NUMERICAL STUDIES ON UNSTEADY MIXED CONVECTION FLOWS A THESIS submitted by DEVARAPU ANILKUMAR for the award of the degree of DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF TECHNOLOGY MADRAS CHENNAI - 600 036 DECEMBER 2004
NUMERICAL STUDIES ON UNSTEADY MIXED
CONVECTION FLOWS
A THESIS
submitted by
DEVARAPU ANILKUMAR
for the award of the degree
of
DOCTOR OF PHILOSOPHY
DEPARTMENT OF MATHEMATICSINDIAN INSTITUTE OF TECHNOLOGY MADRAS
CHENNAI - 600 036
DECEMBER 2004
THESIS CERTIFICATE
This is to certify that the thesis entitled “NUMERICAL STUDIES ON
UNSTEADY MIXED CONVECTION FLOWS” submitted by
Mr. Devarapu Anilkumar to the Indian Institute of Technology, Madras for the award
of the degree of Doctor of Philosophy is a bonafide record of research work carried out
by him under my supervision. The contents of this thesis, in full or in parts, have not
been submitted to any other Institute or University for the award of any degree or diploma.
Chennai - 600 036 Research Guide
December 2004
(Dr. Satyajit Roy)
ii
ACKNOWLEDGEMENTS
First and foremost I thank my advisor, Dr. Satyajit Roy, for his great mentorship,
cool-nature and immense patience. Working under his guidance has been a very pleasant
and rewarding experience.
I am thankful to Professor S. Kalpakam, Head of the Department of Mathematics,
for the facilities provided in the department. My thanks are also due to the Doctoral
Committee members for their valuable suggestions.
To Professor A. Avudainayagam, for providing me with much support in different
ways throughout my work, I am very thankful.
To Professor T. Amaranath, my advisor at University of Hyderabad in the pre-IIT
years, was the first one to show me what research was all about, I owe a special thank you.
To Dr. C. Vani, for her special interest on my research career, I am indebted you.
Special thanks to my colleagues and friends with whom I shared many insightful
discussions and good times.
Last, but not the least, I owe a lot to my parents and and all other family members
for their affection, encouragement and support.
D. Anilkumar
iii
ABSTRACT
KEYWORDS: Mixed convection; Self-similar solution; Non-similar solution;
Heat and mass transfer; Quasilinearization.
This thesis presents a detailed numerical study on unsteady mixed convection
flows such as flow over a rotating sphere, rotating cone in a co-rotating fluid and a moving
vertical slender cylinder.
A new self-similar solution of the unsteady mixed convection boundary layer flow
in the stagnation point region of a rotating sphere has been obtained with constant wall
temperature and constant heat flux conditions. The basic governing partial differential
equations with three independent variables have been reduced to the ordinary differen-
tial equations with one independent variable using suitable similarity transformations.
The resulting ordinary differential equations are solved by converting them into a matrix
equation through the application of an implicit finite difference scheme in combination
with the quasilinearization technique. The results indicate that the buoyancy forces cause
considerable overshoot in velocity profiles for low Prandtl number fluid and for a fixed
buoyancy force, the thermal boundary layer thickness reduces with the increase of Prandtl
number due to the lower thermal conductivity. The surface shear stress and heat trans-
fer parameters for the constant heat flux case are more than those of the constant wall
temperature case.
A detailed numerical study to obtain a new self-similar solutions for unsteady
mixed convection flow on a rotating cone in a rotating fluid has been carried out with
prescribed wall temperature and prescribed heat flux conditions. The transformed system
of non-linear coupled ordinary differential equations has been solved numerically using an
implicit finite difference scheme with the combination of quasilinearization technique. The
higher buoyancy force suppresses the oscillations in the velocity profiles which appears
due to surplus convection of angular momentum in lower buoyancy force. The increase
in Prandtl and Schmidt numbers causes a reduction in the thickness of thermal and
concentration boundary layers, respectively. Due to the increase in the ratio of buoyancy
forces, the skin friction coefficients, Nusselt and Sherwood numbers increase.
iv
Unsteady mixed convection flow on a rotating cone in a rotating fluid due to the
combined effects of thermal and mass diffusion has been studied numerically to obtain
semi-similar solutions for both prescribed wall temperature and prescribed heat flux con-
ditions. Accelerating and decelerating angular velocities are considered in the analysis.
The semi-similar solution of the coupled non-linear partial differential equations governing
the mixed convection flow has been obtained numerically using the method of quasilin-
earization technique and an implicit finite difference scheme. The results show buoyancy
force enhances the skin friction coefficients, and Nusselt and Sherwood numbers. For a
fixed buoyancy force, the Nusselt and Sherwood numbers increase with Prandtl number
and Schmidt number, respectively. It is observed that the skin friction coefficients increase
with time for an increasing angular velocity but for decreasing angular velocity the trend
is reverse.
Non-similar solution of an unsteady mixed convection flow over a continuously
moving slender cylinder has been obtained for both accelerating and decelerating free-
stream velocities using an implicit finite difference scheme in combination with the quasi-
linearization technique. Results show that both the skin friction and heat transfer coeffi-
cients increase with suction, but decrease with the increase of injection. In contrast, the
effect of wall velocity is to decrease the skin friction but to increase the heat transfer rate
due to a slip over the surface. Further, it is noted that the curvature parameter steepens
both the velocity and temperature profiles. The heat transfer rate is found to depend
strongly on viscous dissipation but the skin friction is little affected by it.
v
CONTENTS
LIST OF TABLES ix
LIST OF FIGURES x
NOTATION xv
1 INTRODUCTION 1
1.1 HISTORICAL DEVELOPMENT AND THE CONCEPT OF BOUND-
ARY LAYER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 UNSTEADY LAMINAR BOUNDARY LAYERS . . . . . . . . . . . . . . 5
1.3 ROTATING BOUNDARY LAYERS . . . . . . . . . . . . . . . . . . . . . 6
1.4 HEAT TRANSFER AND CONVECTION . . . . . . . . . . . . . . . . . . 8
1.5 MIXED CONVECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 MASS TRANSFER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 MATHEMATICAL BASIS . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 BOUNDARY LAYER EQUATIONS . . . . . . . . . . . . . . . . . . . . . 16
1.9 METHOD OF SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.10 OBJECTIVES AND SCOPE OF THE THESIS . . . . . . . . . . . . . . . 26
2 SELF-SIMILAR SOLUTION TO UNSTEADY MIXED CONVECTION
FLOW ON A ROTATING SPHERE 29
2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . 32
2.3 PROBLEM FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 METHODS OF SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 OUTLINE OF QUASILINEARIZATION TECHNIQUE . . . . . . 38
2.4.2 APPLICATION OF QUASILINEARIZATON TECHNIQUE WITH
IMPLICIT FINITE DIFFERENCE SCHEME . . . . . . . . . . . . 41
2.5 RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 SELF-SIMILAR SOLUTION TO UNSTEADY MIXED CONVECTION
FLOW ON A ROTATING CONE 62
3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . 64
3.3 MATHEMATICAL FORMULATION . . . . . . . . . . . . . . . . . . . . . 65
3.4 METHOD OF SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 UNSTEADY DOUBLE DIFFUSIVE CONVECTION FROM A
ROTATING CONE IN A ROTATING FLUID 91
4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . 92
4.3 MATHEMATICAL FORMULATION OF THE PROBLEM . . . . . . . . 93
4.4 METHOD OF SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5 UNSTEADY MIXED CONVECTION FROM A MOVING VERTICAL
SLENDER CYLINDER 124
5.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . 127
vii
5.3 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 METHOD OF SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . 138
5.6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
BIBLIOGRAPHY 151
viii
LIST OF TABLES
2.1 Comparison of the results (f ′′(0),−g′(0),−θ′(0)) with those of Lee et al. [70] 48
2.2 Surface shear stresses and heat transfer parameters (f ′′(0),−g′(0),−θ′(0))
for CWT case when Pr = 0.7 and A∗ = 1 . . . . . . . . . . . . . . . . . . . 53
2.3 Surface shear stresses and heat transfer parameters (f ′′(0),−g′(0),−θ′(0))
for CWT case when Pr = 0.7 and λ1 = 1 . . . . . . . . . . . . . . . . . . . 56
2.4 Surface shear-stresses and heat transfer parameter (F ′′(0),−G′(0), 1Θ(0)
) for
CHF case when Pr = 0.7 and A∗ = 1 . . . . . . . . . . . . . . . . . . . . . 61
3.1 Comparison of the results (−f ′′(0),−g′(0),−θ′(0)) with those of Himasekhar
et al.[48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2 For CWT case, Skin friction coefficients, Nusselt number and Sherwood
number (CfxRe1/2x , CfyRe
1/2x , NuRe
−1/2x , ShRe
−1/2x ) when N=1,Pr=0.7,Sc=0.94
and s=0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 Skin friction coefficients, Nusselt and Sherwood numbers (CfxRe1/2x , CfyRe
1/2x ,
NuRe−1/2x , ShRe
−1/2x ) when N=1, α1 = 0.5, s=0.5 and λ1 = 1 for PWT case. 82
3.4 Skin friction coefficients, Nusselt number and Sherwood number (CfxRe1/2x ,
CfzRe1/2x , NuRe
−1/2x , ShRe
−1/2x ) for PHF case when N∗ = 1, s=0.25 and
α1 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1 Comparison of the steady state results (fηη(ξ, 0),−θη(ξ, 0)) with those of
Chen and Mucoglu[22], Takhar et al.[130]. . . . . . . . . . . . . . . . . . . 139
LIST OF FIGURES
1.1 Orthogonal coordinates with their velocities . . . . . . . . . . . . . . . . . 28
2.1 Physical model and co-ordinate system. . . . . . . . . . . . . . . . . . . . . 33
2.2 Comparison of the results (f ′, g) for A∗ = λ = 1, λ1 = 0, P r = 0.7 . . . . . 50
2.3 Effect of A∗ on velocity profiles (f ′) for CWT case when λ = λ1 = 1, P r = 0.7 50
2.4 Effect of A∗ on velocity profiles (g) for CWT case when λ = λ1 = 1, P r = 0.7 51
2.5 Effect of A∗ on velocity profiles (θ) for CWT case when λ = λ1 = 1, P r = 0.7 51
2.6 Effect of λ on velocity profiles (f ′, g) for CWT case when A∗ = λ1 =
1, P r = 0.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.7 Effect of λ on velocity profiles (θ) for CWT case when A∗ = λ1 = 1, P r =
0.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.8 Effects of λ1 and Pr on velocity profiles (f ′) for CWT case when A∗ = λ = 1 54
2.9 Effects of λ1 and Pr on velocity profiles (g) for CWT case when A = λ = 1 54
2.10 Effects of λ1 and Pr on velocity profiles (θ) for CWT case when A∗ = λ = 1 55
2.11 Variations of surface shear stress parameter (f ′′(0)) with A∗ for
CWT case when λ1 = 1 and Pr = 0.7 . . . . . . . . . . . . . . . . . . . . 55
2.12 Variations of surface shear stress parameter (−g′(0)) with A∗ for
CWT case when λ1 = 1 and Pr = 0.7 . . . . . . . . . . . . . . . . . . . . 58
2.13 Variations of heat transfer parameter (−θ′(0)) with A∗ for CWT
case when λ1 = 1 and Pr = 0.7 . . . . . . . . . . . . . . . . . . . . . . . . 58
2.14 Effects of λ1∗ and Pr on F ′ and G for CHF case when A∗ = λ = 1. . . . . 60
2.15 Effects of λ1∗ and Pr on temperature profile (Θ), for CHF case when
A∗ = λ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.1 Physical model and co-ordinate system. . . . . . . . . . . . . . . . . . . . . 66
3.2 Effect of α1 on velocity profiles (f ′) for PWT case when s = 0.5, N = λ1 =
1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Effect of α1 on velocity profiles (f ′) for PWT case when s = 0.5, N =
1, λ1 = 3, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 Effect of α1 on velocity profiles (g) for PWT case when s = 0.5, N = λ1 =
1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5 Effect of α1 on velocity profiles (f ′) for PWT case when s = 0.5, N =
1, λ1 = 3, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . 80
3.6 Effects of Pr and Sc on θ for CWT case when λ1 = 1, α1 = 0.25, s =
0.5 and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.7 Effects of Pr and Sc on φ for CWT case when λ1 = 1, α1 = 0.25, s =
0.5 and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.8 Effect of N on skin friction coefficient (CfxRe1/2x ) for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . 84
3.9 Effect of N on skin friction coefficient (CfyRe1/2x ) for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . 84
3.10 Effect of N on heat transfer coefficient (NuRe−1/2x ) for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . 85
3.11 Effect of N on mass transfer coefficient (ShRe−1/2x ) for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . 85
3.12 Effects of λ1∗ and Pr on Θ for PHF case when α1 = 0.5, s = 0.5, N∗ =
1 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.13 Effects of λ1∗ and Sc on Φ for PHF case when α1 = 0.5, s = 0.5, N∗ =
1 and Pr = 0.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
xi
3.14 Effects of Pr and Sc on Θ for PHF case when λ∗1 = 1, α1 = 0.5, s =
0.5 and N∗ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.15 Effects of Pr and Sc on Φ for PHF case when λ∗1 = 1, α1 = 0.5, s =
0.5 and N∗ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1 Comparison of the results (CfxRe1/2x , 2−1CfyRe
1/2x , Re
−1/2x Nux). . . . . . . 109
4.2 Effects of λ1 and α1 on velocity profiles (−fη) for PWT case when t∗ =
1, N = 1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . 110
4.3 Effects of λ1 and α1 on velocity profiles (g) for PWT case when t∗ = 1, N =
1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4 Effects of λ1 and α1 on temperature profiles (θ) for PWT case when t∗ =
1, N = 1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . 111
4.5 Effects of λ1 and α1 on concentration profiles (φ) for PWT case when
t∗ = 1, N = 1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . 111
4.6 Effects of λ1 and α1 on skin friction coefficients (CfxRe1/2x ) for PWT case
when ε = 0.2, N = 1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . 112
4.7 Effects of λ1 and α1 on skin friction coefficients (2−1CfyRe1/2x ) for PWT
case when ε = 0.2, N = 1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . 112
4.8 Effects of λ1 and α1 on Re−1/2x Nux for PWT case when s = 0.5, N = λ1 =
1, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.9 Effects of λ1 and α1 on Re−1/2x Shx for PWT case when s = 0.5, N = 1, λ1 =
3, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.10 Effect of N on skin friction coefficient (CfxRe1/2x ) for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . 115
4.11 Effect of N on skin friction coefficient (CfyRe1/2x ) for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . 115
xii
4.12 Effect of N on heat transfer coefficient (NuRe−1/2x ) for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . 116
4.13 Effect of N on mass transfer coefficient (ShRe−1/2x ) for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75, P r = 0.7 and Sc = 0.94 . . . . . . . . . . . . . 116
4.14 Effects of Pr and Sc on θ for CWT case when λ1 = 1, α1 = 0.25, s =
0.75 and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.15 Effects of Pr and Sc on φ for CWT case when λ1 = 1, α1 = 0.25, s =
0.75 and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.16 Effects of Pr and Sc on Re−1/2x Nux for PWT case when λ1 = 1, α1 =
0.25, s = 0.75 and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.17 Effects of Pr and Sc on Re−1/2x Shx for PWT case when λ1 = 1, α1 =
0.25, s = 0.75 and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.18 Effects of λ1∗ and Pr on Θ for PHF case when α1 = 0.5, s = 0.75, N∗ =
1 and Sc = 0.94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.19 Effects of λ1∗ and Sc on Φ for PHF case when α1 = 0.5, s = 0.75, N∗ =
1 and Pr = 0.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.20 Effects of Pr and λ∗1 on Re1/2x Cfx for PHF case when α1 = 0.5, Sc =
0.94 and N∗ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.21 Effects of Pr and λ∗1 on Re1/2x Cfy for PHF case when α1 = 0.5, Sc =
0.94and N∗ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.1 Physical model and coordinate system . . . . . . . . . . . . . . . . . . . . . 127
5.2 Effects of λ1 and Pr on F for φ(t∗) = 1 + εt2, ε = 0.5 when Ec = 0.1, α2 =
1, ξ = 0.5, A = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 Effects of λ1 and Pr on θ for φ(t∗) = 1 + εt2, ε = 0.5 when Ec = 0.1, α2 =
1, ξ = 0.5, A = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
xiii
5.4 Effect of ξ on F and θ for φ(t∗) = 1 + εt2, ε = 0.5 when λ1 = 1, P r =
0.7, Ec = 0.1, α2 = 1, A = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.5 Effects of λ1 and ξ on Re1/2x Cf for φ(t∗) = 1 + εt2, ε = 0.5 when Pr =
0.7, Ec = 0.1, α2 = 1, A = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.6 Effects of λ1 and ξ on Re−1/2x Nu for φ(t∗) = 1 + εt2, ε = 0.5 when Pr =
0.7, Ec = 0.1, α2 = 1, A = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.7 Effects of A and α2 on Re1/2x Cf for φ(t∗) = 1 + εt2, ε = 0.5 when λ1 =
1, P r = 0.7, Ec = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.8 Effects of A and α2 on Re1/2x Cf and Re
−1/2x Nu for φ(t∗) = 1 + εt2, ε = 0.5
when λ1 = 1, P r = 0.7, Ec = 0.1. . . . . . . . . . . . . . . . . . . . . . . . 144
5.9 Effects of A and α2 on F for φ(t∗) = 1 + εt2, ε = 0.5 when λ1 = 1, P r =
0.7, ξ = 0.5, Ec = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.10 Effect of Pr on Re1/2x Cf for φ(t∗) = 1 + εt2, ε = 0.5 and ε = −0.5 when
λ1 = 1, Ec = 0.1, α2 = 1, A = 0, ξ = 0.5. . . . . . . . . . . . . . . . . . . . . 146
5.11 Effect of Pr on Re−1/2x Nu for φ(t∗) = 1 + εt2, ε = 0.5 and ε = −0.5 when
λ1 = 1, Ec = 0.1, α2 = 1, A = 0, ξ = 0.5. . . . . . . . . . . . . . . . . . . . . 146
5.12 Effect of Ec on Re1/2x Cf for φ(t∗) = 1 + εt2, ε = 0.5 and ε = −0.5 when
λ1 = 1, P r = 0.7, α2 = 1, A = 0, ξ = 0.5. . . . . . . . . . . . . . . . . . . . . 147
5.13 Effect of Ec on Re−1/2x Nu for φ(t∗) = 1 + εt2, ε = 0.5 and ε = −0.5 when
λ1 = 1, P r = 0.7, α2 = 1, A = 0, ξ = 0.5. . . . . . . . . . . . . . . . . . . . . 147
5.14 Effects of Pr on θ for φ(t∗) = 1 + εt2, ε = 0.5 when λ1 = 1, A = 0, α2 =
1, ξ = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.15 Effects of Ec on θ for φ(t∗) = 1 + εt2, ε = 0.5 when λ1 = 1, A = 0, α2 =
1, ξ = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
xiv
NOTATION
A,A∗ dimensionless mass transfer and acceleration parameter, respectively
C, D species concentration and mass diffusivity, respectively
Cf , Cfx, Cfy skin friction coefficients for CWT case
Cfx, Cfy skin friction coefficients for CHF case
Cp specific heat at constant pressure
Ec Eckert number
f ′, g dimensionless velocity components for CWT case
F ′, G dimensionless velocity components for CHF case
g∗ acceleration due to gravity
Gr1, Gr2 Grashof numbers due to temperature and concentration
distributions, respectively, for PWT case.
Gr1∗, Gr2
∗ Grashof numbers due to temperature and concentration
distributions, respectively, for PHF case.
k thermal conductivity
L characteristic length
N,N∗ ratio of the Grashof numbers for PWT and PHF cases, respectively
Nux, Nux Nusselt numbers for PWT and PHF cases, respectively
Pr Prandtl number
r(x) radius of the section normal to the axis of the sphere at a distance x
R radius of the slender cylinder
R(t∗) function of t∗ with first order continuous derivative
Rex Reynolds number based on x
s unsteady parameter
Sc Schmidt number
Shx, Shx local Sherwood numbers for PWT and PHF cases, respectively
t, t∗ dimensional and dimensionless times, respectively
T temperature
u, v, w dimensional velocity components in x-, y- and z- directions,
respectively
xv
Greek symbols
α∗ semi vertical angle of the cone
α1 ratio of the angular velocities of the cone to the free stream
α2 ratio of the wall velocity to the fluid velocity
β, β∗ volumetric coefficients of the thermal and
concentration expansions, respectively
ε constant
θ, φ dimensionless temperature and concentration, respectively, for PWT
Θ, Φ dimensionless temperature and concentration,respectively, for PHF
4η,4x step sizes in η- and x- directions, respectively
η∞ edge of the boundary layer
η, ξ similarity variables
λ1, λ2 buoyancy parameters due to the temperature and concentration
gradients, respectively, for PWT case
λ1∗, λ2
∗ buoyancy parameters due to the temperature and concentration
gradients, respectively, for PHF case
µ, ν dynamic and kinematic viscosities, respectively
ρ density
ψ dimensional stream function
Ω1, Ω2 angular velocity of the cone and the free stream fluid, respectively
Ω(= Ω1 + Ω2) composite angular velocity
Subscripts
i initial condition
o value at the wall for t∗ = 0
t, x, z denote the partial derivatives w.r.t to these variables, respectively.
∞ conditions in the free stream
e, w conditions at the edge of the boundary layer and on the surface,
respectively
t∗, η, ξ partial derivatives with respect to these variables
xvi
CHAPTER 1
INTRODUCTION
1.1 HISTORICAL DEVELOPMENT AND THE CONCEPT OF BOUND-
ARY LAYER
The importance of fluid dynamics was understood even as early as in 1588 during
the naval battle between English and Spanish ships. This naval battle is of particular
importance because it was the first in history to be fought by ships on both sides powered
completely by sail and it taught the world that political power was going to be synonymous
with naval power. In turn, naval power was going to depend greatly on the speed and
maneuverability of ships. To increase the speed of a ship, it is important to reduce the
resistance created by the water flow around the ships hull. Suddenly, the drag on ship
hulls became an engineering problem of great interest, thus giving impetus to the study
of fluid mechanics.
