oscillation generated. Liu et al. (2013) solved the energy equation of circular micro-

channels, which considers axial heat conduction, velocity slip, temperature jump,

viscous dissipation and thermal entrance effect. The design criterion for whether the

axial heat conduction and viscous dissipation should be considered in engineering is

given by studying their contributions to average Nusselt number.

Yu et al. (2004) extended the work of Yin and Ma (2013) by presenting a

20

new analytical solution of velocity, temperature distributions along with a Nusselt

number to have an oscillating laminar flow inside a round pipe that driven by a

sinusoidal waveform. The flow is considered not symmetrical, and their solution

could overcome the limitation presented by (Yu et al. (2004)). The Prandtl number

was confirmed as the second essential aspect affecting heat transfer performance

within an oscillating flow. The best possible Prandtl number for maximum Nusselt

number is found. Subsequently, Yin and Ma (2014) derived analytical solution of

heat transfer of oscillating flow at a triangular pressure waveform and found that the

heat transfer coefficient of the oscillating flow depends on the fluid properties and

oscillating waveform. Also, the triangular waveform of oscillating motion can result

in a higher heat transfer coefficients.

All the above-quoted analysis of oscillating laminar flow in a pipe driven by

oscillating pressure waveform are based on the Newtonian fluids. According to our

knowledge, theoretical study of oscillating laminar flow of differential non-Newtonian

fluid of grade two with convective heat transfer in a capillary tube generated solely

due to a sinusoidal waveform has not been undertaken before. Theoretical studies

of oscillating laminar flow in a pipe driven by oscillating pressure gradient of non-

Newtonian fluids are crucial in bioengineering such as in blood vessels of animals and

human and other several industrial processes. In view of these, in the first part of

Chapter 8, analytical solutions for convective heat transfer of unsteady second-grade

fluid in a capillary tube driven by sinusoidal pressure waveform was derived.

2.6 .2 Free-convection due to reactive fluid nature

Free convective flow of reactive fluid is very important for efficient design of

equipment in a variety of engineering systems. Therefore, many lubricants used in both

engineering and industrial processes are reactive such as hydrocarbon oils, polyglycols,

synthetic esters and polyphenylethers. Their efficiency is dependent largely on

the temperature variation is very often. Makinde (2005) investigated exothermic

explosions inside a cylinder pipe for viscous fluid under biomolecular Arrhenius

21

and sensitized reactive rates, by neglecting the intake of the material. Makinde

(2008) also analyzed reactive viscous fluid in a horizontal channel with sliding wall.

Further, Makinde (2009) investigated the thermal stability of a reactive viscous flow

through a horizontal channel embedded in porous medium with convective boundary

condition. In this case, the Brinkman model is employed and important properties of

the temperature field including bifurcations and thermal criticality are discussed. In all

three cases Makinde (2005, 2008 and 2009) obtained the solution analytically, using

perturbation technique together with a special type of Hermite-Pade approximation.

Jha et al. (2011a) analyzed transient free convective flow of reactive viscous fluid

in vertical tube. The significant results from their study are that both velocity and

temperature increase with the increase in the value of reactant consumption parameter

and dimensionless time until they reach steady-state value. Jha et al. (2011b) also

investigated the transient free convection flow of reactive viscous fluid in vertical

channel. The results showed that it takes longer time to attain steady-state in the

case of water than air. In both studies, Jha et al. (2011a and 2011b) obtained the

transient solutions for the problems, using numerical schemes and steady-state solution

by regular perturbation method.

Theoretical studies of transient free convection flow of reactive non-Newtonian

fluids are important due to the physical character and chemical composition from the

broadly used industrial scale lubrication (mostly reactive polymer fluids). Sivaraj

and Rushi K (2013) analyzed the chemically responding dusty viscoelastic fluid

(Walters’s liquid model-B) flow within an irregular channel with convective boundary.

