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
v
ABSTRACT
vi
ABSTRAK
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
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
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
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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 7THE STEADY HEAT TRANSFER FLOW OFA 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
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7.3 .2 Solutions for a third-grade fluid
0
1367.4 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
1759.3 Solution of the problem 177
9.3 .1 Solution for a third-grade fluid
0
1789.3 .2 Solution for a second-grade fluid
0
1819.3 .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
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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
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 =−5.54965331and
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 whenγ = 0, ξ = 0, β = 0.5,Pr = Gr = Re = Br = 1, P = Q = 0, C1 = −0.2843403486
andC2 =−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 =
−0.40714571andC2 =−0.35572542 140
8.1 First few eigenvalues ofβ2
cn 157
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8.5 Nusselt number(N u)whenα= 0.4andPr = 10 167 9.1 Numerical results show the comparison of present
results with related published work for small value
of timet(t= 0.2)andβ = 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 timet(t= 2)andβ = 0, ζ = 3, α= 0.5 196 9.3 Variation of frictional force on the total wire with
change in parametersα1, α2andζforM = 0.5, t= 2 201 9.4 Variation of wall shear stress at the surface of the die
with change in parametersα1, M andζfort= 2 201 9.5 Variation of wall shear stress at the surface of the
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 timet = 2,whenα = 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 timet = 4,whenα = 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 timet = 0.2,whenω = 0.5, ξ = 0.3,
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4.8 Velocity profiles versus y with suction for various values of second-grade parameter α in the case of large value of timet = 5, whenω = 0.5, ξ = 0.3,
are fixed (a) Cosine oscillation and (b) Sine oscillation 75 4.9 Profiles of starting and steady solutions versusy for
various values of the transpiration parameterξwhen
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 parameterξwhen
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 parameterξwhen
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 ξwith an impulsive start for small value of timet = 1,when α = 0.4, ω = 0.5,
are fixed 79
4.13 Velocity profile versus y for various values of transpiration parameter ξwith an impulsive start for large value of time t = 2, whenα = 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) whenβ∗∗ = 2.5, β∗ = 1.5, M∗ = 1 ϕ∗ =
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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
andα∗ = 0.5are 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∗ = 0.5andα∗ = 0.5are 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 platey, whenβ∗ = 1.5, β∗∗ = 0.5, M∗ = 1 ϕ∗ = 0.5, ξ = −1,
t = 1, ν∗ = 0.5 andα∗ = 0.5are fixed (a) Cosine
oscillation and (b) Sine oscillation 97
5.6 Frequency series of the flow velocity with various values of the material parametersβ∗, whenβ∗∗ = 1, y = 0, M∗ = 1ϕ∗ = 0.5, ξ = 0.5, ν∗ = 0.5, t = 1
andα∗ = 0.5are fixed (a) Cosine oscillation and (b)
Sine oscillation 98
5.7 Frequency series of the flow velocity with various values of the material parametersβ∗∗, when β∗ = 1, y = 0, M∗ = 1ϕ∗ = 0.5, ξ = 0.5, ν∗ = 0.5, t = 1
andα∗ = 0.5are 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
xviii
5.9 Velocity profiles versus yof 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∗ = 0.5andα∗ = 0.5
are fixed 102
5.10 Velocity profiles versus yof 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∗ = 0.5andα∗ = 0.5are
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∗ = 1ϕ∗ = 0.5, ξ = 0.5, t= 1, ν∗ = 0.5and
α∗ = 0.5are 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 timet= 0.1 118
6.3 Comparison of the present results with HPM and
OHAM for large value of timet = 2 119
6.4 Velocity profiles versusyfor various values ofω for small time t = 1 andα = 0.5, ξ = −0.7,andβ =
0.5with (a) Cosine oscillation and (b) Sine oscillation 120 6.5 Velocity profiles versusyfor various values ofβ for
small value of timet = 1andα= 0.5,andξ =−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
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7.3 Effects of Forchheimer parameter Q on wall shear
stressτω 141
7.4 Effects of Darcian parameterP on Nusselt numberN u 141 7.5 Effects of Forchheimer parameter Q on Nusselt
numberN u 142
7.6 Velocity profiles versus y for various values of
fourth-grade parameterγ on velocityu 142 7.7 Effects of Darcian parameterP on velocityu 143 7.8 Temperature profiles versusywith zero transpiration
(ξ= 0) for different values of the constant pressure
gradientA 143
7.9 Temperature profiles versus y with suction (ξ= 2)
for different values of the constant pressure gradientA 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 whenα = 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
whenϵ = 0.01,Pr = 10, α = 0.4andGr = Ec =
Re= 1.at (a)t = 0.2and (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= 0.2and (b)t = 6 170 8.6 Velocity profiles versus r for various values of α
whenϵ = 0.01,Pr = 10, λ = 0.1andEc = Gr =
Re= 1.at (a)t = 0.2and (b)t= 6 171