The impetus hit its stride almost a century later, when, in 1687, Isaac Newton
published his famous “principia”, in which the entire second book was devoted to fluid
mechanics. Newton encountered the same difficulty as others before him, namely, that the
analysis of fluid flow is conceptually more difficult than the dynamics of solid bodies. A
solid body is usually geometrically well defined and its motion is therefore relatively easy
to describe. On the other hand, a fluid is a “squishy” substance, and in Newton’s time it
was difficult to decide even how to qualitatively model its motion let alone to quantitative
relationships. Newton considered a fluid flow as a uniform, rectilinear stream of particles,
much like a cloud of pellets from a shotgun blast. Newton assumed that upon striking
a surface inclined at an angle θ to the stream, the particle would transfer their normal
momentum to the surface but their tangential momentum would be preserved. Hence,
after collision with the surface , the particles would then move along the surface. This led
to an expression for the hydrodynamic force on the surface which varies as sin2θ. This
1
is Newton’s famous sine-squared law. Although its accuracy left much to be desired, its
simplicity led to wide application in naval architecture.
Later, in 1977, a series of experiment was carried out by Jean le Rond D’Alembert,
in order to measure the resistance of the ships in canals. The result showed that the rule
that of oblique planes resistance varies with the sine squared of the angle of incidence
holds good only for angles between 50 degree to 90 degree and must be abandoned for
laser angles. Also, in 1781, Leonard Euler pointed out the physical inconsistency of
Newton’s model consisting of a rectilinear stream of particles impacting without warning
on a surface. In contrast to this model, Euler noted that the fluid moving toward a body
before reaching the later, bends its direction and its velocity so that when it reaches the
body it flows past it along the surface and exercises no other force on the body except the
pressure corresponding to the single points of contact. Euler went on to present a formula
for resistance which attempted to take into account the shear stress distribution along
the surface, as well as the pressure distribution. This expression became proportional to
sin2θ for large incident angles, where as it was proportional to sinθ at small incidence
angles. Euler noted that such a variation was in reasonable agreement with the ship-hull
experiments carried out by D’Alembert. This early work in fluid dynamics has now been
superseded by modern concepts and techniques.
The major point here is that the rapid rise in the importance of naval architec-
ture after the sixteenth century made fluid dynamics an important science, occupying
the minds of Newton, D’Alembert, Euler and among many others. At the end of the
seventeenth century, the discovery of fundamental laws and equations of dynamics by
Newton prepared the ground for a great advance in the devolvement of mechanics of
fluids. The extension of these fundamental laws to continuous media and particularly
to fluid led to the formation of an independent branch of theoretical mechanics, namely
hydrodynamics. In a way the word “fluid” is used to denote either a liquid or a gas.
When a force is applied tangentially to the surface of a solid, the solid will experienced a
finite deformation, and the tangential force per unit area-the shear stress-will usually be
proportional to the amount of deformation. In contrast, when a tangential shear stress is
applied to surface of fluid, the fluid will experience a continuously increasing deformation,
2
and the shear stress usually will be proportional to the rate of change of the deformation.
The most fundamental distinction between solids, liquids and gases is at the atomic and
molecular level. The movement of molecules in both gases and liquids leads to similar
physical characteristics, the characteristics of a fluid-quite different from those of solid.
Therefore, it makes sense to classify the study of the dynamics of the liquids and gases
under the same general heading, called “fluid dynamics” or in early development known
as Hydrodynamics.
Until the end of the nineteenth century, Fluid Mechanics was mostly concerned
with an “ideal” or inviscid fluid which is governed by the Euler equation of motion.
Although the mathematical theory due to the Euler equation provided us with useful
concepts and elegant theorems on fluid motions, it often brought about serious discrep-
ancies between the physical reality such as the mathematical singularities of the solutions
and the un-physical results like the famous D’Alembert’s paradox of the absence of drag
on a moving body. In fact, the results of this classical hydrodynamics stood in glaring
contradiction to the experimental results and it can’t explain the phenomena of drag,
of pressure loss in pipes, or of flow separation, all of which are direct or indirect effects
of the viscosity of the flow media. For example, according to the inviscid theory, any
body moving uniformly through a fluid medium which extends to infinity experiences no
drag!(D’Alembert’s paradox). Thus, the main limitations of ideal fluid mechanics are (1)
the theory is unable to predict flow separation, (2) the theory is unable to explain the
existence of wake, (3) the pressure distribution according to this theory produces no total
force, (4) there is no viscous force, (5) the analysis of heat transfer is unsatisfactory and
(6) the solutions are unable to fulfill the required boundary conditions.
The engineers of eighteenth and nineteenth centuries knew how to help themselves
by accounting for the viscosity effects through empirical laws. They developed the engi-
neering science of hydraulics as a practical extension of the classical hydrodynamics. Not
that there were no efforts being made to calculate theoretically the influence of viscosity
on the motion of fluids. As early as in the middle of the nineteenth century, Navier and
Stokes succeeded in formulating the conditions for the equilibrium of forces(viscous forces,
inertial forces, pressure and body forces) mathematically. The results was a system of a
3
non-linear partial differential equations of second order. However, extraordinary mathe-
matical difficulties(with the exception of a small number of particular cases) prevented
any useful application of these equations for a half century.
It was in 1904, Prandtl decisively influenced the theory of viscous flows by his
proposal to simplify the Navier-Stokes equations. With the aid of theoretical considera-
tions and several simple experiments, he proved that viscosity, as long as it is small(more
precisely for large Reynolds number, Re), affects the flow only in a relatively thin bound-
ary layer in the vicinity of solid walls. In this boundary layer, the velocity rises from zero
at the wall(no slip) to practically the full value of the unperturbed external velocity. The
increase of velocity with increasing distance from the solid surface involves relative move-
ment between the particles in the boundary layer, and the shear stresses are in evidence.
Since the layer is usually very thin the velocity gradients is high, and the shear stresses
are therefore important. It amounts to the statement that there exists two regions in the
flow field namely, a thin inner region which accounts for the effect of viscous force and
an outer region where the flow can be considered as if it is inviscid. This subdivision of
the flow field such as inner and outer regions reduces the highly non-linear Navier Stokes
equations to what are known as “boundary layer equations” which are considerably sim-
pler to manipulate than the full equations of motion. In fact, the equation for stream
function is reduced from an elliptic to a parabolic form. Boundary layer theory explains
the many important flow phenomena like skin friction, heat transfer, flow separation and
no slippage between the body and the fluid. Prandtl’s concept of the boundary layer,
where the influence of viscosity is confined, has bridged the gap between classical hydro-
dynamics and the observed behaviour of the real fluids. The boundary layer is usually
very thin but may sometimes be observed with the naked eye. Close to the side of a ship,
for example, is a narrow band of water with a velocity relative to the ship clearly less
than that of water further away. Some other examples of boundary layer will make the
concept of clear. (1) The jet warm air created by blowing hot air through an orifice into
a stagnant reservoir or co-current stream of cold air, is an example of boundary layer.
(2) When air flows steadily over an aerofoil there is always a boundary layer near the
leading edge. The rapid advances in mechanics of fluids in the twentieth century are
4
largely due to this important concept of boundary layer . The boundary layer theory was
first developed for the case of laminar incompressible fluids but later, it was extended to
include compressible flows also. The development of the theory of boundary layer flow
in a compressible stream was stimulated by the progress in aeronautical engineering and
in recent times, by the development of rockets and artificial satellites. The idea of com-
pressible boundary layer follows the same path as for the incompressible boundary layer ,
namely that the thickness of the boundary layer is assumed to be small compared with
a characteristic dimensions in the flow direction. However, the work of compression and
energy dissipation produces considerable increase in temperature as the flight velocities
are attained to the value of the order of multiples of velocity of sound. Hence in case
of compressible flow, thermal effect plays an essential part and consequently both the
physics and mathematics of the boundary layer theory are more complicated than for
incompressible flow.
In the following sections, we shall give a brief outline of few important concepts of
boundary layer theory which have appeared sequentially in the literature in the last hun-
dred years since the inception of Prandtl’s concept in 1904. An excellent account of the
development of the boundary layer , as it is called, has been presented by Tani[131] in a re-
view article. For details, readers may refer to several excellent books written by Evans[30],
Moore[90], Pai[98], Rosenhead[107], Schetz[111], Schlichting[112] and Sobey[119].
1.2 UNSTEADY LAMINAR BOUNDARY LAYERS
In unsteady or non-steady flow, the flow variables at a particular point do vary with time.
Such variations add considerable difficulties in solving the problems of unsteady flows.
The study of unsteady boundary layer owes its importance to the fact that all boundary
layers which occur in practice are, in a sense, unsteady. One or more of the following
circumstances may prevail: either the time which has elapsed following the initiation of
the motion is not large or there are fluctuations in the mainstream velocity(which itself
may have zero mean), or the boundary layer is unstable. But at present the analysis
considers the effects of unsteadiness for longer periods of time. Examples of unsteady
5
motion are flow through turbo-machinery blades, dynamic stall of lifts surfaces, stall
flutter of helicopter rotor blades, rotating stall in engine compressors, accelerated and
decelerated rocket missiles and nozzles.
Problems of unsteady flow may be classified into three categories, according to
the rate at which the changes occur. In the first group are problems in which the changes
of mean velocity although significant, take place slowly enough for the forces causing the
temporal acceleration to be negligible compared with other forces involved. An example
of this sort of problem is the continuous filling or emptying of a reservoir. The second
category embraces problems in which the flow changes rapidly enough for the forces pro-
ducing temporal acceleration to be important: this happens in reciprocating machinery,
such as positive displacement pumps and in hydraulic and pneumatic servo-mechanisms.
In the third group may be placed those instances in which the flow is changed so quickly,
as for example by the sudden opening or closing of a valve, that elastic forces become
significant. Unsteady flow, where the unsteadiness in the flow field is created by impulsive
motion of a body, is an important class of flows in this group. The study of flow due to
impulsive motion is dealt in references[39, 100, 105, 132].
Unsteadiness can also occur due to the time dependent body motion or distur-
bances in the surrounding flow filed. This happens when the free stream velocity or the
velocity distribution of the body motion or the surface mass transfer or the wall tem-
perature or all vary arbitrarily with time. Our studies, presented in this thesis, pertain
to the unsteady flow of this type. Many detailed studies of unsteady flows for both in-
compressible and compressible fluids have been reviewed by Coustiex[26], McCroskey[81],
Riley[105] and Telionis[132]. These have provided essentially complete theoretical un-
derstanding of unsteady laminar boundary layer flows. Pop[100] presented an excellent
survey on the theory of unsteady flows.
1.3 ROTATING BOUNDARY LAYERS
Rotational flows of fluids are encountered very often in natural phenomena such as tor-
nadoes, cyclones, whirlpool and dust devils. Earth rotation along with its oceans and
6
atmosphere is also an example. In man-made machines, the flow and heat transfer char-
acteristics of spinning bodies of revolution in a forced flow stream are important in a
variety of engineering applications such as re-entry of missiles, projectile motion, fibre
coating, centrifugal pumping and rotating machinery design. It is, therefore, understand-
able why rotational or vortex flows have attracted the attention of many investigators and
comprehensive reviewes on this topic were reported in the early studies on fluid dynamics
by Greenspan[38], Kreith[65] and Lighthill[74].
The rotational flows of fluids may be broadly classified into two groups, depending
on whether the flow is internal or external. If the rotating fluid is confined within a solid
enclosures, it is termed as confined vortex. The other class of rotational motion involves
a rotating body placed in an unbounded fluid. One interesting by-product of a rotational
motion in a fluid is the secondary flow. Such flows occur, for example, when a fluid rotates
over a stationary disc. Centrifugal force balances the radial pressure gradient in the fluid
at a distance from the disc. Near the wall, in the thin boundary layer, viscous forces
slow down the rotation. Therefore, the reduced centrifugal force can no longer balances
the radial pressure gradient towards the axis. As a result, fluid particles move inward
and because of continuity, an axial flow is generated. This flow in the boundary layer,
which is distinct from the external flow, is known as the secondary flow. An example of
a consequence of secondary flow is the accumulation of tea leaves in a heap around the
center of a stirred tea-cup. In the converse case of a disc rotating in a fluid at rest, the
secondary flow takes place because fluid particles adjacent to the rotating disc are ejected
radially outward by a centrifugal action. In the study of vortex motions, geophysical
vortices form a special class which is concerned with different atmospheric and oceanic
vortices. Theoretical investigations of such vortices may help scientists to understand the
process that goes on in the atmosphere and enable the meteorologists to make a long-term
weather prediction. The concept of a stratified fluid is useful in the study of geophysical
vortices. The review by Monin[88] and a book written by Azad[5] give a good overview of
the subject. Most recent developments in the study of rapidly rotating spherical systems
with applications to magnetohydrodynamics are reviewed by Zhang and Schubert[150].
Motivated by the practical importance of the study of rotational flows, one such problem
7
in rotating boundary layers, which has been presented in the second and third chapter of
this thesis, has been investigated.
1.4 HEAT TRANSFER AND CONVECTION
People have always understood that something flows from hot objects to cold ones known
as flow of heat. Heat transfer generally defined as the transmission of energy from one
region in a medium to another as a consequence of a temperature difference between them.
Heat transfer occurs in three modes- conduction, convection and radiation. Conduction
perhaps the simplest of all. In a solid, atoms are bonded together in a nice, stable manner.
If one side of a solid is kept ‘hot’, and the other side is kept cold, it is expected that the
heat transferred through the solid from the hot side to the cold side. Hot atoms at the
hot side are wiggling around more than atoms on the cold side, due to the extra thermal
energy they have. These atoms bump into the atoms next to them, losing a bit of their
energy, giving it to their neighbours, and, in turn, heating those atoms up a bit. The chain
reaction proceeds in this manner across the solid, and eventually a nice, smooth variations
from hot to cold is established. Different materials have varying abilities to conduct heat.
Conduction occurs similarly in liquids and gases, though fluid motion, when it occurs,
complicates things a bit. Convection seems very different than conduction but, as its
heart, convection relies quite fundamentally on conduction. Convection involves a liquid
or gas moving past a surface that is either hotter or colder than the fluid. In the convection
process(assuming a hot surface and cold air), heat is conducted from the surface to the air
molecules that are on the surface. Since the air is moving, these molecules are constantly
being replaced by new cold air molecules. The faster the air is moving, the better the heat
transfer . Heat transfer occurs by radiation across the vast nothingness of space from the
Sun to Earth’s atmosphere. Radiation does not rely on any type of medium to transport
heat. Thermal radiation is an electromagnetic phenomenon and occurs in the very high
temperatures. In many practical situations, it is due to the joint action of several of these
mechanism, and not just one mode, the heat is transmitted.
Convection deals with the problems of fluid flow involving heat transfer between
8
a surface and a moving fluid maintained at different temperatures. In reality, this is a
combination of diffusion and bulk motion of molecules. Viscous forces result in the fluid
being brought to rest at the wall, where conduction comes into play to transfer heat to
or from the fluid. A study of such flows assumes significance in view of its applicability
in a broad spectrum of fields such as meteorology, geophysics, jet propulsion, high speed
flight, turbo-machinery and nuclear reactors. Heat transfer in a fluid is usually caused
by an interplay between conduction and convection. If the body is at high temperature
than the fluid, in the neighbourhood of the wall heat is transferred from body to fluid by
conduction, and then the heated particles of the fluid carry heat (convection) to transfer
it to cooler fluid particles. When the conductivity of the fluid is small, which is true
in ordinary fluids, the heat transport due to conduction is comparable to that due to
convection only across a thin layer near the surface of the body. This means that the
temperature field which spreads from the body extends, essentially, over a narrow zone
in the immediate neighbourhood of its surface, whereas the fluid at a larger distance
from the surface is not materially affected by the hot body. Thus, for fluids flowing
past heated or cooled bodies the heat transfer occurs principally within a narrow region
(thin layer) of fluid adjacent to the surface of the body, called thermal boundary layer
distinguishing itself from momentum boundary layer. Within this layer, similar to that in
the momentum boundary layer, the temperature changes from its value at the wall to the
free stream value at the edge of the boundary layer. The temperature distribution in this
layer is determined by the combined effects of the motion of the fluid, viscous dissipation
and heat conduction. Further, the temperature distribution is governed by the thermal
energy equation[112]. The relative thickness of momentum and thermal boundary layer is
characterized by the non-dimensional parameter, Prandtl number, represented by Pr. The
Prandtl number is defined as a ratio of coefficient for diffusion of momentum to diffusion
of heat. In most applications, it is essential to determine the quantity of heat exchanged
between the body and the free stream. Convection is an important mode of heat transfer .
It’s diverse applications include meteorology, jet propulsion, geophysics, high-speed flight,
turbo-machinery and molecular reactors.
Although initially the field of heat transfer was concerned with power plant and
9
air craft development, there has been an extensive diversification in the last three to four
decades. The severity of the energy and ecology problem facing society has looked for
alternative sources and could be said to be one of the major reasons for the growing
interest in buoyancy driven transport phenomena. Momentum and heat transfer are
generally studied together due to the close linkage existing between fluid friction and
convection phenomena. For example, in heat exchangers in industrial boilers and reentry
vehicles, the interaction between the velocity field and the temperature field could be
either one-way or two-way. The convective mode of heat transfer is generally divided into
two basic processes. If the motion of the fluid arises from an external agent then the
process is termed as forced convection . If, on the other hand, no such externally induced
flow is provided and the flow arises from the effect of a density difference resulting from
a temperature or concentration difference in a body force field such as the gravitational
field, then the process is termed as the natural or free convection. The density difference
gives rise to buoyancy force which drives the flow and the main difference between the free
and forced convection lies in the nature of the fluid flow generation. In forced convection,
the externally imposed flow is generally known, whereas in free convection it results from
an interaction between the density difference and the gravitational field and is therefore
invariably linked with, and is dependent on the temperature field. Thus, the motion
that arises is not known at the onset and has to be determined. The governing non-
dimensional parameter for free convection flow is the Grashof number(Gr). On the other
hand, inertial force dominate in forced convection flows. The governing non-dimensional
parameter for such flows is Reynolds number(Re). In many instances, in spite of the
presence of a forced velocity, the buoyancy forces considerably modify the flow field and
hence the heat transfer rate from the surface of the body. In such situations, both the
buoyancy and inertial forces are of comparable magnitudes and such flows are termed
as mixed or combined convection flows. For details of boundary layer studies dealing
with convective heat transfer, readers may refer to excellent books written by Bejan[8],
Cebeci[12], Incropera and DeWitt[54] and Pop and Ingham [99].
10
1.5 MIXED CONVECTION
The requirement of modern technology have stimulated interest in fluid flows which in-
volve the interaction of several phenomena. A convection situation in which effects of
both forced and free convection are significant is commonly referred as mixed convec-
tion or combined convection. The effect is especially pronounced in situations where the
forced fluid flow velocity is moderate and/or the temperature difference is large. In mixed
convection flows, the forced convection effects and the free convection effects are of com-
parable magnitude. Thus, in this situation forced and free convection acts simultaneously,
that is, mixed convection occurs in which the effect of buoyancy forces on a forced flow
or the effect of forced flow on a buoyant flow is significant. Mixed convection occurs in a
variety of technological and industrial applications, such as electronic devices cooled by
fans, heat exchanges in low velocity environment, high temperature nuclear reactors, gas-
cooled during emergency shutdown and solar central receivers exposed to wind currents
and in many other cases of practical interest. The governing non-dimensional parameter
for such flows would be a ratio of powers of the Grashof number and Reynolds number.
Analysis indicate that the parameter that characterizes mixed convection flow is
GrRen , where the Grashof number (Gr) and the Reynolds number (Re) represent the vigor of
the natural convection and forced flow effects , respectively. The limiting case of GrRen → 0
and GrRen → ∞ correspond to the forced and natural convection limits, respectively. The
exponent n depend on the geometry , the thermal boundary condition and the fluid. To
analyze mixed convection flow, one requires to understand the concept of two limiting
regimes, natural or free convection and forced convection. The complexity of transport
is largely due to the interaction of the buoyancy force with the externally induced flow
field. If both of these effects are in the same direction, a higher transport rate will result.
However for some angles between the buoyancy force and the forced flow, the resultant
transport may be less than that which would arise with either effect alone. Boundary
layer approximations have been widely applied to analyze mixed convection flows. A
boundary layer mechanism often arises depending on the direction and magnitude of
the two interacting convective effects. References[36, 99] have discussed in detail some
particular applications of mixed convection flows. A scan of the literature and reviews
11
cited in references[12, 148] shows that unsteady mixed convection flows have not been
studied extensively. Hence, in the present thesis we concentrated on unsteady mixed
convection flows over various of geometries of engineering interest like rotating sphere,
rotating cone, slender cylinder, etc.
1.6 MASS TRANSFER
The convective heat transfer phenomena in nature are often accompanied by mass trans-
fer . That is, by the transport of a certain substance that acts as a component(constituent,
species) in the fluid mixture. Circulation of atmospheric air is in many cases driven by
differential heating but in an industrial area, these flow will also acts as a carrier for many
species masses put on by factories into the atmosphere. By the same token, certain ocean
currents driven by differential heats also act as freight trains for salt(in the form of saline
water). Beyond these environment engineering applications, convection mass transfer pro-
cesses alone(in the absence of heat transfer ) constitute the backbone of many operations
in the chemical industry. This seems like enough reason to include mass transfer in the
study of convective heat transfer .
Mass transfer concerns mixture of two or more species. Mass transfer is a move-
ment of particles of a given species through a mixture as a result of a gradient in the
concentration of that species. Diffusion is a mode of mass transfer arising from molec-
ular motion. Macroscopic motion, which we normally describe as “flow”, also provides
a mechanism of mass transfer . When we stir sugar into tea, or simply pour cream into
coffee, a process of homogenization occurs at a rate that depends primarily on the fluid
dynamics of the system and is not strongly dependent on diffusion. We refer to this mode
of mass transfer as convection.