They solved the combined partial differential equations analytically using perturbation

technique. Their results revealed that the rate profiles of dusty fluid are much more

greater compared to dust contaminants. Makinde et al. (2011) analyzed an unsteady

flow of the reactive variable viscosity non-Newtonian fluid via a porous saturated

medium with asymmetric convective boundary conditions. They concluded that

exothermic chemical responses occur inside the flow system with the asymmetric

convective heat exchange using the ambient in the surfaces follow Newton’s law of

cooling. The coupled nonlinear partial differential equations of the problem are derived

and then solved numerically using a semi-implicit finite difference scheme. Rundora

22

and Makinde (2013) looked into the results of suction/injection on unsteady reactive

variable viscosity non-Newtonian fluid flow inside a channel of porous medium

and under convective boundary conditions. The governing flow equations are also

solved using a semi-implicit finite difference scheme. The result revealed that, the

suction/injection Reynolds number, the porous medium parameter and also the Prandtl

number possess a diminishing impact on the velocity and heat transfer in the channel

walls.

In the above theoretical study, the development of modelling, analytical

techniques and flow characteristics of the free-convection of a reactive second-grade

fluid in a capillary tube has not been studied. The main aim of Chapter 8 is to extend

the analysis of (Jha et al. (2011a)) to viscoelastic second-grade fluid.

2.7 Unsteady flow past an inner cylinder

Application of wire coating process discovered to be relevance in lots of engineering

devices. Some recent contributions dealing with the wire coating analysis for different

fluids are (Sajid et al. (2007)). They studied the wire coating with Oldroyd 8-constant

fluid without pressure gradient. Similarly, Sajid and Hayat (2008) examined the wire

coating analysis by withdrawal from a bath of Sisko fluid. In all two cases, Sajid et al.

(2007) and Sajid and Hayat (2008) have solved the problems using the HAM and gave

the solution for the velocity field in the form of a series. A similar model of of Oldroyd

8-constant fluid with pressure gradient has been used by (Shah et al. (2012)). They

applied OHAM and obtained explicit analytical expression for the velocity field. The

result showed that as order of the problem increases, the accuracy increases and the

solution converges to the exact solution by choosing the appropriate auxiliary constants

and increasing the order. Recently, Shah et al. (2013) derived mathematical model

describing the behavior of unsteady second-grade fluid in a wire coating process inside

a straight annular die. An exact solution of the problem have been derived using the

method of separation of variables.

23

No attempt has been made to study unsteady flows of second-grade fluid

in the context of magnetic and heat transfer near inner cylinder moving axial-wire

coating die. Explicit analytical expressions for the transient, steady-state velocity and

temperature fields have been derived using a new modified HPM for unsteady and

steady-state flow problems.

Moreover, no attempt has been made to model unsteady flow of third-grade

arising in coating metallic wire process inside a cylindrical roll die. Explicit analytical

expression for the velocity field has been derived using a new modified HPM for

unsteady flow problems. The problems of unsteady MHD second-grade with heat

transfer and unsteady third-grade, fluid have been reported in Chapter 9.

CHAPTER 3

ANALYTICAL APPROACHES AND CONSTITUTIVE MODELS

3.1 Introduction

This chapter consist related analytical methods and fundamental governing equations

regarding the flow problem in this thesis divided into two parts. In the first part

a review of various different approaches for solving nonlinear partial differential

equations is presented. Within these analytical approaches, the most important and

recently developed methods are considered. These include: Laplace transformation

combined with perturbation technique; Bessel transformation; homotopy perturbation

method (HPM); optimal homotopy asymptotic method (OHAM); He polynomials and

homotopy perturbation transform method (HPTM). Further, new modified analytical

algorithms using the ideas of OHAM, HPM and He polynomials are proposed to

handle nonlinear problems. In the second part, the constitutive models of second-

grade, third-grade and fourth-grade fluids are given. Finally, a general concept of basic

flow equations and dimensionless numbers are demonstrated.

3.2 Analytical methods

Analytical solutions are calculated using techniques that provide an exact solutions.

Exact solution generally refers to a solution that captures the entire physical and

mathematical aspects of a problem as opposed to one that is approximate, perturbation.

These unlike numerical solution that gives a quantitative approximation to the exact

213

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