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9.2 Comparison of velocity profiles with related published paper for the case of small value of timet
whent = 0.2andβ = 0, ζ = 3, α= 0.5are fixed 197 9.3 Comparison of velocity profiles with related
published paper for the case of large value of timet
whent = 2andβ = 0, ζ = 3, α= 0.5are fixed 198
9.4 Velocity profiles versusrfor various values of third-grade parameterβfor the case of small value of time
twhent = 0.2, ζ = 3andα = 0.5are fixed 198
9.5 Velocity profiles versusrfor various values of third-grade parameterβfor the case of large value of time
twhent = 2, ζ = 3andα= 0.5are fixed 199
9.6 Velocity profiles versus r for various values of distance ζ parameter for the case of small value of
timetwhent= 0.2, β = 0.5andα= 0.5are fixed 199 9.7 Velocity profiles versus r for various values of
distance ζ parameter for the case of large value of
timetwhent= 2, β = 0.5andα= 0.5are fixed 200 9.8 Velocity profiles versus r for various values of time
parametertwhenβ = 0.5, α= 0.5andζ = 3are fixed 200 9.9 Temperature profiles versus r for various values of
PrwhenM = 0.5, Br= 1, α= 0.5andt= 2 202
9.10 Temperature profiles versusrfor various values oft
whenM = 0.5, Br= 1, α= 0.5andPr = 10 203
9.11 Temperature profiles versus r for various values of
Brwhen whenM = 0.5,Pr = 10andα= 0.5 203 9.12 Temperature profiles versus r for various values of
M whenα= 0.5,Pr = 10, Br= 1andt= 2 204
9.13 Nusselt number versus t for various values of Pr
whenr = 1,M = 0.5, α= 0.5, Br= 1andζ = 3 204
9.14 Nusselt number versustfor various values ofαwhen
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
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NOMENCLATURE
Roman Letters
a - Lower limit of range integration
an - Constant defined by equation (8.24) az - Unit vector alongz−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
xxiii
C0 - Initial concentration of the reactant species
d
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 ordern 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
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 thanD
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
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
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 ofΓ (z)
Γ (z) - Gamma function
Γ - Boundary of domainΩ,
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 ofΓ (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
xxvii
Subscripts
s - Steady state
t - Transient state
m - Mean value
w - Wall
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.
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.
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)).
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;
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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;
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(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
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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
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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)
[image:32.595.236.417.174.583.2]Objective (vi)
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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)).
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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.
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
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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. Erdo˘gan (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 timest, the transient solutions tend to the steady-state solutions. However, the results presented by (Erdo˘gan (2000)) are not directly presented as a sum of steady-state and transient solutions. (Fetecau et al. (2008)) reinvestigated the work of (Erdo˘gan (2000)) and presented new exact solutions. These solutions, unlike those obtained by (Erdo˘gan (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, Erdo˘gan and ˙Imrak (2009) reinvestigated the same problem by (Erdo˘gan (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.
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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
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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.
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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.
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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.
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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.
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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.
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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 whenω∗ = 0.1,γ0 = 0.01andPr = 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.
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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
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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.
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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
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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.
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
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