The phenomenon of mass transfer is familiar in daily life. A drop of dye in water
spreads, dilutes, and soon colors the water in the entire container. An odor in one corner
of a room is soon noticeable throughout the room. A wet surface soon dries out, even
if there is no ambient air motion. Helium in a glass container seems to leak because it
is migrating by mass transfer through the solid walls. All of these are examples of the
12
diffusion of mass due to differences in concentration of substances in a mixture. Mass
transfer tends toward making a mixture uniform, just as heat transfer tries to make the
temperature uniform. Many industrial operations involve mass transfer . Examples in the
aerospace industry involve ablative, film, and transpiration cooling of vehicles and rocket
engines. Air and water pollution processes are diffusion controlled. Bio-engineering design
such as blood oxygenators, respirators, and artificial kidneys involve mass transfer . The
chemical industries have perhaps the strongest mass transfer emphasis: separation, extrac-
tion, gas absorption, distillation, adsorption, crystallization, ore and isotope production,
drying, humidification, chromatography and sublimation all depend strongly upon mass
transfer processes.
Mass transfer is one of several sciences which results from the interplay of conser-
vation laws and transfer laws. The phenomenon of mass transfer observed to be similar
to that of momentum and heat transfer . Clearly the science of mass transfer is a close
neighbour of thermodynamics and of heat transfer , indeed, heat transfer will be almost
appear to be a branch of mass transfer . Since material transport takes place most easily
between fluids in motion, and since its rate depends on the details of the flow pattern.
The territory of fluid mechanics partially overlaps to that of mass transfer . Mass transfer
often occurs in the presence of convective flow, and often the mass transfer is dominated
by the details of the flow field. Another way to look at this point is to emphasize that
we have the potential to control mass transfer by exercising some control over the fluid
dynamics. This is a very powerful design tool then, and so we need to learn something
about the concentration of fluid dynamics to mass transfer . The coupling between the
concentration and velocity fields is similar to that between the temperature and velocity
fields. Thus the processes of viscous resistance, convective heat transfer and mass trans-
fer are all interrelated. In fact, the concentration equation in the absence of chemical
reaction and the energy equation in the absence of internal heat generation are very sim-
ilar. The governing non-dimensional parameter Schmidt number(Sc), corresponds to the
Prandtl number . The Schmidt number is defined as a ratio of coefficient for diffusion of
momentum to diffusion of mass.
The practice of mass transfer as an industrial art is being observed in many prac-
13
tical applications. When a space-vehicle enters the dense layers of the atmosphere, ex-
tremely high temperatures are created due to the stagnation flow effect which is produced
at the nose or in the boundary layer along the wall. The quantity of heat transferred to
the vehicles can be reduced by injecting a light gas or fluid through a porous wall, which
would create a thin film along the walls. This is one example wherein both heat and mass
transfer occur simultaneously due to the combined effects of thermal diffusion of mass
species. In many natural and technological processes, temperature and mass or concen-
tration diffusion act together to create a buoyancy force which drives the fluid and this
is known as double-diffusive convection, or combined heat and mass transfer convection.
In oceanography, convection processes involve thermal and salinity gradients of temper-
ature and this is referred to as thermohaline convection, whilst surface gradients of the
temperature and the solute concentration are referred to as Maragoni’s convection. The
term double-diffusive convection is now widely accepted for all processes which involves
simultaneous thermal and concentration(solutal) gradients and provides as explanation
for a number of natural phenomena. Because of the coupling between the fluid velocity
field and the diffusive(thermal and concentration) fields, double-diffusive convection is
more complex than the convective flow which is associated with a single diffusive scalar,
and many different behaviours may be expected. Such double-diffusive processes occur
in many fields, including chemical engineering(drying, cleaning operations, evaporations,
condensation, sublimation, deposition of thin films, energy storage in solar ponds, roll-over
in storage tanks containing liquefied natural gas, solution mining of salt caverns for crude
oil storage, casting of metal alloys and photosynthesis), solid-state physics(solidification of
binary alloys and crystal growth), oceanography(melting and cooling near ice surfaces, sea
water intrusion into fresh water lakes and the formation of layered or columnar structures
during crystallization of igneous intrusions in the Earth’s crust), geophysics(dispersion
of dissolvent materials or particular matter in flows), etc. A clear understanding of the
nature of the interaction between thermal and mass or concentration buoyancy forces
are necessary in order to control these processes. A substantial literature survey on the
subject of double-diffusive convection has been made by Bejan[8], Brandt[10], Huppert
and Turner[52], Mahajan and Angirasa[79], Mongruel et al[89], and Ostrach[94],.
14
1.7 MATHEMATICAL BASIS
Ludwing Prandtl laid the foundation of his monumental theory in 1904 popularly known
as Boundary Layer Theory. Since it’s inception in literature, exactly a period of hun-
dred years is past from 1904 during which boundary layer theory has exerted a wide
and exceeding fruitful influence on almost all branches of fluid dynamics. In spite of the
overwhelming practical success of the boundary layer theory, not many attempts were
made to verify the soundness of the theory from a rigorous mathematical view point. Ar-
guments were based more on physical observations and to some extent on intution. Even
though solutions for several boundary layer problems were obtained, the questions of
existence and uniqueness of the solutions and the well-posedness of the problems were not
answered satisfactorily. It was only in last three decades that the questions of existence
of the solution, its uniqueness and well-posedness of the problem have been attempted
to answer for some simple cases. In fact, situations improved when it was verified that
Prandtl’s boundary layer equations can be derived mathematically by the singular per-
turbation method as a first order approximation of Navier-Stokes equations. In recent
years, progress has been made due to the application of the theory of parabolic differ-
ential inequalities to Prandtl equations and the questions of existence, uniqueness and
well-posedness have been very well answered for at least few simple cases.
Among the pioneer investigators Chorin and Marsden[25], Ladyzhenskaya[68] and
Shinbrot [115] gave a good insight of the mathematical aspects such as existence, unique-
ness and boundedness of the solution of Navier-Stokes equations and boundary layer equa-
tions. Further, Heywood[44], and Liu and Lee[76] have proved the existence and unique-
ness of steady laminar incompressible two dimensional boundary layer equations and
have presented very important properties of the velocity profiles. Also, Solonnikov and
Kazhikhov[120], Wang[141], and Ou and Sritharan[96] have presented an excellent survey
on the progress towards establishing a mathematical foundation of Navier-Stokes equa-
tions and Prandtl’s boundary layer equations. References (Ansorge and Blanck[4], Chong
et al.[24], Heywood and Padula[45], Hoff[49], Wang[142]) present some of the recent work
done in this area.
15
1.8 BOUNDARY LAYER EQUATIONS
Unsteady viscous compressible flows are governed primarily by the equation of continuity,
momentum and energy. In the absence of external body forces, these equations can be
written in the following form[90]
∂ρ
∂t+ (ρuj), j = 0 (1.1)
ρ
(∂ui
∂t+ ujui, j
)= −p, i + (τij), j (1.2)
ρ
(∂I
∂t+ ujI, j
)=
∂p
∂t+ (uiτij), j + (kT, j), j (1.3)
where ρ is the density, ui (i = 1, 2, 3) are the velocity components, p is the pressure of
the fluid flow at an instant of time t, τij is the shear stresses, I is the total enthalpy, T is
the temperature, and k is the coefficient of thermal conductivity. The shear stresses and
total enthalpy may be expressed in terms of velocity components as follows:
τij = (µ′ − 2
3µ)δij um,m + µ(ui, j + uj, i) (1.4)
I = CpT +1
2uiui (1.5)
where µ and µ′ are the coefficient of dynamic and bulk viscosity, respectively, δij is Kro-
necker delta and Cp is the specific heat at constant pressure.
The above system of partial differential equation’s is highly non-linear and is ex-
tremely difficult to solve except for some very simple flow configurations in which the
non-linear terms are dropped out or because simple enough. This suggest that problems
with more complicated configurations can be treated at least approximately by omitting
mathematically unpleasant terms of secondary importance. Such simplifications are pos-
sible in the limiting case of very large Reynolds number. The limiting case of very small
viscous forces (i.e. very large Reynolds number) is of great practical importance. It is not
advisable simply to omit the viscous terms as this would reduce the order of the complete
flow equations and hence the solution of the simplified equations could not be made to
16
satisfy the boundary conditions. On the other hand, this would lead to a singular pertur-
bation problem. Prandtl devised a method of solving the aforesaid equations and thereby
derived the boundary layer equations. The method has great mathematical generality and
can be used to solve many other singular perturbation problems of mathematical physics.
Prandtl understood that the region with high gradient in velocity and tempera-
ture fields are confined to a narrow region near the wall, known as the boundary layer.
Further, he has shown that on the basis of order of magnitude, one could neglect in the
boundary layer all gradients occurring in the viscous and conduction terms except the
component normal to the wall and also any pressure change across the boundary layer
could be neglected. These simplifications yield to the boundary layer equations which are
in orthogonal curvilinear coordinates (Figure 1.1) read as[90]:
∂ρ
∂t+
1
h1h2h3
[∂(h2h3ρu1)
∂x1
+∂(h1h3ρu2)
∂x2
+∂(h1h2ρu3)
∂x3
]= 0 (1.6)
∂u1
∂t+
u1
h1
∂u1
∂x1
+u2
h2
∂u1
∂x2
+u3
h3
∂u1
∂x3
+u1u2
h1h2
∂h1
∂x2
− u22
h1h2
∂h2
∂x1
+1
ρh1
∂p
∂x1
=1
ρh3
∂( µh3
∂u1
∂x3)
∂x3
(1.7)
∂u2
∂t+
u1
h1
∂u2
∂x1
+u2
h2
∂u2
∂x2
+u3
h3
∂u2
∂x3
+u1u2
h1h2
∂h2
∂x1
− u22
h1h2
∂h1
∂x2
+1
ρh2
∂p
∂x2
=1
ρh3
∂( µh3
∂u2
∂x3)
∂x3
(1.8)
1
ρ
∂p
∂x3
= 0 (1.9)
ρCp
∂T
∂t+
u1
h1
∂T
∂x1
+u2
h2
∂T
∂x2
+u3
h3
∂T
∂x3
−
∂p
∂t+
u1
h1
∂p
∂x1
+u2
h2
∂p
∂x2
=1
h3
∂(µCp
Pr1h3
∂T∂x3
)
∂x3
+ µ
(1
h3
∂u1
∂x3
)2
+
(1
h3
∂u2
∂x3
)2
(1.10)
where Pr = µCp
kis the Prandtl number. Here x1 and x2 are coordinates which lie and
are defined on the surface over which the fluid under boundary layer approximation is
flowing while the coordinate x3 extends into the boundary layer and u1, u2 and u3 are
the velocity components in the x1, x2 and x3 directions, respectively, and h1, h2 and h3
are such that a general element of length ds is given by (see Figure 1.1)
(ds)2 = h21(dx1)
2 + h22(dx2)
2 + h23(dx3)
2 (1.11)
17
At the outer edge of the boundary layer, the inner flow must be continuous with
the predetermined invisicid motion, hence all the dependent variables should approach
the free stream values asymptotically. At the base of the boundary layer, the equations
satisfy conditions prescribed at the wall, typical boundary conditions for boundary layer
flow obeying equations (1.6) to (1.10)are the following
ui(x1, x2, 0, t) = uiw(x1, x2, t), i = 1, 2, 3 (1.12)
T (x1, x2, 0, t) = Tw(x1, x2, t) (1.13)
ui(x1, x2,∞, t) = uie(x1, x2, t), i = 1, 2 (1.14)
p(x1, x2,∞, t) = pe(x1, x2, t) (1.15)
T (x1, x2,∞, t) = Te(x1, x2, t) (1.16)
Similarly, the typical initial conditions relevant to equations (1.6) to (1.10) can be ex-
pressed as
ui(x1, x2, x3, 0) = uio , i = 1, 2, 3 (1.17)
T (x1, x2, x3, 0) = To(x1, x2, x3) (1.18)
where the subscript ‘w’ denotes the conditions at the wall, the subscript ‘e’ denotes the
conditions at the edge of the boundary layer, the subscript ‘o’ refers to initial conditions
(i.e. conditions at t = 0) and ‘∞’ represents the edge of the boundary layer.
Boundary layer theory is developed mainly to study the flow in the boundary layer
by solving the boundary layer equations. These equations, in general, present considerable
mathematical difficulties owing to their non-linearity. Apart from the very few closed
form solutions that are available, these equations have to be solved either by power series
expansions or by numerical techniques. The term “approximate solution” is used if it is
derived from integral relations such as momentum and energy integral equations or power
series expansion.
18
The quantities of physical interest like shear stresses and heat flux appear in
conservation form. For laminar boundary layer, these quantities are expressed through
Newton’s law of viscosity and Fourier’s law of heat conduction. We define the wall velocity
gradients (Fη)w and (Sη)w and temperature gradient (Gη)w as
(Fη)w =
(∂F
∂η
)
w
=
(∂( u
ue)
∂η
)
w
(Sη)w =
(∂S
∂η
)
w
=
(∂( v
ve)
∂η
)
w
(Gη)w =
(∂G
∂η
)
w
=
(∂( T−T∞
Tw−T∞)
∂η
)
w
where the subscript ‘w’ signifies that the quantities are evaluated at the wall. These
wall velocity gradients ((Fη)w, (Sη)w) are measures of the viscous shear stress, frequently
called skin friction exerted on the wall by the fluid. The temperature gradients ((Gη)w)
measures the heat flux at the wall, that is, the quantity of heat transmitted from the wall
to the fluid per unit surface.
The present thesis provides a rigorous study with a detailed discussions on un-
steady mixed convection flow over various geometries, viz, rotating sphere, rotating cone,
slender cylinder, etc. As an introduction, author would like to present a simple example
for illustrating the basic governing equations of mixed convection flows: Consider a verti-
cal impermeable flat plate of finite height, which is immersed in a binary fluid/solute flow,
where the temperature and concentration at the wall, Tw and Cw, respectively, and in the
ambient field, T∞ and C∞, respectively, are constant. We assumed that the combination
of the buoyancy forces tend to induce an upward motion near the wall and that the ther-
mal buoyancy tends to induce a downward motion in the far field. The basic equations
for the double-diffusive mixed convection boundary layer flow over a vertical flat plate
are the conservation of mass, momentum, energy with the Boussinesq approximation for
the variation of the density with temperature and concentration. On making use of the
boundary layer approximations, these equations become for steady flow, see Gebhart and
19
Pera[35],
∂u
∂x+
∂v
∂y= 0 (1.19)
u∂u
∂x+ v
∂u
∂y= ue
∂ue
∂x+ ν
∂2u
∂y2+ g∗β(T − T∞) + g∗β∗(C − C∞) (1.20)
u∂T
∂x+ v
∂T
∂y=
ν
Pr
∂2T
∂y2(1.21)
u∂C
∂x+ v
∂C
∂y=
ν
Sc
∂2C
∂y2(1.22)
Here ν is the kinematic viscosity, g∗ is the acceleration due to gravity, β and β∗ are
the thermal and concentration expansion coefficients respectively. ue is the free stream
velocity. Pr is the Prandtl number and Sc is the Schmidt number. Equations (1.19)-(1.22)
have to be solved subject to the boundary conditions
u = 0, v = 0, T = Tw, C = Cw on y = 0, x > 0
u → ue, T → T∞, C → C∞ as y →∞, x > 0
In taking v(x, 0) = 0 we suppress any mass flux across the wall, as might occur in a
dissolution or melting process. Under this rather mild restriction the roles of T and C
are entirely interchangeable.
1.9 METHOD OF SOLUTION
The boundary layer equations are a set of coupled non-linear parabolic partial differential
equations. They are considerably simpler than the Navier-Stokes equations, which are
elliptic in nature. Nevertheless, considerable mathematical difficulty is associated with
the solutions of the boundary layer equations, mainly because of their non-linearity.
Hence, number of approximate analytical and numerical methods have been developed to
solve these boundary layer equations. In this section, some of the methods used to solve
the boundary layer equations are briefly reviewed.
20
The idea of transforming the boundary layer equations to another set of equations
amicable to known techniques, led to an interesting class of solutions called the similar
solutions. Similarity solutions exist only in certain flows in which the velocity and enthalpy
profiles at different stations can be made congruent if they are plotted in co-ordinates
which have been made dimensionless with reference to the scale factor. These similarity
solutions are sometimes called affine. The similarity or affinity relates to the internal or
self-similitude. These can be categorized mainly in two ways as geometrical and dynamical
similarities. Two phenomena are said to be geometrically similar if the ratio of any length
in one system to the corresponding length in the other system is same everywhere. This
ratio is usually known as the “scale factor”. On the other hand, when the dimensionless
form of each physical variable has the same value at corresponding points, it is said to be
dynamically similar.
The concept of similar boundary layer’s is no more than a means of simplifying
the problem of finding solutions to the boundary layer equations. In cases, when the con-
ditions for similarity are satisfied, the set of partial differential equations governing the
velocity and enthalpy of the fluid in a laminar boundary layer transforms to a group of
ordinary differential equations, which evidently, constitute a considerable mathematical
simplification of the problem. This reduction of the system to a set of ordinary differen-
tial equations can be achieved by transforming the dependent and independent variables
through similarity variables with the aid of a transformation known as the similarity trans-
formation. The conditions imposed on the coefficient of all the terms in the equations
obtained after the application of the transformations in order to obtain ordinary differ-
ential equations, are the similarity conditions or the similarity requirements. However,
the application of similarity transformations to three dimensional and unsteady boundary
layer flows does not always ensure the reduction of the system of partial differential equa-
tions to a set of ordinary differential equations as in the case of steady two-dimensional
or axisymmetric boundary layers. This is due to the fact that the later involves only two
independent variables. Fortunately, there are a number of circumstances, for instance,
flow past a yawed infinite wing, three dimensional stagnation point flows, three dimen-
sional axisymmetric flows with swirl etc., where the final reduction to a single independent
21
variable is possible. A solution is called self-similar if a system of partial differential equa-
tions can be reduced to a system of ordinary differential equations. In this case, certain
restrictions need to be imposed on the form for the velocity in the free stream. If the
transformations are only able to reduce the number of independent variables then the
transformed equations are known as semi-similar and the corresponding solutions are the
semi-similar solutions [139, 145]. Solutions are termed non-similar if the number of inde-
pendent variables cannot be reduced. The non-similarity may be due to the surface mass
transfer, the free stream velocity or due to the curvature of the body or even possibly due
to all. Thermal and concentration boundary layers non-similarity could also be caused by
a streamwise variation in the surface temperature or surface heat flux. Non-similarities
are dealt in references[21, 57, 58].
The advantages of the similarity assumptions are well known. Similar solutions
are the only accurate solutions covering an extensive range of such parameters as skin
friction, heat transfer , boundary layer displacement thickness, etc. Accuracy of approx-
imate procedures can be judged from this. Also, as shown by Lees[72], the application
of similarity is good in the neighbourhood of a stagnation point as well as for the case
of slowly varying external flow properties or a highly cooled surfaces. Similarity solu-
tions can also be employed in analyzing the non-similar flows containing limited regions
of locally similar flows. The similar solutions for both incompressible and compressible
fluids have been obtained by several investigators [78, 146]. However, it is not always
possible to respect the similarity requirements, that is, the initial boundary layer profiles,
the external stream and the flow at the surface all be compatible with similarity and at
the same time satisfy the conservation equations. For example, Wu and Libby[146] have
demonstrated that for a range of values of the pressure gradient parameter β, no similarity
solution exists for the Falkner-Skan equation.
In a number of practical situations, the natural demand of a detailed analysis of
boundary layer flows taking non-similarity into account is fulfilled by using the momentum
integral method despite certain inherent limitations. The basic idea behind this method
is that certain assumptions are made so as to the profiles of the flow variables and the
equations used are obtained by taking suitable integrals of the boundary layer equations
22
across the boundary layer. The integral approach of Kendall and Bartlett[62], Fletcher
and Holt[33] and Thomas and Ammingar[135, 136] are a few methods employed to solve
the non-linear boundary layer equations.
Boundary layer equations, which are parabolic in nature, after some transfor-
mations (self-similar/semi-similar/non-similar) can be reduced to a non-linear boundary
value or initial-boundary value problem and analytical treatment of such type of cou-
pled non-linear ordinary differential equations or partial differential equations for most
of the cases is totally impossible. To overcome this difficulty, one has to search for nu-
merical methods to perform the integration for obtaining an exact solution. There are
many numerical techniques in handling such type of non-linear problems and several ad-
vanced numerical methods are described in recent books on computational fluid dynamics,
amongst those are the books by Anderson et al. [3], Fletcher[32] , Farrell et al.[31] , Ce-
beci and Cousteix[16], Caughey and Hafez[13]. Few recent review articles (Rubin and
Tannehill[110], Fischer and Patera[34], Roache[106], Agarwal[1]) also explains the impor-
tance and applications of computational fluid dynamics.
The development of the high-speed digital computers significantly enhanced the
use of numerical methods in various branches of science and engineering. Many compli-
cated problems can now be solved at very little cost and in a very short time with the
available computing power. Presently, the Finite Difference Method(FDM), the Finite El-
ement Method(FEM) and Finite Volume Method(FVM) are widely used for the solution
of partial differential equations of heat, mass and momentum transfer. Extensive amount
of literature exist on the application of these methods for the solution of such problems.
Each method has it’s advantage depending on the nature of the physical problem to be
solved; but there is no best method for all problems. Finite difference method are simple
to formulate and can readily be extended to two or three-dimensional problems, are easy
to learn and apply. However, with the advent of numerical grid generation technique, the
finite difference method now possesses the geometrical flexibility. There are several exist-
ing numerical methods and their generalization for the solution of non-linear boundary
value problem/ initial boundary value problems are available in the literature, some of
which have computational advantage in certain situations.
23
One of the simple and most widely used numerical method of solving the boundary
value problems is the finite difference method. Finite difference method can be classified
into two types, i.e., one is explicit scheme and the other one is implicit scheme. The
former one is conditionally stable and requires a very small mesh size to be chosen. On
the otherhand, later is unconditionally stable. The Crank-Nicolson scheme is one of the
most popular implicit schemes. The convergence and stability of various finite difference
schemes are discussed in books by Mitchell and Griffiths[86], Smith[118] and Davis[27].
