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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
5.2 Formulation of the problem 82
5.3 Solution of the problem 86
5.3 .1 Start-up phase solution 87
5.3 .2 Impulsive start 90
5.3 .3 Steady-state solution 90
5.4 Results and discussion 93
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
6.2 Formulation of the problem 107
6.3 Solution of the problem 109
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
6.4 Results and discussion 116
6.5 Conclusions 123
CHAPTER 7 THE STEADY HEAT TRANSFER FLOW OF A FOURTH-
GRADE FLUID IN A DARCY FORCHHEIMER
POROUS MEDIA 125
7.1 Introduction 125
7.2 Formulation of the problem 125
7.3 Solution of the problem 128
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
7.5 Results and discussion 138
7.6 Conclusion 145
CHAPTER 8 THE UNSTEADY FLOW OF A SECOND-GRADE
FLUID IN A CAPILLARY TUBE 146
8.1 Introduction 146
8.2 Problem formulation due to sinusoidal pressure effect 146
8.3 Solution of the problem 149
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
8.5 Results and discussion 166
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
9.1 Introduction 175
9.2 Formulation of the problem when 0 175
9.3 Solution of the problem 177
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 Solution of the problem 185
9.4 .1 Solutions for steady-state velocity and
temperature fields 186
9.4 .2 Solutions for transient velocity and
temperature fields 188
xii
9.5 Results and discussion 195
9.6 Conclusion 206
CHAPTER 10 CONCLUSIONS AND FUTURE RESEARCH 209
10.1 Introduction 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
4.4 Velocity profiles versus y for various values of
transpiration parameter ξ in the case of large value
of time t = 4, when α = 0.4, ω = 0.5, are fixed (a)
Cosine oscillation and (b) Sine oscillation 71
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)
Cosine oscillation and (b) Sine oscillation 73
4.7 Velocity profiles versus y with suction for various
values of second-grade parameter α in the case of
small value of time t = 0.2, when ω = 0.5, ξ = 0.3,
are fixed (a) Cosine oscillation and (b) Sine oscillation 74
xvi
4.8 Velocity profiles versus y with suction for various
values of second-grade parameter α in the case of
large value of time t = 5, when ω = 0.5, ξ = 0.3,
are fixed (a) Cosine oscillation and (b) Sine oscillation 75
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
various values of the transpiration parameter ξ when
y = 1.2, ω = 0.5, α = 0.1, are fixed (a) Cosine
oscillation and (b) Sine oscillation 77
4.11 Profiles of starting and steady solutions versus t for
various values of the transpiration parameter ξ when
y = 1.2, ω = 1, α = 0.1, are fixed (a) Cosine
oscillation and (b) Sine oscillation 78
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
5.4 Frequency series of the flow velocity with various
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
oscillation and (b) Sine oscillation 96
5.5 Frequency series of the flow velocity with various
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
oscillation and (b) Sine oscillation 97
5.6 Frequency series of the flow velocity with various
values of the material parameters β∗, when β∗∗ = 1,
y = 0, M∗ = 1 ϕ∗ = 0.5, ξ = 0.5, ν∗ = 0.5, t = 1
and α∗ = 0.5 are fixed (a) Cosine oscillation and (b)
Sine oscillation 98
5.7 Frequency series of the flow velocity with various
values of the material parameters β∗∗, when β∗ = 1,
y = 0, M∗ = 1 ϕ∗ = 0.5, ξ = 0.5, ν∗ = 0.5, t = 1
and α∗ = 0.5 are fixed (a) Cosine oscillation and (b)
Sine oscillation 100
5.8 Frequency series of the flow velocity with various
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
are fixed (a) Cosine oscillation and (b) Sine oscillation 101
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)
Cosine oscillation and (b) Sine oscillation 122
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
7.6 Velocity profiles versus y for various values of
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
8.3 Temperature profiles versus r for various values of
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
9.9 Temperature profiles versus r for various values of
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
9.11 Temperature profiles versus r for various values of
Br when when M = 0.5, Pr = 10 and α = 0.5 203
9.12 Temperature profiles versus r for various values of
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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