Moreover, Keller[59] presented an efficient finite difference method for the solution of two-
dimensional, time dependent and three dimensional flows (Cebeci[14], Keller[60], Keller
and Lentini[61]). This implicit method, known as Keller-Box method, is very much useful
for stiff boundary value problems because many points can be added where the solution
undergoes large changes and different discretization schemes may be used in different
regions. Details of this technique are given in the books by Cebeci and Bradshaw[15] and
Press et al. [102] as well as in a review article by Keller[60]. Another approach in tackling
two-point non-linear initial boundary value problems is to try to linearize the equations by
some means and quasilinearization technique is one of the popular and powerful method
of this sort. Quasilinearization method can be viewed as a generalization of the Newton-
Raphson approximation technique in functional space. An iterative sequence of linear
equations are carefully constructed to approximate a non-linear equation for achieving
quadratic convergence and monotonicity. The efficiency and accuracy of the method have
been illustrated through its applications to many boundary value problems in the books by
Bellman and Kalaba[9], Lee[69] and Radbill and McCue[103]. Later, Inouye and Tate[56]
introduced a method which involves an implicit finite difference scheme in combination
with the quasilinearization technique. In this combined approach, the non-linear equation
is linearized by perturbing about a known solution obtained from the previous iteration
and then solved by an implicit finite difference scheme. It may be noted that this combined
approach has been employed to solve few problems in the present thesis and the detailed
discussions of this method are presented in Chapter 2.
The numerical simulation of fluid dynamics and heat transfer problems has become
a routine part of engineering practice as well as a focus for fundamental and applied
24
research. Though there are still various topical areas where our physical understanding
and/or ineffective numerical algorithms limit the investigations, a large number of complex
phenomena can now be confidently studied via numerical simulations. Though finite
difference methods have been and will continue to play a major role in computational
fluid dynamics and heat transfer , Finite Element technique have spurred the explosive
development of “general purpose” methods and the growth of commercial software. The
inherent strength of the finite element method such as unstructured meshes, element-by-
element formulation and processing, and the simplicity and rigor of boundary condition
application are being coupled with modern development in automatic mesh generation
and adaptive meshing, to produce accurate and reliable simulation packages that are
widely accessible. The finite element method in fluid dynamics and heat transfer has
rapidly caught up with the well established solid mechanics community in simulation
capabilities. The method is a generalization of the classical variational(i.e., Rayleigh-Ritz)
and weighted-residual(eg: Galerkin, least squares, collocation, etc.) methods, which are
based on the idea that the solution of a differential equation can be represented as a linear
combination of unknown parameters and approximately selected functions in the entire
domain of the problem. Most of the real world problems are defined on regions that are
geometrically complex, and therefore it is difficult to generate approximate functions that
satisfy different types of boundary conditions on different portions of the boundary of a
complex domain. The basic idea of the finite element method is to view a given domain as
an assemblage of simple geometric shapes, called finite elements, for which it is possible
to systematically generate the approximate functions needed in the solution of differential
equations by any of the variational and weighted-residual methods. Though this method
was originally developed to solve structural engineering problems, it is being equally used
at present days to solve fluid flow problems (Baker[7],Backstrom[6],Lohner[77],Reddy[104],
Lewis[73]) including boundary layer flow problems.
Finite Volume Method(FVM), often called control volume methods, are formu-
lated from the inner product of the governing partial differential equations with a unit
function. This process results in the spatial integration of the governing equations. The
integrated terms are approximated by either finite difference or finite elements, discretely
25
summed over the entire domain. One of the most important features of finite volume
method is their flexibility for unstructured grids. The traditional curvilinear transfor-
mation required for finite difference method is no longer needed. Designation of the
components of a vector normal to boundary surfaces in finite volume method accommo-
dates the unstructured grid configuration with each boundary surface integral constructed
between nodal points. Finite volume method is applied to variety of heat transfer and
boundary layer problems.
1.10 OBJECTIVES AND SCOPE OF THE THESIS
The idea of transforming the boundary layer equations to another set of equations am-
icable to known techniques, led to an interesting class of solutions called the similar
solutions. In cases when the conditions for similarity are satisfied, the set of partial dif-
ferential equations governing the velocity and enthalpy of the fluid in a laminar boundary
layer transforms to a set of ordinary differential equations, which evidently, constitute a
considerable mathematical simplification of the problem. This reduction of the system
to a set of ordinary differential equations can be achieved by transforming the dependent
and independent variables through similarity variables with the aid of a transformation
known as the similarity transformation. A solution is called self-similar if a system of
partial differential equations can be reduced to a system of ordinary differential equa-
tions. In the present thesis, we intend to develop a self-similar solutions for the unsteady
mixed convection boundary layer flow in the stagnation point region of a rotating sphere
in a viscous incompressible fluid(Chapter 2) as well as in the stagnation point region of
a rotating cone in a co-rotating viscous incompressible fluid(Chapter 3). It is found that
a self-similar solution exists if the free stream velocity varies directly as the distance and
inversely as the time, and also the angular velocity of the sphere varies inversely as time
(Chapter 2). In the case of a rotating cone in a co-rotating fluid(Chapter 3), it is observed
that self-similar solution is possible when the free stream angular velocity and the angular
velocity of the cone vary inversely as a linear function of time.
However, the application of similarity transformations to three dimensional and
26
unsteady boundary layer flows does not always ensure the reduction of the system of
partial differential equations to a set of ordinary differential equations. If the transforma-
tions are only able to reduce the number of independent variables then the transformed
equations are known as semi-similar and the corresponding solutions are the semi-similar
solutions. Solutions are termed non-similar if the number of independent variables cannot
be reduced. The non-similarity may be due to the surface mass transfer, the free stream
velocity or due to the curvature of the body or even possibly due to all. Thermal and
concentration boundary layers non-similarity could also be caused by a streamwise varia-
tion in the surface temperature or surface heat flux. In a number of practical situations,
nature demands a detailed analysis of boundary layer flows taking non-similarity into
account. In Chapter 4, the semi-similar solutions are obtained for an unsteady mixed
convection flow over a rotating cone in a co-rotating fluid to analyze the combined effects
of thermal and mass diffusion. The unsteadiness in the flow field is due to angular ve-
locity of the cone and the freestream angular velocity which vary arbitrarily with time.
In the last chapter (Chapter 5) of this thesis, non-similar solutions are obtained for the
unsteady mixed convection flow along a heated vertical slender cylinder which is moving
in the same direction of free stream velocity. The unsteadiness is introduced by the time
dependent velocity of the slender cylinder as well as that of the free stream. The effects
of transverse curvature, viscous dissipation and surface mass transfer(injection/suction)
are also included in the analysis.
27
(ds)2 = h21dx2
1 + h22dx2
2 + h23dx2
3
and dx1 =∂x1
∂u1
du1 +∂x1
∂u2
du2 +∂x1
∂u3
du3, etc
Figure 1.1: Orthogonal coordinates with their velocities
28
CHAPTER 2
SELF-SIMILAR SOLUTION TO UNSTEADY
MIXED CONVECTION FLOW ON A ROTATING
SPHERE
2.1 INTRODUCTION
Interest in studying the flow and heat transfer characteristics over a rotating body of
revolution in forced flow stream stems from its practical importance in problems involving
projectile motion, re-entry missile behaviour, fiber coating applications, roto-dynamic
machine design and in modeling of many geophysical models. When an axisymmetric body
rotates, the fluid near the surface of the body is forced outward in radial direction due to
centrifugal action. This fluid is replaced by the one moving in the axial direction. Thus,
the axial velocity of the fluid in the neighbourhood of spinning body has a higher value
compared to that of a non-spinning body. This increase in the axial velocity results in an
enhancement of the convective heat transfer between the body and the fluid. Applicability
of this principle to develop practical systems has long been a subject of investigation.
The flow field in the vicinity of a rotating sphere in a uniform flow stream with its axis
of rotation parallel to the free stream velocity has been investigated by Hoskin[53] and
the temperature field by Siekmann [114]. Both authors have used four-term Blasius series
method to solve the governing boundary layer equations. The drawback of the Blasius
series method has been pointed out by Gortler[37]. Chao and Greif [17] re-studied the
temperature field using an improved method where the velocity field is assumed to be
quadratic and the temperature field is expressed as a universal function. These authors
noticed that for small values of the Prandtl number and for large values of rotation
parameter, the quadratic velocity profile is inadequate in determining the temperature
field. Subsequently, Chao [18] improved the above method [17] by taking more terms
29
of the velocity profiles to apply in the case of a rotating disc. Later, Lee et al. [70] re-
studied both the flow and temperature field using approximate series solution as defined by
Chao and Faghenle [19]. The investigation of flow and heat transfer in rotating systems
done prior to 1968 can be found in the review article reported by Kreith [65]. Later,
Kumari and Nath [66] have investigated the heat and mass transfer problem for the
steady incompressible boundary layer flow over a rotating sphere including the effect of
viscous dissipation using an implicit finite difference method. The analogous unsteady
problem was also considered by Kumari and Nath [67] with the time dependent free
stream velocity. Recently, the unsteady flow and heat transfer of a viscous incompressible
electrically conducting fluid in the forward stagnation point region of rotating sphere in
the presence of a magnetic field is investigated by Takhar and Nath [128].
The flow and convective heat transfer characteristics of a heated body rotating
about its own axis in an otherwise quiescent medium have been the subject of several
investigators in recent years. The interest is possibly due to the extensive applications of
rotating systems in many industrial processes as in chemical or electrochemical engineering
and also in geophysical and meteorological problems. When a surface is in contact with a
fluid whose temperature is different from that of the surface, the change in density occurs
which gives rise to buoyancy forces. In early study, Chiang et.al.[20] studied the flow
field in the vicinity of a fixed sphere for pure free convection with various surface thermal
conditions. Merkin[83] analyzed the free convective boundary layer on a stationary sphere
in a saturated porous medium. Suwono[126] investigated the buoyancy effect on flow
caused by rotating axisymmetric bodies of uniform surface temperature in the absence of
a uniform flow from infinity. Later, Lien et al.[75] studied the influence of heat transfer and
surface mass transfer (blowing and suction) on a steady laminar free convective flow over a
rotating sphere for the cases of uniform surface temperature and uniform heat flux. On the
other hand, Huang and Chen[51] investigated the flow and heat transfer characteristics in
the case of a steady laminar free convective flow over a fixed sphere. They considered the
case of non-uniform temperature and non-uniform heat flux. Further, a numerical study
on the free convection flows over a rotating sphere have also been reported by Takhar et
al.[129].
30
Our survey shows that heat transfer from a sphere has been the subject of nu-
merous analytical and computational investigations from the stand point of either pure
free convection or pure forced convection. The requirements of modern technology have
stimulated interest in fluid flows which involve the interaction of several phenomenon. At-
tention is directed recently towards to the situation where forced and free convection acts
simultaneously, that is, mixed convection occurs in which the effect of buoyancy forces on
a forced flow or the effect of forced flow on a buoyant flow is significant. The predictions
of heat transfer in combined convection from a sphere are of great practical interest as in-
dicated by studies of combustion, condensation, absorption and other processes involving
liquid drops or small particles. The phenomena is also observed in applications such as
rotating machinery design, fibre coating and cooling of rotating machinery parts etc. A
detailed review of the literature dealing with mixed convection flows including exhaustive
list of references can be found in the book by Gebhart et al.[36] and Bejan[8]. Steady
mixed convection flow over stationary spheres was studied by Chen and Mucoglu[23] using
Keller box scheme. Mucoglu and Chen[92] followed this up with a study of the same prob-
lem with uniform heat flux on the surface rather than uniform wall temperature. Some of
the recent studies on mixed convection flows over bodies of various shapes include those
of Amin and Riley[2], Seshadri et al.[113] and Merkin and Pop [84].
In approaching such problems, the study of similar solutions may be directly us-
able in important technical applications or may provide a standard of comparison for
approximate methods of calculating more complex non-similar cases. Moreover, the gen-
eral trends may provide valuable insight in understanding the physical occurrences which
take place in such combined convection flows. Similarity solutions have long played an
important role in exploring the influence of physical, dynamical and thermal parame-
ters on the behaviour of boundary layer flows without introducing the complications of
non-similar solutions and in providing bases for approximate methods of calculating more
complex non-similar cases. In case when the conditions for similarity are satisfied, the
complex set of partial differential equations governing the velocity and total enthalpy
of the fluid in a laminar boundary layer transforms to a group of ordinary differential
equations which evidently constitute a considerable mathematical simplification of the
31
problem. Most existing exact solutions in fluid mechanics are similarity solutions in the
sense that the number of independent variable is reduced by one or more. Similarity
solutions are the only accurate solutions covering an extensive range of such parameters
as skin friction, heat transfer, boundary layer displacement thickness etc. Accuracy of
approximate procedures can be judged from this. Also, as shown by Lees[72], the applica-
tion of similarity is good in the neighbourhood of a stagnation point as well as for the case
of slowly varying external flow properties or a highly cooled surface. Similarity solutions
can also be employed for analyzing the non-similar flows containing limited regions of
locally similar flows and when dealing with non-similar and semi-similar boundary layer
flows by finite difference, perturbation, finite elements and spectral methods.
The similarity solutions of unsteady laminar incompressible boundary layer flow
in the immediate neighbourhood of the forward stagnation point of a blunt nosed cylinder
have been obtained by Yang[147]. In the study by Ma and Hui[78], the method of a Lie
group transformations is used to derive all group invariant similarity solutions of the
unsteady two dimensional laminar incompressible boundary layers. The numerical results
have been presented for the unsteady two dimensional laminar incompressible stagnation
point flow. Similar solutions of the steady boundary layer flow over two dimensional and
axisymmetric bodies are also well documented in literature. However, the similar solutions
analogous to those of Ma and Hui[78] for unsteady axisymmetric mixed convection flows
have not been reported so far.
2.2 STATEMENT OF THE PROBLEM
The purpose of the present investigation is to develop a self-similar solution for the un-
steady mixed convection boundary layer flow in the stagnation point region of a rotating
sphere. The unsteadiness in the flow and the temperature fields is introduced by the free
stream velocity and by the angular velocity of the sphere which vary continuously with
time. Both constant wall temperature(CWT) and constant heat flux (CHF) conditions
have been considered. It is found that a self-similar solution exists if the free stream
velocity varies directly as the distance x and inversely as the time t, and also the angu-
32
lar velocity of the sphere varies inversely as t. The basic governing partial differential
equations with three independent variables have been reduced to the ordinary differential
equations with one independent variable using suitable similarity transformations. The
resulting ordinary differential equations are solved by converting them into a matrix equa-
tion through the application of an implicit finite difference scheme in combination with
the quasilinearization technique[9, 56, 69].
It appears that the present analysis is more general than those of previous in-
vestigators. The results for few special cases have been compared with the available
results[70, 128] and they are found to be in excellent agreement.
2.3 PROBLEM FORMULATION
Consider an unsteady mixed convection boundary layer flow in the forward stagnation
point region of a heated sphere which is rotating with time dependent angular velocity
Ω(t) in a viscous fluid (Figure 2.1). The temperature at the wall or the heat flux at
the wall is taken as a constant. It is assumed that the viscous dissipation terms are
negligible. The fluid has constant properties except the density changes which produce
buoyancy forces.
I
W
±
-
Ω0
o
U∞
r(x)
x
z
y
¾ g∗
Figure 2.1: Physical model and co-ordinate system.
33
Under the above assumptions along with Boussinesq approximation, the unsteady
laminar boundary layer equations governing the mixed convection flow are given by [36,
128, 129]
(ru)x + (rv)y = 0, (2.1)
ut + uux + vuy −(
w2
r
)rx = (ue)t + ue(ue)x + νuyy + g∗β(T − T∞), (2.2)
wt + uwx + vwy +(uw
r
)rx = νwyy, (2.3)
Tt + uTx + vTy = αTyy. (2.4)
The initial conditions are
u(0, x, y) = ui(x, y); v(0, x, y) = vi(x, y);
w(0, x, y) = wi(x, y); T (0, x, y) = Ti(x, y); (2.5)
and the boundary conditions are given by,
u(t, x, 0) = 0; v(t, x, 0) = 0, w(t, x, 0) = Ω(t)r,
T (t, x, 0) = Tw or − kTy(t, x, 0) = qw,
u(t, x,∞) = ue(x, t), w(t, x,∞) = 0, T (t, x,∞) = T∞ = constant. (2.6)
Here x, y and z are co-ordinates measured from the forward stagnation point along the
surface, normal to the surface and in the rotating direction, respectively; r(x) is the
radial distance from a surface element to the axis of symmetry (r ≈ x in the neighbor-
hood of the stagnation point); u, v and w are the velocity components along the x, y and
z- directions, respectively; t is the time; g∗ is the acceleration due to gravity; α and ν are
thermal diffusivity and kinematic viscosity, respectively; k is the thermal conductivity; T
is the temperature in the boundary layer; β is the volumetric co-efficient of thermal ex-
pansion; qw is the heat flux at the wall; the subscripts t, x and y denote partial derivatives
34
with respect to the corresponding variables and the subscripts e, i, w and ∞ denote the
condition at the edge of the boundary layer, initial condition, condition at the wall and
free stream condition, respectively.
It has been found that equations(2.1)-(2.4) admit a similarity solution in the
stagnation point region if the velocity at the edge of the boundary layer ue varies as in a
particular manner i.e., ue(t, x) = A∗xt
, A∗ > 0, x > 0, t > 0 [78]. Applying the following
transformation for constant wall temperature (CWT) case:
ue =A∗x
t, A∗ > 0, x > 0, t > 0, Ω(t) =
B
t,B > 0, r ≈ x,
dr
dx≈ 1,
η = 21/2(νt)−1/2y, u =
(A∗x
t
)f ′(η),
w =
(Bx
t
)g(η), v = −21/2
(ν
t
)1/2
A∗f(η),
T − T∞ = (Tw − T∞)θ(η), λ =
(B
A∗
)2
=
(ww
ue
)2
,
P r =ν
α, λ1 =
Grx
Re2x
, Grx =g∗β(Tw − T∞)x3
ν2, Rex =
uex
ν(2.7)
and for the constant heat flux (CHF) case:
ue =A∗x
t, A∗ > 0, x > 0, t > 0, Ω(t) =
B
t,B > 0, r ≈ x,
dr
dx≈ 1,
η = 21/2(νt)−1/2y, u =
(A∗x
t
)F ′(η),
w =
(Bx
t
)G(η), v = −21/2
(ν
t
)1/2
A∗F (η),
T − T∞ = 2−1/2(qw
k
)(νt)1/2Θ(η), λ =
(B
A∗
)2
,
P r =ν
α, λ1
∗ =Grx
∗
Re5/2x
, Grx∗ =
g∗βqwx4
kν2, Rex =
uex
ν(2.8)
35
to equations(2.1)-(2.4), we find that Eq. (2.1) is identically satisfied, and equations(2.2)-
(2.4) for the constant wall temperature (CWT) case reduce to
f ′′′ + A∗ff ′′ +(
A∗
2
)[1− f ′2 + λg2] +
A∗
2λ1θ − 1
2[1− f ′ − η
f ′′
2] = 0 (2.9)
g′′ + A∗(fg′ − f ′g) +1
2(g +
1
2ηg′) = 0 (2.10)
θ′′ + A∗Prfθ′ + Prη
4θ′ = 0 (2.11)
For CHF case, the equations corresponding to (2.9)-(2.11) are given by
F ′′′ + A∗FF ′′ +A∗
2[1− F ′2 + λG2] +
(A∗
2
) 32
λ1∗Θ− 1
2[1− F ′ − η
2F ′′] = 0 (2.12)
G′′ + A∗(FG′ − F ′G) +1
2(G +
η
2G′) = 0 (2.13)
Θ′′ + A∗PrFΘ′ + Prη
4Θ′ − Pr
4Θ = 0. (2.14)
The boundary conditions for the CWT case can be expressed as
f(0) = 0 = f ′(0), g(0) = θ(0) = 1;
f ′(∞) = 1, g(∞) = θ(∞) = 0. (2.15)
The boundary conditions for the CHF case are reduced to
F (0) = 0 = F ′(0), G(0) = 1, Θ′(0) = −1;
F ′(∞) = 1, G(∞) = Θ(∞) = 0. (2.16)
Here η is the similarity variable; f and F are the dimensionless stream functions for the
CWT and CHF cases, respectively; f ′ and g are, respectively, the dimensionless velocity
along x- and z - directions for the CWT case; F ′ and G are the corresponding velocities for
the CHF case; θ and Θ are the dimensionless temperature profiles for the CWT and CHF
cases, respectively; Grx and (Grx)∗ are the local Grashof number for the CWT and CHF
cases, respectively; Rex is the local Reynolds number; Pr is the Prandtl number; λ1 and
36
λ1∗ are, respectively, the buoyancy parameters for CWT and CHF cases (λ1, λ1
∗ > 0 for
aiding flows and < 0 for opposing flows); Prime denotes derivative with respect to η. λ is
the dimensionless rotation parameter and λ = 0 implies that the sphere is stationary. In
this case, there is no rotational motion of the sphere to induce a circumferential velocity
in the field through the action of viscosity, the velocity component w will be identically
zero. Hence, the momentum equations in z- direction (2.10) and (2.13) vanish for CWT
and CHF cases, respectively.
It may be remarked that the equations (2.9)-(2.11) for λ1 = 0 (no buoyancy force)
of the CWT case are the same as those of Takhar et al.[128] with M = 0 (no magnetic
field) who considered the forced convection flow in the stagnation point region of a rotating
sphere with a magnetic field. Further, the steady state version of equations (2.9)-(2.11)
were solved for A∗ = 1 by Lee et al.[70].
The set of partial differential equations(2.1)-(2.4) governing the flow has to be
solved subject to the initial conditions (2.5) and boundary conditions (2.6). Since we
are interested in the similar solutions, we solve the ordinary differential equations(2.9)-
(2.11) under boundary conditions (2.15) for CWT case or equations(2.12)-(2.14) under
boundary conditions (2.16) for CHF case.
The surface shear stresses in x- and z- directions for the CWT case are, respec-
tively given by
Cf x =
[2µ
(∂u∂y
)]y=0
ρu2e
= 232 Rex
− 12 (A∗)−
12 f ′′(0),
Cf z =
[−2µ
(∂w∂y
)]y=0
ρu2e
= −232 Rex
− 12 λ
12 (A∗)−
12 g′(0), (2.17)
where Rex =uex
ν=
(A∗x2
νt
). Similarly, the surface shear stresses in x- and z- directions
for CHF case are, respectively, given by
Cf x = 232 Rex
− 12 (A∗)−
12 F ′′(0),
Cf z = −232 Rex
− 12 λ
12 (A∗)−
12 G′(0). (2.18)
37
The surface heat transfer rate in terms of Nusselt number for the CWT case can be
expressed as
Nu =−x
(∂T∂y
)y=0
(Tw − T∞)= −2
12 Rex
12 (A∗)−1/2θ′(0) (2.19)
The surface heat transfer rate for the CHF case is given in the form
Nu = 212 Rex
12 (A∗)−1/2 1
Θ(0)(2.20)
From equations (2.17)-(2.20), it is clear that f ′′(0),−g′(0),−θ′(0) for CWT case and
F ′′(0),−G′(0), 1Θ(0)
for CHF case are the crucial parameters which characterize the surface
shear stresses and the surface heat transfer rate of the fluid flow.
2.4 METHODS OF SOLUTION
The set of nonlinear ordinary differential equations (2.9)-(2.14) under the boundary con-
ditions (2.15) and (2.16) have been solved numerically using an implicit finite difference
scheme in combination with quasilinearization technique[9, 56]. At first the nonlinear
coupled ordinary differential equations were replaced by an iterative sequence of linear
ordinary differential equations following quasilinearization technique. The resulting se-
quence of linear ordinary differential equations were expressed in difference form using
central difference scheme. In each iteration step, the difference equations were then re-
duced to a system of linear algebraic equations with block tri-diagonal structure which is
solved using an algorithm due to Varga[137]. The detailed description of the implicit fi-
nite difference scheme with quasilinearization technique and its application to the present
problem are given in the next two Sections 2.4.1 and 2.4.2, respectively.
2.4.1 OUTLINE OF QUASILINEARIZATION TECHNIQUE
The finite difference version of the quasilinearization technique is briefly described in the
present section. Since a majority of the boundary layer flow problems are of the two-point
boundary value types, a second order vector system in the domain [a, b] is considered here
38
to describe the present numerical approach. For simplicity, a system with one independent
variable has been chosen. However, extension to systems with more independent variables
is simple and obvious.
Let the vector equation read as
Z ′′ = ψ(ξ, Z, Z ′); a ≤ ξ ≤ b; Z(a) = ao, Z(b) = bo (2.21)
where ξ is the independent variable, ZT = (Z1, Z2, ..Zm) and ψT = (ψ1, ψ2, ..ψm). The
prime denote differentiation with respect to ξ and ao and bo are constant vectors whose
values are known. Applying the method of quasilinearization to the above system, a linear
system of equations, in vector notation, is obtained as follows:
Z ′′(i+1) = ψ(i) +m∑
l=1
(∂ψ
∂Z ′l
)i
[Z′(i+1)l − Z
′(i)l ] +
m∑
l=1
(∂ψ
∂Zl
)i
[Z(i+1)l − Z
(i)l ] (2.22)
The equation (2.22) can be rearranged and written as an explicit linear system in ((i+1)th)
approximations as
Z ′′ = PZ ′ + QZ + R (2.23)
Here, the iterative indices are dropped from the dependent variables and the coefficients
P, Q and R are supposed to be known functions as they contain only ith iteration values.
For the application of finite difference discretization on the above system of equa-
tions (2.23), the interval a ≤ ξ ≤ b is divided into N sub-intervals with constant mesh
size each of width d = b−aN
. The derivatives can be written in a central difference scheme
for 2 ≤ n ≤ N ,
Z ′ =Zn+1 − Zn−1
2d,
Z ′′ =Zn+1 − 2Zn + Zn−1
d2. (2.24)
The boundary conditions become Z1 = ao and ZN+1 = bo. Substitution of (2.24) into
39
(2.23) gives a linear system of equations expressible in the following form:
B1 C1 0 0 . . . 0 0 0
A2 B2 C2 0 . . . 0 0 0
0 A3 B3 C3 . . . 0 0 0
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
0 0 0 0 . . . AN BN CN
0 0 0 0 . . . 0 AN+1 BN+1
Z1
Z2
Z3
.
.
.
ZN
ZN+1
=
D1
D2
D3
.
.
.
DN
DN+1
(2.25)
Matrix inversion method can be used to solve the above set of linear algebraic equa-
tion(2.25). The method, which has been used by us, is known as Varga algorithm[137].
For the problems presented in this thesis, the maximum number of components of the
vector Z are four dependent variables F,G, Θ and Φ. It is also possible to form a matrix
equation of the form
AnZn−1 + BnZn + CnZn+1 = Dn (2 ≤ n ≤ N) (2.26)
where the vectors and coefficient matrices are of the form
Zn =
F
G
Θ
Φ
n
, Dn =
d1
d2
d3
d4
n
, An =
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
n
Bn =
b11 b12 b13 b14
b21 b22 b23 b24
b31 b32 b33 b34
b41 b42 b43 b44
n
Cn =
c11 c12 c13 c14
c21 c22 c23 c24
c31 c32 c33 c34
c41 c42 c43 c44
n
The system of linear algebraic equations (2.25) with block tri-diagonal structure is solved
by using the following algorithm due to Varga[137]
Zn = −EnZn+1 + Jn, 1 ≤ n ≤ N
40
where En = (Bn − AnEn−1)−1Cn,
Jn = (Bn − AnEn−1)−1(Dn − AnJn−1), 2 ≤ n ≤ N
E1 = EN+1 =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
and J1 =
F (a)
G(a)
Θ(a)
Φ(a)
, JN+1 =
F (b)
G(b)
Θ(b)
Φ(b)
The column matrix J1 and JN+1 are known as they represent the boundary conditions.
In order to achieve numerically stable results, the matrix in equations (2.25)
or (2.26) must be block diagonally dominant with respect to the matrix norm ‖.‖. If
‖Aj‖2 = maximum eigen value of AjATj , then to restore the stability of (2.25) the condition
‖Bn‖ ≥ ‖An‖+ ‖Cn‖ must be satisfied.
The above prescribed method was first outlined by Inouye and Tate[56] to demon-
strate the capability of solving the two point boundary value problem. It may be noted
that this method can also be applied to nonlinear partial differential equations.
2.4.2 APPLICATION OF QUASILINEARIZATON TECHNIQUE WITH
IMPLICIT FINITE DIFFERENCE SCHEME
For the constant wall temperature case, equations (2.9)-(2.11) under the boundary con-
ditions (2.15) have been linearized by using the quasilinearization technique as described
in the Section 2.4.1 and the following set of linear ordinary differential equations are
obtained.
f ′′′(i+1)+ X i
1f′′(i+1)
+ X i2f′(i+1)
+ X i3g
(i+1) + X i4θ
(i+1) = X i5 (2.27)
g′′(i+1)+ Y i
1 g′(i+1)+ Y i
2 g(i+1) + Y i3 f ′(i+1)
= Y i4 (2.28)
θ′′(i+1)+ Zi
1θ′(i+1)
+ Zi2θ
(i+1) = Zi3 (2.29)
41
The coefficient functions with iterative index i are known and the functions with iterative
index i + 1 are to be determined. The boundary conditions become
f ′(i+1)= 0, g(i+1) = 1 = θ(i+1) at η = 0
f ′(i+1)= 1, g(i+1) = 0 = θ(i+1) at η = η∞
where η∞ is the edge of the boundary layer. The coefficients in equations (2.27) - (2.29)
are given by
X i1 = A∗f + 2−2η
X i2 = −A∗f ′ + 2−1
X i3 = A∗λg
X i4 = (
A∗
2)λ1
X i5 = 2−1 + 2−1[A∗g2 − f ′2 − 1]
Y i1 = A∗f + 2−2η
Y i2 = −A∗f ′ + 2−1
Y i3 = −A∗g
Y i4 = −A∗gf ′
Zi1 = Pr(A∗f +
η
4)
Zi2 = 0
Zi3 = 0
The linear system with boundary conditions is discretized by introducing a mesh
in the η- direction. The boundary η →∞ is replaced by a finite boundary η = η∞. Now
42
the interval [0 η∞] is divided into N equal subintervals of height 4η. In accordance with
the above mesh system , the derivatives of the dependent variables are approximated by
a central difference formula as
f ′′′ =(f ′n+1 − 2f ′n + f ′n−1)
(4η)2,
f ′′ =(f ′n+1 − f ′n−1)
2(4η), (2.30)
Similar expressions can be written for g and θ. The finite difference method in combination
with quasilinearization technique was outlined by Inouye and Tate[56] to solve the two
point boundary value problems.Thus, applying the finite difference scheme, we get a
system of equations which can be written in the matrix form as
AnWn−1 + BnWn + CnWn+1 = Dn, 2 ≤ n ≤ N (2.31)
where the vectors and coefficient matrices are given by
Wn =
f ′
g
θ
n
, Dn =
d1
d2
d3
n
, An =
a11 a12 a13
a21 a22 a23
a31 a32 a33
n
,
Bn =
b11 b12 b13
b21 b22 b23
b31 b32 b33
n
, Cn =
c11 c12 c13
c21 c22 c23
c31 c32 c33
n
43
The elements of matrices An, Bn, Cn and Dn are
a11 = 1−X14η2
a12 = 0 a13 = 0
a21 = 0 a22 = 1− Y14η2
a23 = 0
a31 = 0 a32 = 0 a33 = 1− Z14η2
b11 = −2 + X2(4η)2 b12 = X3(4η)2 b13 = X4(4η)2
b21 = Y3(4η)2 b22 = −2 + Y2(4η)2 b23 = 0
b31 = 0 b32 = 0 b33 = −2
c11 = 1 + X14η2
c12 = 0 c13 = 0
c21 = 0 c22 = 1 + Y14η2
c23 = 0
c31 = 0 c32 = 0 c33 = 1 + Z14η2
d1 = X5(4η)2 d2 = Y4(4η)2 d3 = Z3(4η)2
W1 and WN+1 can be obtained from boundary conditions at η = 0 and at η = η∞:
W1 =
f ′
g
θ
η=0
=
0
1
1
and WN+1 =
f ′
g
θ
η=η∞
=
1
0
0
(2.32)
The equations (2.31) together with the boundary conditions (2.32) can be solved by
Varga’s algorithm
Wn = −EnWn+1 + Jn, 1 ≤ n ≤ N
where En = (Bn − AnEn−1)−1Cn
Jn = (Bn − AnEn−1)−1(Dn − AnJn−1), 2 ≤ n ≤ N
E1 = EN+1 =
0 0 0
0 0 0
0 0 0
and J1 =
0
1
1
, JN+1 =
1
0
0
The computations are carried out using the following initial profiles
f ′ = 1− exp(−η); g = exp(−η) θ = exp(−η) (2.33)
44
The profiles are chosen such that they satisfy the given boundary conditions (2.32).
Similarly for the case of constant heat flux, equations (2.12)-(2.14) under the
boundary conditions (2.16) have been linearized by using the quasilinearization technique
as described in the Section 2.4.1 and the following set of linear ordinary differential equa-
tions are obtained.
F ′′′(i+1)+ X i
1F′′(i+1)
+ X i2F
′(i+1)+ X i
3G(i+1) + X i
4Θ(i+1) = X i
5 (2.34)
G′′(i+1)+ Y i
1 G′(i+1)+ Y i
2 G(i+1) + Y i3 F ′(i+1)
= Y i4 (2.35)
Θ′′(i+1)+ Zi
1Θ′(i+1)
+ Zi2Θ
(i+1) = Zi3 (2.36)
The coefficient functions with iterative index i are known and the functions with iterative
index i + 1 are to be determined. The boundary conditions become
F ′(i+1)= 0, G(i+1) = 1, Θ′(i+1)
= −1 at η = 0
F ′(i+1)= 1, G(i+1) = 0 = Θ(i+1) at η = η∞
where η∞ is the edge of the boundary layer. The coefficients in equations (2.34) - (2.36)
are given by
X i1 = A∗F + 2−2η
X i2 = −A∗F ′ + 2−1
X i3 = A∗λG
X i4 = (
A∗
2)
32 λ1
∗
X i5 = 2−1 + 2−1A∗[λG2 − F ′2 − 1]
Y i1 = A∗F + 2−2η
45
Y i2 = −A∗F ′ + 2−1
Y i3 = −A∗G
Y i4 = −A∗F ′G
Zi1 = Pr(A∗F +
η
4)
Zi2 = −Pr
4
Zi3 = 0
The linear system with boundary conditions is discretized by introducing a mesh
in the η- direction. The boundary η →∞ is replaced by a finite boundary η = η∞. Now
the interval [0 η∞] is divided into N equal subintervals of height 4η. In accordance with
the above mesh system , the derivatives of the dependent variables are approximated by
a central difference formula as
F ′′′ =(F ′
n+1 − 2F ′n + F ′
n−1)
(4η)2,
F ′′ =(F ′
n+1 − F ′n−1)
2(4η), (2.37)
Similar expressions can be written for G and Θ. The finite difference method in combi-
nation with quasilinearization technique was outlined by Inouye and Tate[56]to solve the
two point boundary value problems.Thus,applying the finite difference scheme,we get a
system of equations which can be written in the matrix form as
AnWn−1 + BnWn + CnWn+1 = Dn, 2 ≤ n ≤ N (2.38)
where the vectors and coefficient matrices are given by
Wn =
F ′
G
Θ
n
, Dn =
d1
d2
d3
n
, An =
a11 a12 a13
a21 a22 a23
a31 a32 a33
n
,
46
Bn =
b11 b12 b13
b21 b22 b23
b31 b32 b33
n
, Cn =
c11 c12 c13
c21 c22 c23
c31 c32 c33
n
.
The elements of matrices An, Bn, Cn and Dn are
a11 = 1−X14η2
a12 = 0 a13 = 0
a21 = 0 a22 = 1− Y14η2
a23 = 0
a31 = 0 a32 = 0 a33 = 1− Z14η2
b11 = −2 + X2(4η)2 b12 = X3(4η)2 b13 = X4(4η)2
b21 = Y3(4η)2 b22 = −2 + Y2(4η)2 b23 = 0
b31 = 0 b32 = 0 b33 = −2 + Z2(4η)2
c11 = 1 + X14η2
c12 = 0 c13 = 0
c21 = 0 c22 = 1 + Y14η2
c23 = 0
c31 = 0 c32 = 0 c33 = 1 + Z14η2
d1 = X5(4η)2 d2 = Y4(4η)2 d3 = Z3(4η)2
W1 and WN+1 can be obtained from boundary conditions at η = 0 and at η = η∞: The
linear system of equations (2.38) together with the boundary conditions can be solved by
Varga’s algorithm
Wn = −EnWn+1 + Jn, 1 ≤ n ≤ N
where En = (Bn − AnEn−1)−1Cn
Jn = (Bn − AnEn−1)−1(Dn − AnJn−1), 2 ≤ n ≤ N
E1 = EN+1 =
0 0 0
0 0 0
0 0 0
and J1 = W1, JN+1 =
1
0
0
The computations are carried out using the following initial profiles
f = fw+η2
2η∞; F ′ = 1−e−η; G = e−η, Θ =
(0.5− η
η∞+
(η√2η∞
)2)
η∞. (2.39)
47
2.5 RESULTS AND DISCUSSIONS
The nonlinear coupled ordinary differential equations (2.9)-(2.11) with boundary condi-
tions (2.15) for constant wall temperature case [Equations (2.12)-(2.14) for constant heat
flux case with boundary conditions (2.16)] have been solved numerically using an implicit
finite difference scheme in combination with quasilinearization technique and its detailed
description is presented in the previous section. To ensure the convergence of the numer-
ical solution to the true solution, the step size ∆η and the edge of the boundary layer η∞
have been optimized. The results presented here are independent of ∆η and η∞ at least
up to the fourth decimal place. The computations have been carried out with ∆η = 0.01
for various values of A∗(0 < A∗ ≤ 3.0), λ(0 < λ ≤ 10), P r(0.7 ≤ Pr ≤ 10), λ1(−2 ≤ λ1 ≤10) and λ∗1(−2 ≤ λ∗1 ≤ 10). In all numerical computations the edge of the boundary layer
η∞ is taken as 6. In order to assess the accuracy of our method, we have compared our
results for CWT case with those of Lee et al.[70] and Takhar and Nath[128]. In both the
cases the results are found in excellent agreement. Some of the comparisons are shown in
Table 2.1 and Figure 2.12.
Present Results Lee’s Result[70] .
Pr λ f ′′(0) −g′(0) −θ′(0) f ′′(0) −g′(0) −θ′(0)
1 1.11300 0.78499 0.55383 1.1129 0.7849 0.5536
1 4 1.62320 0.84637 0.58982 1.6233 0.8463 0.5897
10 2.52161 0.93626 0.64343 2.5216 0.9362 0.6432
1 1.11300 0.78501 1.29115 1.1129 0.7849 1.2911
10 4 1.62328 0.84631 1.41802 1.6233 0.8463 1.4180
10 2.52170 0.93620 1.60035 2.5216 0.9362 1.6003
Table 2.1: Comparison of the results (f ′′(0),−g′(0),−θ′(0)) with those of Lee et al. [70]
48
The results for constant wall temperature(CWT) case are presented in Figures
2.3-2.13 and for constant heat flux(CHF) case in Figures 2.14-2.15.
Case(i): Constant Wall Temperature.
For the constant wall temperature(CWT) case, the effect of the acceleration pa-
rameter A∗ on the velocity profiles in x- and z- directions (f ′, g) and the temperature
profiles (θ) for λ = λ1 = 1, P r = 0.7 is shown in Figures 2.3-2.5. It is observed that the
velocity profiles in the x- direction (f ′) increases everywhere with increasing A∗. Due to
the increase in the value of the acceleration parameter A∗, the velocity at the edge of the
boundary layer ue increases. Hence, the fluid inside the boundary layer gets accelerated
in the x- direction which increases the velocity f ′ and reduces the viscous boundary layer
thickness. On the other hand, increase of A∗ tends to oppose the fluid motion in the
rotational direction(i.e., z-direction) and consequently the velocity in the rotational di-
rection is lowered everywhere but for lower values of acceleration parameter A∗, rotation
significantly affects the velocity in the rotational direction leading to a velocity overshoot
in the rotational direction velocity profiles (g) as shown in Figure 2.3 and Figure 2.4.
Further, it appears in Figure 2.3 that for values A∗ < 0.115 reverse flow in the velocity
profiles (f ′) occur near the wall. Similar observations for the case of the unsteady forced
convection boundary layer flow over rotating sphere have been reported by Takhar and
Nath [128]. Figure 2.5 shows that the thermal boundary layer thickness decreases with
the increase of the acceleration parameter A∗.
For the CWT case, the variation of the velocity profiles in x- and z- directions
(f ′, g) with the rotation parameter λ is displayed in Figure 2.6. The x- direction velocity f ′
is strongly affected by the variation of the rotational parameter λ where as the z- direction
velocity component g is very little affected by it because λ affects the x- momentum
equation directly. The increase in λ injects an additional momentum into the boundary
layer which accelerates the fluid. Hence, the velocity profiles in x- direction steepen and
the boundary layer thickness decreases. Since λ does not affects the energy equation
directly, the effect of λ on the temperature profile θ is also found to be small, and the
temperature profile θ are shown in Figure 2.7.
49
0 1 2 3 40
0.5
1
η
f| , s
f|
s
* Takhar et al
Present results
Figure 2.2: Comparison of the results (f ′, g) for A∗ = λ = 1, λ1 = 0, P r = 0.7
0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
η
fη
A*=0.31.02.00.1150.08
Figure 2.3: Effect of A∗ on velocity profiles (f ′) for CWT case when λ = λ1 = 1, P r = 0.7
50
0 1 2 3 4−0.1
0.3
0.7
1.1
η
g
A*= 0.3 1.0 2.0 0.2 0.15
Figure 2.4: Effect of A∗ on velocity profiles (g) for CWT case when λ = λ1 = 1, P r = 0.7
0 1 2 3 4
0.5
1
η
θ
A*=0.31.02.0
Figure 2.5: Effect of A∗ on velocity profiles (θ) for CWT case when λ = λ1 = 1, P r = 0.7
51
0 1 2 3
0.5
1
η
f η, g
1
3
5
10
1
λ =10
fη
g
Figure 2.6: Effect of λ on velocity profiles (f ′, g) for CWT case when
A∗ = λ1 = 1, P r = 0.7
0 1 2 30
0.5
1
η
θ
λ =135
Figure 2.7: Effect of λ on velocity profiles (θ) for CWT case when A∗ = λ1 = 1, P r = 0.7
52
For the CWT case, Figures 2.8-2.10 display the effect of buoyancy parameter λ1
and Prandtl number Pr on the velocity profiles (f ′, g) and the temperature profiles (θ).
The action of the buoyancy force in buoyancy assisting flows shows the overshoot in the
velocity profiles (f ′) near the wall for lower Prandtl number (Pr = 0.7) but for higher
Prandtl number (Pr = 7.0) or in buoyancy opposing flows the velocity overshoot in f ′
is not observed as shown in Figure 2.8. The magnitude of the overshoot increases with
the buoyancy parameter λ1. The reason that the buoyancy force (λ1) effect is larger in a
low Prandtl number fluid (Pr = 0.7, air) is due to the lower viscosity of the fluid which
enhances the velocity as the assisting buoyancy force acts like a favorable pressure gradient
and the velocity overshoot occurs. For higher Prandtl number fluid (Pr = 7.0, water)
the velocity overshoot is not present because higher Prandtl number fluid implies more
viscous fluid which makes it less sensitive to the buoyancy parameter λ1. The effect of
λ1 is comparatively less on the velocity component (g) and temperature (θ) as shown in
Figures 2.9 and 2.10.
λ1 λ f ′′(0) −g′(0) −θ′(0)
1 1 1.28271 0.64575 0.58957
3 1.65024 0.69610 0.60766
5 1.99019 0.73751 0.62273
10 2.76174 0.81974 0.65392
3 1 1.86961 0.74949 0.62963
3 2.19547 0.78497 0.64316
5 2.50472 0.81630 0.65496
10 3.22435 0.88234 0.68048
5 1 2.40004 0.82724 0.66140
3 2.69939 0.85503 0.67212
5 2.98728 0.88050 0.68190
10 3.66805 0.93574 0.70355
10 1 3.58401 0.96785 0.72129
3 3.84439 0.98629 0.72865
5 4.09863 1.00370 0.73573
10 4.71170 1.04360 0.75157
Table 2.2: Surface shear stresses and heat transfer parameters (f ′′(0),−g′(0),−θ′(0)) for
CWT case when Pr = 0.7 and A∗ = 1
53
0 1 2 3 4
0.5
1
1.4
η
f η
−2.0 for Pr =0.71.0 "3.0 "5.0 "10.0 "1.0 for Pr = 7.0
λ1
Figure 2.8: Effects of λ1 and Pr on velocity profiles (f ′) for CWT case when A∗ = λ = 1
0 1 2 3
0.5
1
η
g
1.0 for Pr =0.75.0 "10.0 "1.0 for Pr =7.010.0 "
λ1
Figure 2.9: Effects of λ1 and Pr on velocity profiles (g) for CWT case when A = λ = 1
54
0 1 2 3
0.5
1
η
θ
10
λ1 =1
10
1
Pr = 0.7
Pr = 7.0
Figure 2.10: Effects of λ1 and Pr on velocity profiles (θ) for CWT case when A∗ = λ = 1
1 2 30
1
2
4
5
f|| (0)
A*
λ =1.0 3.0 5.0 10.0
Figure 2.11: Variations of surface shear stress parameter (f ′′(0)) with A∗ for CWT case
when λ1 = 1 and Pr = 0.7
55
For the CWT case, the effects of acceleration parameter A∗, the rotation pa-
rameter λ and the buoyancy parameter λ1 in the buoyancy assisting flow on the sur-
face shear stress parameters (f ′′(0),−g′(0) ) and the surface heat transfer parameter
(−θ′(0)) for Pr = 0.7 are presented in Figures 2.11-2.13 and in Tables 2.2 and 2.3 .
λ A∗ f ′′(0) −g′(0) −θ′(0)
1 0.1 -0.02246 -0.74511 0.34855
0.3 0.50094 0.05390 0.40955
0.5 0.79946 0.30351 0.46743
1.0 1.28271 0.64575 0.58957
2.0 1.91728 1.05422 0.77954
3 0.1 0.31953 -0.45973 0.36052
0.3 0.78468 0.12185 0.42224
0.5 1.09894 0.35565 0.48159
1.0 1.65024 0.69610 0.60766
2.0 2.40387 1.11308 0.80359
5 0.1 0.56075 -0.35182 0.36723
0.3 1.02437 0.16838 0.43198
0.5 1.36665 0.39587 0.49304
1.0 1.99019 0.73751 0.62273
2.0 2.85942 1.16339 0.82466
10 0.1 0.99988 -0.22044 0.37730
0.3 1.53273 0.24704 0.44987
0.5 1.95849 0.47017 0.51545
1.0 2.76174 0.81974 0.65392
2.0 3.90632 1.26603 0.86787
Table 2.3: Surface shear stresses and heat transfer parameters (f ′′(0),−g′(0),−θ′(0)) for
CWT case when Pr = 0.7 and λ1 = 1
56
The surface shear stress parameters (f ′′(0),−g′(0)) and the surface heat transfer
parameter (−θ′(0)) are decrease with decreasing A∗. The physical reason for this behavior
is that both the viscous and thermal boundary layer thicknesses increase with decreasing
A∗ as it was mentioned earlier. For example, for λ = λ1 = 1, P r = 0.7, f ′′(0),−g′(0) and
−θ′(0) decrease by about 58%, 71% and 40%, respectively, when acceleration parameter
A∗ decreases from 2 to 0.5 (See Table 2.3). It is observed that the surface shear stress pa-
rameter in x- direction f ′′(0) reaches zero for some value of the parameter A∗ = A∗0 which
depends on λ and λ1. Dependence of A∗0 on rotational parameter λ and buoyancy param-
eter λ1 can be noticed from the approximate data that A∗0 = 0.115, 0.026, 0.010 for λ =
1, 3, 5, respectively, when λ1 = 1 and A∗0 = 0.115, 0.075, 0.056 for λ1 = 1, 3, 5, respec-
tively, when λ = 1. It may be remarked that for unsteady flows the vanishing of the
surface shear stress does not imply separation. The unsteady separation is characterized
by the vanishing of both shear and velocity at an interior point of the boundary layer as
seen in a co-ordinate system which is moving with separation [55, 117]. It is also found
that the value of f ′′(0) becomes negative for further reduction in the values of A∗ ( i.e.,
A∗ < A∗0). The flow reversals are observed near the wall and the back flow profiles are
already presented in Figure 2.3.
The surface shear stress parameters (f ′′(0),−g′(0)) and the surface heat transfer
parameter (−θ′(0)) are increase with λ or λ1 due to the reduction of viscous and ther-
mal boundary layer thicknesses as mentioned earlier. However, the effects of rotational
paramter λ and buoyancy parameter λ1 on −g′(0) and − θ′(0) are small. For example,
for A∗ = λ1 = 1, P r = 0.7, f ′′(0),−g′(0) and − θ′(0) increase, respectively, by about
114% , 26% and 11% when λ increase from 1 to 10. Also for A∗ = λ = 1, P r =
0.7, f ′′(0),−g′(0) and − θ′(0) increase about 178%, 49% and 22%, respectively, as λ1
increases from 1 to 10.
57
0.1 1 2 3
−0.5
0.5
1.5
A*
−g| (0
)
1.0 = λ 3.0 5.0 10.0
Figure 2.12: Variations of surface shear stress parameter (−g′(0)) with A∗ for CWT case
when λ1 = 1 and Pr = 0.7
0.1 1 2 3
0.7
1
− θ
| (0)
A*
1.0 = λ3.05.010.0
Figure 2.13: Variations of heat transfer parameter (−θ′(0)) with A∗ for CWT case
when λ1 = 1 and Pr = 0.7
58
Case(ii): Constant Heat Flux.
For the constant heat flux ( CHF) case, the effects of buoyancy parameter λ∗1 and
Prandtl number Pr on the surface shear stresses and the surface heat transfer parameters
(F ′′(0),−G′(0), 1Θ(0)
) are presented in Table 2.4. Figures 2.14 and 2.15 display the effects
of λ∗1 and Pr on velocity and temperature profiles (F ′, G, Θ). Similar to the CWT case,
the velocity profile (F ′) shows an overshoot near the wall in buoyancy assisting flows for
lower Prandtl number fluid (Pr = 0.7) and its magnitude increases with λ∗1 but for higher
Prandtl number fluid (Pr = 7.0) or in buoyancy opposing flows, the velocity overshoot is
not present in velocity profiles F ′(see Figure 2.14).
An interesting feature to be noted is that while in the CWT case, the temperature
profile θ varies from ‘1′ on the wall to ‘0′at the edge of the boundary layer η∞, in the
CHF case the temperature on the wall is different from ‘1′. This is to be expected in
view of the boundary condition on the wall being imposed on Θ′(η) rather than Θ(η).
Since Θ′(0) = −1 on the wall, all the temperature profiles Θ are equally inclined to the
Θ- axis at η = 0 as shown in Figure 2.15. Further, it is noticed in Table 2.4 that for
buoyancy assisting flow (λ1 > 0), the surface shear stress parameters (F ′′(0),−G′(0))
decrease with increasing Prandtl number Pr because the higher Prandtl number fluid
implies more viscous fluid which increases the boundary layer thickness and consequently
reduces the shear stresses. On the other hand, the surface heat transfer parameter 1Θ(0)
increases significantly with Pr as the higher Prandtl number fluid has a lower thermal
conductivity which results in thinner thermal boundary layer and hence a higher surface
heat transfer rate. For example for A∗ = λ = 1, λ1∗ = 10 as Pr increase from 0.7 to 7.0,
F ′′(0) and − G′(0) decrease by about 49% and 16%, respectively, but 1Θ(0)
increases by
87%. From the engineering view point, the heat transfer rate should not be large. This
can be achieved by (a) using a low Prandtl number fluid (such as air, Pr = 0.7), (b)
maintaining the surface at a constant temperature instead of at a constant heat flux, and
(c) by imposing the buoyancy force in the opposing direction to that of forced flow.
59
0 1 2 3 3.5
0.5
1
1.4
η
F| , G Pr = 0.7
Pr = 7.0, λ1* = 10
_______
− − − − −
λ1* = 10
5
1
−1
1
10 G
F|
Figure 2.14: Effects of λ1∗ and Pr on F ′ and G for CHF case when A∗ = λ = 1.
0 1 2 3
1
1.8
η
Θ
λ1* = 1
1
10
5
10
Pr = 0.7
Pr = 7.0
Figure 2.15: Effects of λ1∗ and Pr on temperature profile (Θ), for CHF case when
A∗ = λ = 1.
60
Since the structure of the equations in both the cases (CWT and CHF) are almost
similar, it is natural to expect the effects of A∗ and λ to be similar in the present CHF
case as in the CWT case. Therefore, the results due to the effects of A and λ are found
to be qualitatively the same as those for the CWT case and the detailed description is
not repeated here.
Pr λ1∗ F ′′(0) −G′(0) 1
Θ(0)
0.7 1 1.3326 0.6555 0.6060
3 1.9426 0.7604 0.6451
5 2.4442 0.8323 0.6779
10 3.5456 0.9624 0.7143
7.0 1 1.0982 0.6263 1.1140
3 1.2947 0.7016 1.1791
5 1.4632 0.7393 1.2525
10 1.8365 0.7974 1.3357
Table 2.4: Surface shear-stresses and heat transfer parameter (F ′′(0),−G′(0), 1Θ(0)
) for
CHF case when Pr = 0.7 and A∗ = 1
2.6 CONCLUSIONS
A new self-similar solution of the unsteady mixed convection boundary layer flow in the
stagnation point region of a rotating sphere has been obtained with the constant wall
temperature and constant heat flux conditions. The present results have been compared
with the existing results and found to be in good agreement. The surface shear stress and
heat transfer parameters are enhanced by the buoyancy force, the acceleration of the free
stream and the rotation of the sphere. The effect of the buoyancy force is more pronounced
for small Prandtl numbers than for large Prandtl numbers. For a fixed buoyancy force,
the heat transfer parameter increases but the shear stress parameters decrease with the
increase of Prandtl number. The surface shear stress and heat transfer parameters for
the constant heat flux case are more than those of the constant wall temperature case.
For a certain value of the acceleration parameter, the surface shear stress in x- direction
vanishes. However, it does not imply flow separation since the flow is unsteady. Below
this value of the acceleration parameter the surface shear stress in x- direction becomes
negative and reverse flow occurs in the x-component of the velocity profiles.
61
CHAPTER 3
SELF-SIMILAR SOLUTION TO UNSTEADY
MIXED CONVECTION FLOW ON A ROTATING
CONE
3.1 INTRODUCTION
Laminar boundary layer flows exhibiting similarity have long played an important role
in exposing the influence of physical, dynamical and thermal parameters without intro-
ducing the complications of non-similar solutions and in providing bases for approximate
methods of calculating more complex non-similar cases. In case when the conditions for
similarity are satisfied, the complex set of partial differential equations governing the flow
transforms to a system of ordinary differential equations which evidently constitutes a
considerable mathematical simplification of the problem. Most existing exact solutions
in fluid mechanics are similarity solutions in the sense that the number of independent
variables is reduced by one or more. Similar solutions are the only accurate solutions cov-
ering an extensive range of such parameters as skin friction, heat transfer, boundary layer
displacement thickness, etc. The application of similarity is good in the neighbourhood
of a stagnation point as well as for the case of slowly varying external flow properties or
a highly cooled surface. Similarity solutions can also be employed in analyzing the non-
similar flows containing limited regions of locally similar flows. Using the method of Lie
group transformation, Ma and Hui[78] have derived all group invariant similarity solutions
of the unsteady two dimensional incompressible boundary layer equations. The numerical
results have been also presented for the unsteady two dimensional laminar incompressible
stagnation point flow.
In many practical circumstances of moderate flow velocities or large wall-fluid
temperature differences, the influence of buoyancy forces and associated free convec-
62
tion motions may significantly affect both the heat transfer and shear stress. Merk and
Prins[82] obtained a similarity solution for the case of an isothermal cone while Hering
and Grosh[41] have obtained a number of similarity solutions for non-isothermal cone with
prescribed wall temperature being a power function of the distance from the apex along
the generator. Hartnett and Deland[50] studied the influence of Prandtl number(Pr) on
the heat transfer rates from rotating non-isothermal disks and cones where the analysis
was restricted to the forced convection conditions. The same problem was solved for low
Prandtl numbers by Hering and Grosh [43] and Sparrow and Guinle[121] and for high
Prandtl numbers by Roy[108]. Pop and Cheng[101] have used an integral method to
study the transverse curvature effect on free convection of a Darcian fluid about a verti-
cal cone with a power-law variation of wall temperature. Many situations exist wherein
the transport processes are influenced by buoyancy forces arising due to both thermal
and mass diffusion. This phenomenon occurs, for example, in chemical processes such as
drying and dehydration operations in chemical and food processing plants. Gebhart and
Pera[35] have studied the nature of natural convection flows from a vertical cone due to
the combined effects of thermal and mass diffusion.
Cone-shaped bodies are often encountered in many engineering applications and
the heat transfer problem of mixed convection boundary layer flow over a rotating cone,
which occurs in rotating heat exchangers, are extensively used by the chemical and au-
tomobile industries. Moreover, convective heat transfer on a rotating cone has several
important applications such as design of canisters for nuclear waste disposal, nuclear re-
actor cooling system, geothermal reservoirs. The cooling of the nose-cone of a re-entry
vehicle by spinning the nose[95] may also be considered as a possible application of the
present study. The rotational motion of the cone induces a circumferential velocity in the
fluid through the action of viscosity. Further, due to the action of the centrifugal force
field, the fluid is impelled along the cone surface parallel to a cone ray, and to satisfy
conservation of mass, fluid distant from the cone migrates towards it, replacing the fluid
which has been centrifuged along the cone surface. If the cone surface and free stream
fluid temperature differ, not only energy will be transferred to the flow but also density
difference will exist. In a gravitational field, these density differences result in an addi-
63
tional force(buoyancy force) besides that due to the viscous action or the centrifugal force
field. In many practical circumstances of moderate flow velocities and large wall-fluid
temperature differences, the magnitudes of buoyancy force and the centrifugal force are of
comparable order and convective heat transfer process is considered as mixed convection.
Hering and Grosh [41] have obtained a number of similarity solutions for cones with pre-
scribed wall temperature being a power function of the distance from the apex along the
generator. Hartnett and Deland [50] studied the influence of Prandtl number(Pr) on the
heat transfer rates from rotating non-isothermal disks and cones where the analysis was
restricted to the forced convection conditions. Hering and Grosh [42] analyzed the practi-
cal case of mixed convection from a vertical rotating cone for Pr= 0.7. They made use of
a similarity transformation suggested by Tien and Tsuji [134], and found that (Gr/Re2)
is the dominant dimensionless parameter that would categories the three regions, namely
forced, free and mixed convection. An integral analysis of this problem was performed
by Himasekhar and Sarma[46] to obtain explicit expressions for the estimation of heat
transfer and moment co-efficient. Not surprisingly, the problem of mixed convection flow
on a rotating cone has attracted the attention of many investigators [47, 48, 143]. All
these studies pertain to steady flows. In many practical problems, the flow could be un-
steady due to the angular velocity of the spinning body which varies with time or due
to the impulsive change in the angular velocity of the body or due to the free stream
angular velocity which varies with time. The unsteady boundary layer flow of an impul-
sively started translating and spinning rotational symmetric body has been investigated
by Ece[29] who obtained the solution for small values of times. The corresponding heat
transfer problem has been considered by Ozturk and Ece [97]. Therefore, it is important
to develop a similarity solutions for unsteady mixed convection flow on a rotating cone in
a rotating viscous fluid when the angular velocity of the cone and the free stream angular
velocity depend on time in a particular fashion.
3.2 STATEMENT OF THE PROBLEM
The objective of the present analysis is to develop a new self-similarity solution for the
unsteady mixed convection flow on a rotating cone in a rotating viscous incompressible
64
fluid due to the combined effects of thermal and mass diffusion. It is observed that a
similar solution is possible when the free stream angular velocity and the angular velocity
of the cone vary inversely as a linear functions of time. The basic governing partial
differential equations with three independent variables have been reduced to the ordinary
differential equations using suitable similarity transformations. The resulting system of
non-linear coupled ordinary differential equations has been solved numerically using an
implicit finite difference scheme with the combination of quasilinearization technique.
Both prescribed wall temperature and prescribed heat flux conditions are considered.
Our specific aim in this chapter is to investigate the effects of the variation of buoyancy
forces ratio and angular velocities ratio on skin friction coefficients, Nusselt and Sherwood
numbers. The effect of various parameters on the velocity, temperature and concentration
profiles are also considered.
It may be remarked that the present analysis is more general than those of previous
investigators. Particular cases of this present results are compared with those of Hering
and Grosh[42] and Himasekhar et al[48]. Comparisons show that the results are in good
agreement with the results obtained by earlier investigators for few particular cases of the
present study.
3.3 MATHEMATICAL FORMULATION
We consider the unsteady laminar viscous incompressible fluid flowing over an infinite
rotating cone in a rotating fluid. Both the cone and the fluid are rotating about the axis
of the cone with time-dependent angular velocities either in the same direction or in the
opposite direction. This introduces unsteadiness in the flow field. Figure 3.1 shows the
co-ordinate system and the physical model. We have taken the rectangular co-ordinate
system (x, y, z) where x is measured along a meridional section, the y- axis along a circular
section and z- axis normal to the cone surface. Let u, v and w be the velocity components
along x(tangential), y(circumferential or azimuthal) and z (normal) directions, respec-
tively. The buoyancy forces arise due to the temperature and concentration variations in
the fluid and the flow is taken to be axisymmetric. Under the above assumptions and
65
using the Boussinesq approximation, the governing boundary layer momentum, energy
and diffusion equations can be expressed as [36, 48, 42]:
..................................................................................................................................................................................................................................................................................................................
?
-
Y
g∗
z, w
x, uy, v
Tw or qw
T∞
0
Y
α∗
Ω1
?Ω2
Figure 3.1: Physical model and co-ordinate system.
(xu)x + (xw)z = 0 (3.1)
ut +uux +wuz− v2
x= −ve
2
x+νuzz +g∗β cos α∗(T −T∞)+g∗β∗ cos α∗(C−C∞)(3.2)
vt + uvx + wvz +uv
x= (ve)t + νvzz (3.3)
Tt + uTx + wTz = αTzz, (3.4)
Ct + uCx + wCz = DCzz. (3.5)
The initial conditions are
u(0, x, z) = ui(x, z), v(0, x, z) = vi(x, z), w(0, x, z) = wi(x, z),
T (0, x, z) = Ti(x, z), C(0, x, z) = Ci(x, z) (3.6)
66
and the boundary conditions are given by
u(t, x, 0) = w(t, x, 0) = 0, v(t, x, 0) = Ω1x sin α∗(1− st∗)−1;
T (t, x, 0) = Tw and C(t, x, 0) = Cw (for PWT case);
−kTz(t, x, 0) = qw, and − ρDCz(t, x, 0) = mw (for PHF case);
u(t, x,∞) = 0, v(t, x,∞) = ve = Ω2x sin α∗(1− st∗)−1;
T (t, x,∞) = T∞, C(t, x,∞) = C∞ (3.7)
Here α∗ is the semi-vertical angle of the cone; ν is the kinematic viscosity ; ρ is
the density; t and t∗(= Ω sin α∗t) are the dimensional and dimensionless times, respec-
tively ; Ω1 and Ω2 are the angular velocities of the cone and the fluid far away from
the surface, respectively; Ω(= Ω1 + Ω2) is the composite angular velocity; g∗ is the ac-
celeration due to gravity; T is the temperature; C is the species concentration; β is the
volumetric co-efficient of thermal expansion; β∗ is the volumetric co-efficient of expansion
for concentration; α and D are thermal and mass diffusivity, respectively; Subscripts t,
x and z denote partial derivatives with respect to the corresponding variables and the
subscripts e, i, w and ∞ denote the conditions at the edge of the boundary layer, initial
conditions, conditions at the wall and free stream conditions , respectively ; C0 and T0 are
the values of Cw and Tw at t∗ = 0, respectively; C∞ and T∞ are constants; qw and mw
are, respectively, the heat and mass flux at the wall.
Equations (3.1)-(3.5) are a system of partial differential equations with three in-
dependent variables x, z and t. It has been found that these partial differential equations
can be reduced to a system of ordinary differential equations, if we take the velocity at
the edge of the boundary layer ve and the angular velocity of the cone to vary inversely
as a linear functions of time. Consequently, applying the following transformations for
prescribed wall temperature (PWT) case:
ve = Ω2x sin α∗(1− st∗)−1; η = (Ω sin α∗/ν)12 (1− st∗)−
12 z, t∗ = (Ω sin α∗)t;
67
u(t, x, z) = −2−1Ωx sin α∗(1− st∗)−1f ′(η); v(t, x, z) = Ωx sin α∗(1− st∗)−1g(η);
w(t, x, z) = (νΩ sin α∗)12 (1− st∗)−
12 f(η); T (t, x, z)− T∞ = (Tw − T∞)θ(η);
Tw − T∞ = (T0 − T∞)(x
L)(1− st∗)−2; C(t, x, z)− C∞ = (Cw − C∞)φ(η);
Cw − C∞ = (C0 − C∞)(x
L)(1− st∗)−2; Gr1 = g∗β cos α∗(T0 − T∞)
L3
ν2;
ReL = Ω sin α∗L2
ν; λ1 =
Gr1
Re2L
; Gr2 = g∗β∗ cos α∗(C0 − C∞)L3
ν2; λ2 =
Gr2
Re2L
;
α1 =Ω1
Ω; N =
λ2
λ1
; Pr =ν
α; Sc =
ν
D(3.8)
and for the prescribed heat flux (PHF) case:
ve = Ω2x sin α∗(1− st∗)−1; η = (Ω sin α∗/ν)12 (1− st∗)−
12 z, t∗ = (Ω sin α∗)t;
u(t, x, z) = −2−1Ωx sin α∗(1− st∗)−1F ′(η); v(t, x, z) = Ωx sin α∗(1− st∗)−1G(η);
w(t, x, z) = (νΩ sin α∗)12 (1− st∗)−
12 F (η); ReL = Ω sin α∗
L2
ν;
qw = q0(x
L)(1− st∗)−
52 ; T (t, x, z)− T∞ = (Ω sin α∗/ν)−
12 (1− st∗)
12 (
qw
k)Θ(η);
C(t, x, z)− C∞ = (Ω sin α∗/ν)−12 (1− st∗)
12 (
mw
ρD)Φ(η); mw = m0(
x
L)(1− st∗)−
52 ;
Gr∗1 =gβ cos α∗(q0)L
4
kν2; λ∗1 =
Gr∗1
Re52L
; Gr∗2 =gβ∗ cos α∗(m0)L
4
ρDν2;
λ∗2 =Gr∗2
Re52L
; α1 =Ω1
Ω; N∗ =
λ∗2λ∗1
; Pr =ν
α; Sc =
ν
D(3.9)
to equations (3.1)-(3.5), we find that equations (3.1) is identically satisfied, and equations
(3.2)-(3.5) for the prescribed wall temperature (PWT) case reduce to,
68
f ′′′ − ff ′′ + 2−1f ′2 − 2[g2 − (1− α1)2]−
2λ1(θ + Nφ)− s(f ′ + 2−1ηf ′′) = 0 (3.10)
g′′ − [fg′ − gf ′] + s(1− α1 − g − 2−1ηg′) = 0 (3.11)
Pr−1θ′′ − (fθ′ − f ′θ
2)− s(2θ + 2−1ηθ′) = 0 (3.12)
Sc−1φ′′ − (fφ′ − f ′φ
2)− s(2φ + 2−1ηφ′) = 0 (3.13)
For PHF case, the equations corresponding to equations (3.10)-(3.13) are given by
F ′′′ − FF ′′ + 2−1F ′2 − 2[G2 − (1− α1)2]−
2λ∗1(Θ + N∗Φ)− s(F ′ + 2−1ηF ′′) = 0 (3.14)
G′′ − [FG′ − F ′G] + s(1− α1 −G− 2−1ηG′) = 0 (3.15)
Pr−1Θ′′ − (FΘ′ − F ′Θ2
)− s(2Θ + 2−1ηΘ′) = 0 (3.16)
Sc−1Φ′′ − (FΦ′ − F ′Φ2
)− s(2Φ + 2−1ηΦ′) = 0 (3.17)
The boundary conditions for the PWT case can be expressed as
f(0) = 0 = f ′(0), g(0) = α1, θ(0) = φ(0) = 1
f ′(∞) = 0, g(∞) = 1− α1, θ(∞) = φ(∞) = 0 (3.18)
The boundary conditions for the PHF case are reduced to
F (0) = 0 = F ′(0), G(0) = α1, Θ′(0) = Φ′(0) = −1
F ′(∞) = 0, G(∞) = 1− α1, Θ(∞) = Φ(∞) = 0 (3.19)
Here η is the similarity variable; f, F are the dimensionless stream functions
for the PWT and PHF cases, respectively; f ′ and g are, respectively, the dimensionless
velocity along x-and y -directions for the PWT case; F ′ and G are the corresponding
velocities for the PHF case, respectively; θ and φ are the dimensionless temperature
and concentration for the PWT case; Θ and Φ are the dimensionless temperature and
concentration for the PHF case; ReL is the Reynolds number ; Gr1, Gr2, Gr∗1 and Gr∗2
are the Grashof numbers; λ1, λ2, λ∗1 and λ∗2 are the buoyancy parameters; N , N∗ are the
69
ratio of Grashof numbers for PWT and PHF case, respectively; α1 is the ratio of angular
velocity of the cone to the composite angular velocity; Pr and Sc are the Prandtl and
Schmidt numbers, respectively; s is the parameter characterizing the unsteadiness in the
free stream velocity ve = Ω2x sin α∗(1− st∗)−1. The flow is accelerating if s > 0 provided
st∗ < 1 and the flow is decelerating if s < 0. Further, α1 = 0 implies that the cone is
stationary and the fluid is rotating, α1 = 1 represents the case where the cone is rotating
in an ambient fluid, and for α1 = 0.5, the cone and the fluid are rotating with equal
angular velocity in the same direction. For α1 < 0.5, Ω1 < Ω2 and for α1 > 0.5, Ω1 > Ω2.
The ratio of Grashof numbers, denoted by the parameter N(PWT case) or N∗ (PHF case)
measures the relative importance of thermal and species or chemical diffusion in inducing
the buoyancy forces which drive the flow. N or N∗ = 0 for no species diffusion, infinite for
the thermal diffusion, positive for the case when the buoyancy forces due to temperature
and concentration difference act in the same direction and negative when they act in the
opposite direction.
It may be remarked that the equations (3.10)-(3.12) for α1 = 1, s = 0, N =
0 and λ1 = Gr1
Re2L
of the PWT case are the same as those of Hering and Grosh [42],
and Himasekhar et al.[48]. The set of partial differential equations (3.1)-(3.5) governing
the flow has to be solved subjected to initial conditions (3.6) and boundary conditions
(3.7). Since we are interested in the self-similar solutions, we solve the ordinary differen-
tial equations (3.10)-(3.13) under boundary conditions (3.18) for PWT case or equations
(3.14)-(3.17)under boundary conditions (3.19) for PHF case. The initial conditions will
not be used here and they do not effect the solution of equations (3.10)-(3.13) under the
conditions (3.18) or of equations (3.14)-(3.17) under the conditions (3.19). Self-similar
solution implies that the solution at different times may be reduced to a single solutions
i.e., the solution at one value of time t is similar to the solution at any other value of time
t. This similarity property permits a decrease in the number of independent variables
from three to one (in the present case) and yields treatment using ordinary differential
equations instead of partial differential equations.
The quantities of physical interest are as follows [36]:
The surface skin friction co-efficient in x- and y-directions for the PWT case are, respec-
70
tively, given by
Cfx =[2µ(∂u
∂z)]z=0
ρ[Ωx sin α∗(1− st∗)−1]2= −Re
− 12
x f ′′(0)
Cfy = − [2µ(∂v∂z
)]z=0
ρ[Ωx sin α∗(1− st∗)−1]2= −2Re
− 12
x g′(0),
Thus,
CfxRe12x = −f ′′(0)
2−1CfyRe12x = −g′(0), (3.20)
where Rex = Ωx2 sin α∗(1− st∗)−1/ν. Similarly, the surface skin friction coefficients in x-
and z- directions for PHF case are, respectively, given by
CfxRe12x = −F ′′(0)
2−1CfyRe12x = −G′(0) (3.21)
The Nusselt number and Sherwood number for the PWT case can be expressed as
NuRe− 1
2x = −θ′(0)
ShRe− 1
2x = −φ′(0), (3.22)
where Nux = − [x( ∂T∂z
)]z=0
Tw−T∞and Shx = − [x( ∂C
∂z)]z=0
Cw−C∞.
Similarly, the Nusselt number and Sherwood number for the PHF case are, re-
spectively, given in the form
NuRe− 1
2x =
1
Θ(0)
ShRe− 1
2x =
1
Φ(0). (3.23)
71
3.4 METHOD OF SOLUTION
For the constant wall temperature case, equations (3.10)-(3.13) under the boundary con-
ditions (3.18) have been linearized by using the quasilinearization technique as described
in the Section 2.4.2 and the following set of linear ordinary differential equations are
obtained.
h′′(i+1)+ X i
1h′(i+1)
+ X i2h
(i+1) + X i3g
(i+1) + X i4θ
(i+1) + X i5φ
(i+1) = X i6 (3.24)
g′′(i+1)+ Y i
1 g′(i+1)+ Y i
2 g(i+1) + Y i3 h(i+1) = Y i
4 (3.25)
θ′′(i+1)+ Zi
1θ′(i+1)
+ Zi2θ
(i+1) + Zi3h
(i+1) = Zi4 (3.26)
φ′′(i+1)+ M i
1φ′(i+1)
+ M i2φ
(i+1) + M i3h
(i+1) = M i4 (3.27)
where h = f ′ and the corresponding boundary conditions for equations (3.18) become
hi+1 = 0, gi+1 = α1, θi+1 = 1 = φi+1 at η = 0
hi+1 = 0, gi+1 = 1− α1, θi+1 = 0 = φi+1 at η = η∞
The coefficient functions with iterative index i are known and the functions with
iterative index i + 1 are to be determined. The coefficients in equations (3.24) - (3.27)
are given by
X i1 = −f − 2−1ηs Y i
1 = −f − 2−1ηs Z i1 = −Pr(f + ηs
2) M i
1 = −Sc(f + ηs2)
X i2 = h− s Y i
2 = h− s Z i2 = Pr(h/2− 2s) M i
2 = Sc(h/2− 2s)
X i3 = −4g Y i
3 = g Z i3 = 2−1Pr θ M i
3 = 2−1Scφ
X i4 = −2λ1 Y i
4 = gh− s(1− α1) Zi4 = 2−1Pr h θ M i
4 = 2−1Sc h φ
X i5 = −2λ1N X i
6 = 2−1h2 − 2[g2 + (1− α1)2]
The linear system with boundary conditions is discretized by introducing a mesh
in the η direction. The boundary η → ∞ is replaced by a finite boundary η = η∞. Now
the interval [0; η∞] is divided into N equal subintervals of height 4η. In accordance with
72
the above mesh system, the derivatives of the dependent variables are approximated by
a central difference formula as
h′′ =(hn+1 − 2hn + hn−1)
(4η)2,
h′ =(hn+1 − hn−1)
2(4η), (3.28)
Similar expressions can be written for g, θ and φ. The finite difference method
in combination with quasilinearization technique was outlined by Inouye and Tate[56]
to solve the two point boundary value problems. Thus, applying the finite difference
scheme,we get a system of equations which can be written in the matrix form as
AnWn−1 + BnWn + CnWn+1 = Dn, 2 ≤ n ≤ N (3.29)
where the vectors and coefficient matrices are given by
Wn =
h
g
θ
φ
n
, Dn =
d1
d2
d3
d4
n
, An =
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
n
,
Bn =
b11 b12 b13 b14
b21 b22 b23 b24
b31 b32 b33 b34
b41 b42 b43 b44
n
, Cn =
c11 c12 c13 c14
c21 c22 c23 c24
c31 c32 c33 c34
c41 c42 c43 c44
n
73
The elements of matrices An, Bn, Cn and Dn are
a11 = 1−X14η2
a12 = 0 a13 = 0 a14 = 0
a21 = 0 a22 = 1− Y14η2
a23 = 0 a24 = 0
a31 = 0 a32 = 0 a33 = 1− Z14η2
a34 = 0
a41 = 0 a42 = 0 a43 = 0 a44 = 1−M14η2
b11 = −2 + X2(4η)2 b12 = X3(4η)2 b13 = X4(4η)2 b14 = X5(4η)2
b21 = Y3(4η)2 b22 = −2 + Y2(4η)2 b23 = 0 b24 = 0
b31 = Z3(4η)2 b32 = 0 b33 = −2 + Z2(4η)2 b34 = 0
b41 = M3(4η)2 b42 = 0 b43 = 0 b44 = −2 + M2(4η)2
c11 = 1 + X14η2
c12 = 0 c13 = 0 c14 = 0
c21 = 0 c22 = 1 + Y14η2
c23 = 0 c24 = 0
c31 = 0 c32 = 0 c33 = 1 + Z14η2
c34 = 0
c41 = 0 c42 = 0 c43 = 0 c44 = 1 + M14η2
d1 = X5(4η)2 d2 = Y4(4η)2 d3 = Z4(4η)2 d4 = M4(4η)2
W1 and WN+1 can be obtained from boundary conditions at η = 0 and at η = η∞:
W1 =
h
g
θ
φ
η=0
=
0
α1
1
1
WN+1 =
h
g
θ
φ
η=η∞
=
0
1− α1
0
0
(3.30)
The equations (3.29) together with the boundary conditions (3.30) can be solved by
Varga’s algorithm
Wn = −EnWn+1 + Jn, 1 ≤ n ≤ N
where En = (Bn − AnEn−1)−1Cn
74
Jn = (Bn − AnEn−1)−1(Dn − AnJn−1), 2 ≤ n ≤ N
E1 = EN+1 =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
and J1 =
0
α1
1
1
, JN+1 =
0
1− α1
0
0
Note that here h = f ′. The computations are carried out using the following initial
profiles
f =(1− (1 + η)e−η)
η∞; f ′ =
η
η∞e−η
g = α1e−η + (1− α1)
η
η∞; θ = e−η = φ (3.31)
Similarly for the case of constant heat flux case, equations (3.14)-(3.17) under the
boundary conditions (3.19) have been linearized by using the quasilinearization technique
and the following set of linear ordinary differential equations are obtained.
H ′′(i+1)+ X i
1H′(i+1)
+ X i2H
(i+1) + X i3G
(i+1) + X i4Θ
(i+1) + X i5Φ
(i+1) = X i6 (3.32)
G′′(i+1)+ Y i
1 G′(i+1)+ Y i
2 G(i+1) + Y i3 H(i+1) = Y i
4 (3.33)
Θ′′(i+1)+ Zi
1Θ′(i+1)
+ Zi2Θ
(i+1) + Zi3H
(i+1) = Zi4 (3.34)
Φ′′(i+1)+ M i
1Φ′(i+1)
+ M i2Φ
(i+1) + M i3H
(i+1) = M i4 (3.35)
where H = F ′ and the corresponding boundary conditions for equations (3.19) become
H i+1 = 0, Gi+1 = α1, Θ′(i+1)= −1 = Φ′(i+1)
at η = 0
H i+1 = 0, Gi+1 = 1− α1, Θi+1 = 0 = Φi+1 at η = η∞
The coefficient functions with iterative index i are known and the functions with iterative
index i + 1 are to be determined. The coefficients in equations (3.32) - (3.35) are given
75
by
X i1 = −F − 2−1ηs Y i
1 = −F − 2−1ηs Zi1 = −Pr(F + ηs
2) M i
1 = −Sc(F + ηs2)
X i2 = H − s Y i
2 = H − s Z i2 = Pr(H/2− 2s) M i
2 = Sc(H/2− 2s)
X i3 = −4G Y i
3 = G Z i3 = 2−1Pr Θ M i
3 = 2−1Sc Φ
X i4 = −2λ∗1 Y i
4 = GH − s(1− α1) Zi4 = 2−1Pr H Θ M i
4 = 2−1Sc H Φ
X i5 = −2λ∗1N
∗ X i6 = 2−1H2 − 2[G2 + (1− α1)
2]
The linear system with boundary conditions is discretized by introducing a mesh
in the η direction.The boundary η → ∞ is replaced by a finite boundary η = η∞. Now,
the interval [0; η∞] is divided into N equal subintervals of height 4η. In accordance with
the above mesh system , the derivatives of the dependent variables are approximated by
a central difference formula as
H′′
=(Hn+1 − 2Hn + Hn−1)
(4η)2,
H′=
(Hn+1 −Hn−1)
2(4η), (3.36)
Similar expressions can be written for G, Θ and Φ. The finite difference method in com-
bination with quasilinearization technique was outlined by Inouye and Tate[56] to solve
the two point boundary value problems. Thus, applying the finite difference scheme,we
get a system of equations which can be written in the matrix form as
AnWn−1 + BnWn + CnWn+1 = Dn, 2 ≤ n ≤ N (3.37)
where the vectors and coefficient matrix are given by
Wn =
H
G
Θ
Φ
n
, Dn =
d1
d2
d3
d4
n
, An =
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
n
,
76
Bn =
b11 b12 b13 b14
b21 b22 b23 b24
b31 b32 b33 b34
b41 b42 b43 b44
n
, Cn =
c11 c12 c13 c14
c21 c22 c23 c24
c31 c32 c33 c34
c41 c42 c43 c44
n
The elements of matrices An, Bn, Cn and Dn are
a11 = 1−X14η2
a12 = 0 a13 = 0 a14 = 0
a21 = 0 a22 = 1− Y14η2
a23 = 0 a24 = 0
a31 = 0 a32 = 0 a33 = 1− Z14η2
a34 = 0
a41 = 0 a42 = 0 a43 = 0 a44 = 1−M14η2
b11 = −2 + X2(4η)2 b12 = X3(4η)2 b13 = X4(4η)2 b14 = X5(4η)2
b21 = Y3(4η)2 b22 = −2 + Y2(4η)2 b23 = 0 b24 = 0
b31 = Z3(4η)2 b32 = 0 b33 = −2 + Z2(4η)2 b34 = 0
b41 = M3(4η)2 b42 = 0 b43 = 0 b44 = −2 + M2(4η)2
c11 = 1 + X14η2
c12 = 0 c13 = 0 c14 = 0
c21 = 0 c22 = 1 + Y14η2
c23 = 0 c24 = 0
c31 = 0 c32 = 0 c33 = 1 + Z14η2
c34 = 0
c41 = 0 c42 = 0 c43 = 0 c44 = 1 + M14η2
d1 = X5(4η)2 d2 = Y4(4η)2 d3 = Z4(4η)2 d4 = M4(4η)2
W1 and WN+1 can be obtained from boundary conditions at η = 0. Following the sane
procedure, the equations (3.37) together with the boundary conditions has been solved by
Varga’s algorithm and the computations are carried out using the following initial profiles
F =(1− (1 + η)e−η)
η∞; H = F ′ =
η
η∞e−η
G = α1e−η + (1− α1)
η
η∞; Θ =
(0.5− η
η∞+
(η√2η∞
)2)
η∞ = Φ
3.5 RESULTS AND DISCUSSIONS
The nonlinear coupled ordinary differential equations (3.10)-(3.13) with boundary condi-
tions (3.18) for constant wall temperature case [Equations (3.14)-(3.17) for constant heat
77
flux case with boundary conditions (3.19)] have been solved numerically using an implicit
finite difference scheme in combination with quasilinearization technique and its detailed
description is presented in the previous section. To ensure the convergence of the numer-
ical solution to the true solution, the step size ∆η and the edge of the boundary layer η∞
have been optimized. The results presented here are independent of ∆η and η∞ at least
up to the fourth decimal place. The computations have been carried out with ∆η = 0.01
for various values of Pr(0.7 ≤ Pr ≤ 10), λ1(0 ≤ λ1 ≤ 5), α1(−0.25 ≤ α1 ≤ 1.0), Sc(0.22 ≤Sc ≤ 2.57), s(−1 ≤ s ≤ 1), N(−0.5 ≤ N ≤ 1.0), λ∗1(0 ≤ λ∗1 ≤ 5), and N∗(−0.5 ≤ N∗ ≤1.0). The edge of the boundary layer η∞ is taken between 4 and 6 depending on the values
of parameters. In order to verify the correctness of our method, we have compared our
results for PWT case with those of Hering and Grosh [42] and Himasekhar et al. [48].
The results are found to be in excellent agreement and some of the comparisons are shown
in Table 3.1.
Table 3.1: Comparison of the results (−f ′′(0),−g′(0),−θ′(0)) with those of Himasekhar
et al.[48].
Present Results Himasekhar et al. Result [48].
Pr λ1 −θ′(0) −g′(0) −f ′′(0) −θ′(0) −g′(0) −f ′′(0)
0.0 0.4305 0.6160 1.0199 0.4299 0.6158 1.0256
0.4285∗ 0.6159∗ 1.0205∗
0.7 1.0 0.6127 0.8499 2.1757 0.6120 0.8496 2.2012
0.6120∗ 0.8507∗ 2.2078∗
10 1.0175 1.4061 8.5029 1.0097 1.3990 8.5041
1.0173∗ 1.4037∗ 8.5246∗
0.0 0.5180 0.6160 1.0199 0.5184 0.6158 1.0256
1 1.0 0.7005 0.8250 2.0627 0.7010 0.8176 2.0886
10 1.1494 1.3504 7.9045 1.1230 1.3460 7.9425
0.0 1.4072 0.6154 1.0175 1.4110 0.6158 1.0256
10 1.0 1.5885 0.6894 1.5458 1.5662 0.6837 1.5636
10 2.3582 0.9903 5.0531 2.3580 0.9840 5.0821
∗ Values taken from Hering and Grosh [42]
The results for prescribed wall temperature(PWT) case are presented in Figures
3.2-3.11 and Tables 3.2-3.3, and for prescribed heat flux(PHF) case in Figures 3.12-3.15
and Table 3.4.
78
0 2 4 6
−1
−0.5
0
0.5
0.9
η
−f| (η
)
−0.2 0.0 0.2 0.5 1.0
α1
Figure 3.2: Effect of α1 on velocity profiles (f ′) for PWT case when
s = 0.5, N = λ1 = 1, P r = 0.7 and Sc = 0.94
0 1 2 3 4 5
0
0.4
0.8
1.2
η
−f| (η
)
−0.2 0.0 0.2 0.5
α1
Figure 3.3: Effect of α1 on velocity profiles (f ′) for PWT case when
s = 0.5, N = 1, λ1 = 3, P r = 0.7 and Sc = 0.94
79
0 2 4 6−0.5
0
0.5
1
1.5
η
g
−0.50.00.20.50.751.0
α1
Figure 3.4: Effect of α1 on velocity profiles (g) for PWT case when
s = 0.5, N = λ1 = 1, P r = 0.7 and Sc = 0.94
0 2 4 6−0.5
0
0.5
1
1.5
η
g
−0.50.00.20.50.751.0
α1
Figure 3.5: Effect of α1 on velocity profiles (f ′) for PWT case when
s = 0.5, N = 1, λ1 = 3, P r = 0.7 and Sc = 0.94
80
Case(i): Prescribed Wall Temperature.
For the prescribed wall temperature(PWT) case, the effects of the buoyancy pa-
rameter λ1 and the parameter α1, which is the ratio of the angular velocity of the cone
to the composite angular velocity, on the velocity profiles in the tangential and azimuthal
directions (f ′(η), g(η)) for accelerating flows s = 0.5, N = 1, Sc = 0.94 and Pr = 0.7
are shown in Figures 3.2-3.5. Also the effects of λ1 and α1 on the skin friction coeffi-
cients, Nusselt number and Sherwood number (CfxRe1/2x , CfyRe
1/2x , NuRe
−1/2x , ShRe
−1/2x )
are presented in Table 3.2.
λ1 α1 CfxRe1/2x CfyRe
1/2x NuRe
−1/2x ShRe
−1/2x
1 -0.5 -1.27215 -1.33537 0.55580 0.66305
0.0 0.63241 -0.63949 0.81922 0.95065
0.25 1.31339 -0.22765 0.89011 1.02812
0.5 1.84798 0.19806 0.93700 1.07977
0.75 2.24659 0.62679 0.96563 0.11132
3 -0.5 2.43934 -1.43105 0.91210 1.05951
0.0 3.79522 -0.59651 1.02869 1.18645
0.25 4.31854 -0.13691 1.06539 1.22639
0.5 4.73958 0.33552 1.09111 1.25444
0.75 5.05951 0.81201 1.10712 1.27223
5 -0.5 5.18154 -1.55129 1.06503 1.23177
0.0 6.36147 -0.60724 1.14323 1.31640
0.25 6.82071 -0.10547 1.16887 1.34416
0.5 7.19231 0.40602 1.18730 1.36415
0.75 7.47647 0.92102 1.19926 1.37740
Table 3.2: For CWT case, Skin friction coefficients, Nusselt number and Sherwood number
(CfxRe1/2x , CfyRe
1/2x , NuRe
−1/2x , ShRe
−1/2x ) when N=1,Pr=0.7,Sc=0.94 and s=0.5.
For α1 = 0.5, the cone and the fluid are rotating with equal angular velocity
in the same direction and the non-zero velocities in tangential and azimuthal directions
(f ′(η), g(η)) for α1 = 0.5 are only due to the positive buoyancy parameter λ1 = 1,
which acts like a favorable pressure gradient. When α1 > 0.5, the fluid is being dragged
by the rotating cone and due to the combined effects of buoyancy force and rotating
parameter, the tangential velocity (f ′) increases its magnitude but the azimuthal velocity
(g) decreases its magnitude within the boundary layer. On the other hand, when α1 <
0.5, the cone is dragged by the fluid and the combined effects of buoyancy force and
81
rotation parameter is just the opposite. For α1 < 0 and λ1 = 1, the velocity profiles
f ′(η)(Figure 3.2) and g(η)(Figure 3.4) reach their asymptotic values at the edge of the
boundary layer in an oscillatory manner. Physically these oscillations are caused by
surplus convection of angular momentum present in the boundary layer. Similar trend
has been noticed by King and Lewellen [63], and Stewartson and Troesch [125] in the
absence of buoyancy force. But for higher buoyancy parameter(i.e, for λ1 = 3 in Figures
3.3 and 3.5), the oscillations are not observed. Thus, the buoyancy force which acts like
a favorable pressure gradient suppresses the oscillations in velocity profiles (f ′, g). Since
the positive buoyancy force (λ1 > 0) implies favorable pressure gradient, the fluid gets
accelerated which results in thinner momentum, thermal and concentration boundary
layers. Consequently, the skin friction coefficients, Nusselt number and Sherwood number
(CfxRe1/2x , CfyRe
1/2x , NuRe
−1/2x , ShRe
−1/2x ) are increased as shown in Table 3.2.
Pr Sc CfxRe1/2x CfzRe
1/2x NuRe
−1/2x ShRe
−1/2x
0.7 0.22 1.55445 -0.18622 0.92497 0.52524
0.60 1.39265 -0.21528 0.90102 0.83549
0.94 1.31338 -0.22757 0.89011 1.02816
2.57 1.13111 0.25133 0.86679 1.63912
3.0 0.22 1.28590 -0.21429 1.79674 0.50415
0.60 1.12920 -0.24890 1.76657 0.80323
0.94 1.05237 -0.26441 1.75183 0.99103
2.57 0.87275 0.29469 1.71827 1.59183
7.0 0.22 1.13916 -0.22526 2.66305 0.49616
0.60 0.98246 -0.26198 2.62977 0.78886
0.94 0.90546 -0.27866 2.61338 0.97337
2.57 0.72523 0.31244 2.57533 1.56711
Table 3.3: Skin friction coefficients, Nusselt and Sherwood numbers (CfxRe1/2x , CfyRe
1/2x ,
NuRe−1/2x , ShRe
−1/2x ) when N=1, α1 = 0.5, s=0.5 and λ1 = 1 for PWT case.
In Figures. 3.6 and 3.7, the effects of the Prandtl number Pr and Schmidt num-
ber Sc on the temperature and concentration profiles (θ, φ) for λ1 = 1, α1 = 0.25, s =
0.5 and N = 1 are presented. Also, the effects of Pr and Sc on the skin friction coeffi-
cients, Nusselt number and Sherwood number (CfxRe1/2x , CfyRe
1/2x , NuRe
−1/2x , ShRe
−1/2x )
are given in Table 3.3. Figures 3.6-3.7 show that the increase in Pr and Sc causes a
reduction in thermal and concentration boundary layer thicknesses, respectively . Hence
in Table 3.3, NuRe−1/2x increases with Pr and ShRe
−1/2x increases with Sc. For example,
82
for Sc = 0.94, NuRe−1/2x increases by about 193% as Pr increases from 0.7 to 7.0 and for
Pr = 0.7, ShRe−1/2x increases by about 209% as Sc increases from 0.22 to 2.57.
0 2 4 6
0.5
1
Pr = 0.7
7.0
η
θ Sc = 0.22
Sc = 2.57
3.0
Figure 3.6: Effects of Pr and Sc on θ for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5 and N = 1
0 2 4 6
0.5
1
η
φ
Sc = 0.22
0.94
2.57
Pr = 0.7
Pr = 7.0
Figure 3.7: Effects of Pr and Sc on φ for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5 and N = 1
83
−0.5 0 0.5
0.6
1
1.5
2
2.5
s
Cfx
Re x1/
2
−0.5 = N0.00.51.0
Figure 3.8: Effect of N on skin friction coefficient (CfxRe1/2x ) for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5, P r = 0.7 and Sc = 0.94
−0.5 0 0.5
0.1
0.2
0.3
0.4
s
Cfy
Re x1/
2
−0.5=N0.00.51.0
Figure 3.9: Effect of N on skin friction coefficient (CfyRe1/2x ) for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5, P r = 0.7 and Sc = 0.94
84
−0.5 0 0.5−0.2
0.2
0.6
1
s
NuR
ex−1
/2
−0.5 = N0.00.51.0
Figure 3.10: Effect of N on heat transfer coefficient (NuRe−1/2x ) for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5, P r = 0.7 and Sc = 0.94
−0.5 0 0.5−0.2
0.2
0.6
1
s
ShR
ex−1
/2
−0.5 = N0.00.51.0
Figure 3.11: Effect of N on mass transfer coefficient (ShRe−1/2x ) for CWT case when
λ1 = 1, α1 = 0.25, s = 0.5, P r = 0.7 and Sc = 0.94
85
Figures 3.8 and 3.9 display the effects of the ratio of the buoyancy forces N
(which measures the relative importance of the thermal and species diffusion) and the
unsteady parameter s on the skin friction coefficients (CfxRe1/2x , CfyRe
1/2x for λ1 = 1, α1 =
0.25, P r = 0.7 and Sc = 0.94. For the same above data, effects of of the ratio of the
buoyancy forces N and the unsteady parameter s on the skin friction coefficients, Nusselt
and Sherwood numbers NuRe−1/2x , ShRe
−1/2x ) are shown in Figures 3.10 and 3.11. Due
to the increase in N , the velocity gradient in the primary flow as well as in the secondary
flow i.e., skin friction coefficients (CfxRe1/2x , CfyRe
1/2x ), Nusselt and Sherwood numbers
(NuRe−1/2x , ShRe
−1/2x ) increase for both the steady (s=0) and unsteady (s 6= 0) cases. For
example, the percentage increase in CfxRe1/2x , CfyRe
1/2x , NuRe
−1/2x and ShRe
−1/2x due to
the increase in N from -0.5 to 1.0 is about 280%, 30%, 82% and 70%, respectively , for the
steady case (s=0). Further, as the unsteady parameter (s)increases from -0.5 to 0.5, the
skin friction coefficients (CfxRe1/2x , CfyRe
1/2x ) decrease(Figures 3.8 and 3.9)where as the
Nusselt and Sherwood numbers (NuRe−1/2x , ShRe
−1/2x ) increase(Figures 3.10 and 3.11).
Case(ii): Prescribed Heat Flux.
For the prescribed heat flux (PHF) case, the effects of buoyancy parameter λ∗1
and Prandtl number Pr on the skin friction coefficients, Nusselt and Sherwood numbers
(CfxRe1/2x , CfyRe
1/2x , NuRe
−1/2x , ShRe
−1/2x ) are presented in Table 3.4. Figure 3.12 display
the effects of λ∗1 and Pr on temperature profile(Θ) and Figure 3.13 display the effects of
λ∗1 and Sc on concentration profiles (Φ). Further, the effects of Pr and Sc on tempera-
ture and concentration profiles (Θ, Φ) are shown in Figures 3.14 and 3.15. An interesting
feature to be noted is that while in the PWT case, the temperature and concentration
profiles (θ, φ) vary from 1 on the wall to 0 at the edge of the boundary layer η∞, in the
PHF case the temperature and concentration on the wall are different from 1. This is to be
expected in view of the boundary condition on the wall being imposed on Θ′(η) and Φ′(η)
rather than on Θ(η) and Φ(η). Since Θ′(η) = Φ′(η) = −1 on the wall, all the temperature
and concentration profiles Θ(η) and Φ(η) are equally inclined to the vertical axis at η = 0
as shown in Figures 3.12-3.15. Further, it is noticed in Table 3.4 that for the buoyancy
assisting flow (λ∗1 > 0), the skin friction coefficients (CfxRe1/2x , CfyRe
1/2x ) decrease with
increasing Prandtl number because the higher Prandtl number fluid implies more viscous
86
fluid which increases the boundary layer thickness and consequently reduces the shear
stresses. On the other hand, Nusselt and Sherwood numbers (NuRe−1/2x , ShRe
−1/2x ) in-
crease with Pr. The higher Prandtl number fluid has a lower thermal conductivity which
results in thinner thermal boundary layer and hence a higher surface heat transfer rate.
For example for s = 0.25, α1 = 0.5, N∗ = 1, Sc = 0.94 and λ∗1 = 1 as Pr increase from
0.7 to 7.0, CfxRe1/2x and CfyRe
1/2x decrease by about 61% and 76%, respectively, but
NuRe−1/2x and ShRe
−1/2x increase by 101% and 99%, respectively. From the engineering
view point, the heat transfer rate should not be large. This can be achieved by (a) using a
low Prandtl number fluid (such as air, Pr = 0.7), (b) maintaining the surface at a constant
temperature instead of at a constant heat flux, and (c) by imposing the buoyancy force
in the opposing direction to that of forced flow. Since the structure of the equations in
both the cases (PWT and PHF) are almost similar, it is natural to expect the effects of
α1, N, s and Sc to be similar in the present PHF case as in the PWT case and the results
are therefore not presented here.
Pr Sc λ1∗ CfxRe
1/2x CfyRe
1/2x NuRe
−1/2x ShRe
−1/2x
0.7 0.94 1 2.8296 0.2656 0.6675 0.6680
3 6.8983 0.4099 0.6684 0.6691
5 9.4730 0.4622 0.7428 0.7436
2.57 1 2.3499 0.2161 0.7411 0.7443
3 5.7931 0.3548 0.7420 0.7459
5 8.4210 0.4188 0.7709 0.7754
7.0 0.94 1 1.0929 0.0629 1.3394 1.3297
3 2.9309 0.1445 1.3423 1.3306
5 4.5426 0.2010 1.3442 1.3312
2.57 1 0.9446 0.0482 1.4339 1.4273
3 2.5303 0.1121 1.4365 1.4288
5 3.9096 0.1567 1.4383 1.4299
Table 3.4: Skin friction coefficients, Nusselt number and Sherwood number (CfxRe1/2x ,
CfzRe1/2x , NuRe
−1/2x , ShRe
−1/2x ) for PHF case when N∗ = 1, s=0.25 and α1 = 0.5.
87
0 1 2 3
0.5
1
1.5
Pr = 0.7; λ*1 = 1
Pr = 0.7; λ*1 = 5
Pr = 0.7; λ*1 = 10
Pr = 7.0; λ*1 = 1
Pr = 7.0; λ*1 = 10
η
Θ
Figure 3.12: Effects of λ1∗ and Pr on Θ for PHF case when
α1 = 0.5, s = 0.5, N∗ = 1 and Sc = 0.94
0 1 2 3
0.5
1
1.5
Sc = 0.94; λ*1 = 1
Sc = 0.94; λ*1 = 3
Sc = 0.94; λ*1 = 5
η
Φ
Figure 3.13: Effects of λ1∗ and Sc on Φ for PHF case when
α1 = 0.5, s = 0.5, N∗ = 1 and Pr = 0.7
88
0 1 2 3
0.5
1
1.5
η
Θ
Pr = 0.7; Sc = 0.94Pr = 0.7; Sc = 2.57Pr = 7.0; Sc = 0.94Pr = 7.0; Sc = 2.57
Figure 3.14: Effects of Pr and Sc on Θ for PHF case when
λ∗1 = 1, α1 = 0.5, s = 0.5 and N∗ = 1
0 1 2 3
0.5
1
1.5
η
Φ
Pr=0.7; Sc=0.94Pr=7.0; Sc=0.94Pr=0.7; Sc=2.57Pr=7.0; Sc=2.57
Figure 3.15: Effects of Pr and Sc on Φ for PHF case when
λ∗1 = 1, α1 = 0.5, s = 0.5 and N∗ = 1
89
3.6 CONCLUSIONS
A detailed numerical study to obtain a new self-similar solutions for unsteady mixed con-
vection flow on a rotating cone in a rotating fluid has been carried out with PWT and PHF
conditions. The present results are compared with the available results in the literature
and they are found to be in good agreement. The results indicate that skin friction coeffi-
cients Nusselt and Sherwood numbers are enhanced by the positive buoyancy force. The
buoyancy force (which acts like a favorable pressure gradient) suppresses the oscillations
in the velocity profiles which appears due to surplus convection of angular momentum.
The increase in Prandtl and Schmidt numbers causes a reduction in the thickness of ther-
mal and concentration boundary layers, respectively. Due to the increase in the ratio of
buoyancy forces (N, which measures the relative importance of the thermal and species
diffusion) , the skin friction coefficients, Nusselt and Sherwood numbers increase. For a
fixed buoyancy force, heating by prescribed heat flux yields a higher value of surface heat
transfer rate than heating by prescribed wall temperature.
90
CHAPTER 4
UNSTEADY DOUBLE DIFFUSIVE CONVECTION
FROM A ROTATING CONE IN A ROTATING
FLUID
4.1 INTRODUCTION
In Chapter 3, we have presented a new self-similar solution of unsteady mixed convec-
tion flow on a rotating cone in a rotating fluid due to the combined effects of thermal
and mass diffusion. In many cases, unsteady mixed convection flow problems of engi-
neering interest do not admit similarity solution. The non-similarity in these problems
may be due to the free stream velocity or due to the curvature of the body or due to
the variations in wall temperature and concentration or due to all these effects. If, the
similarity transformations are only able to reduce the number of independent variables,
the transform equations are known as semi-similar and the corresponding solutions are
the semi-similar solutions[109, 129, 145]. In the present chapter, we propose to obtain a
semi-similar solution of unsteady mixed convection flow from the same geometry .
Convective heat transfer in rotating flows over a stationary or rotating cone is
important for the thermal design of various types of industrial equipments such as rotat-
ing heat exchangers, spin stabilized missiles, canisters for nuclear waste disposal, nuclear
reactor cooling system and geothermal reservoirs. The cooling of the nose-cone of re-entry
vehicle by spinning the nose [95] may also be considered as another possible application
of the present study. The early works of Tien and Tsuji [134], and Koh and Price [64]
present a theoretical analysis of the forced flow and heat transfer past a rotating cone
and the influence of Prandtl number on the heat transfer on rotating non-isothermal disks
and cones was described by Hartnett and Deland [50]. Also, the similarity solution of the
mixed convection from a rotating vertical cone in an ambient fluid was obtained by Hering
91
and Grosh [42] for Prandtl number Pr = 0.7. In a further study, Himasekhar et al.[48]
found the similarity solution of the mixed convection flow over a vertical rotating cone in
an ambient fluid for a wide range of Prandtl numbers. Wang [143] has also obtained a
similarity solution of boundary layer flows on rotating cones, discs and axisymmetric bod-
ies with concentrated heat sources. Further, Yih[149] has presented non-similar solutions
to study the heat transfer characteristics in mixed convection about a cone in saturated
porous media. All these studies pertain to steady flows. In many practical problems,
the flow could be unsteady due to the angular velocity of the spinning body which varies
with time or due to the impulsive change in the angular velocity of the body or due to
the free stream angular velocity which varies with time. The unsteady boundary layer
flow of an impulsively started translating and spinning rotational symmetric body has
been investigated by Ece[29] who obtained the solution for small values of times. The
corresponding heat transfer problem has been considered by Ozturk and Ece[97]. In a
most recent investigation, Takhar et al.[127] have presented a study on unsteady mixed
convection flow over a vertical cone rotating in an ambient fluid with a time-dependent
angular velocity in the presence of a magnetic field. Therefore, as a step towards the
eventual development of studies on unsteady mixed convection flows, it is interesting as
well as useful to investigate the combined effects of thermal and mass diffusion on a ro-
tating cone in a rotating viscous fluid where the angular velocity of the cone and the free
stream angular velocity vary arbitrarily with time.
4.2 STATEMENT OF THE PROBLEM
The aim of the present study is to analyze the combined effects of thermal and mass
diffusion on an unsteady mixed convection flow over a rotating cone in a co-rotating fluid.
The unsteadiness in the flow field is due to angular velocity of the cone and the freestream
angular velocity which vary arbitrarily with time. The semi-similar solution of the coupled
non-linear partial differential equations governing the mixed convection flow has been
obtained numerically using the method of quasi-linearization technique and an implicit
finite difference scheme. Both the prescribed wall temperature and prescribed heat flux
conditions are considered. Our specific objective in this chapter is to investigate the
92
effects of the variation of buoyancy forces ratio and angular velocities ratio on skin friction
coefficients, Nusselt number and Sherwood numbers. The effect of various parameters on
the velocity, temperature and concentration profiles are also considered.
It may be remarked that the present analysis is more general than those previous
investigators. Particular cases of these present results are compared with those of Hering
and Grosh [42], Himasekhar et al.[48] and Takhar et al.[127]. Comparisons show that the
results are in good agreement with the results obtained by earlier investigators for few
particular cases of the present study.
4.3 MATHEMATICAL FORMULATION OF THE PROBLEM
Consider the unsteady laminar mixed convection flow over an infinite rotating cone in a
rotating viscous fluid. The unsteadiness in the flow field is introduced by rotating the
cone and the surrounding free stream fluid about the axis of the cone with time-dependent
angular velocities either in the same direction or in the opposite direction. Figure 3.1
shows the co-ordinate system and the physical model. The buoyancy forces arise due to
both the temperature and concentration variations in the fluid and the flow is taken to
be axisymmetric. Both the temperature and concentration at the wall vary linearly with
the tangential co-ordinate x. Under the above assumptions and using the Boussinesq
approximation, the governing boundary layer momentum, energy and diffusion equations
can be expressed as [8, 48, 99, 127]:
(xu)x + (xw)z = 0 (4.1)
ut +uux +wuz− v2
x= −ve
2
x+νuzz +g∗β cos α∗(T −T∞)+g∗β∗ cos α∗(C−C∞)(4.2)
vt + uvx + wvz +uv
x= (ve)t + νvzz (4.3)
Tt + uTx + wTz = αTzz, (4.4)
Ct + uCx + wCz = DCzz. (4.5)
93
The initial conditions are
u(0, x, z) = ui(x, z), v(0, x, z) = vi(x, z), w(0, x, z) = wi(x, z),
T (0, x, z) = Ti(x, z), C(0, x, z) = Ci(x, z), (4.6)
and the boundary conditions are given by
u(t, x, 0) = w(t, x, 0) = 0, v(t, x, 0) = Ω1x sin α∗R(t∗),
T (t, x, 0) = Tw and C(t, x, 0) = Cw for PWT case ,
−kTz(t, x, 0) = qw and − ρDCz(t, x, 0) = mw for PHF case,
u(t, x,∞) = 0, v(t, x,∞) = ve = Ω2x sin α∗R(t∗),
T (t, x,∞) = T∞, C(t, x,∞) = C∞. (4.7)
Here α∗ is the semi-vertical angle of the cone; ν is the kinematic viscosity ; ρ is
the density; t and t∗(= Ω sin α∗t) are the dimensional and dimensionless times, respec-
tively ; Ω1 and Ω2 are the angular velocities of the cone and the fluid far away from
the surface, respectively; Ω(= Ω1 + Ω2) is the composite angular velocity; g∗ is the ac-
celeration due to gravity; T is the temperature; C is the species concentration; β is the
volumetric co-efficient of thermal expansion; β∗ is the volumetric co-efficient of expansion
for concentration; α and D are thermal and mass diffusivity, respectively; Subscripts t,
x and z denote partial derivatives with respect to the corresponding variables and the
subscripts e, i, w and ∞ denote the conditions at the edge of the boundary layer, initial
conditions, conditions at the wall and free stream conditions , respectively ; C0 and T0 are
the values of Cw and Tw at t∗ = 0, respectively; C∞ and T∞ are constants; qw and mw
are, respectively, the heat and mass flux at the wall.
Applying the following transformations for prescribed wall temperature (PWT)
case:
η = (Ω sin α∗/ν)12 z, ve = Ω2x sin α∗R(t∗), Ω = Ω1 + Ω2,
94
t∗ = (Ω sin α∗)t, u(t, x, z) = −2−1Ωx sin α∗R(t∗)fη(η, t∗),
v(t, x, z) = Ωx sin α∗R(t∗)g(η, t∗), w(t, x, z) = (νΩ sin α∗)12 R(t∗)f(η, t∗),
T (t, x, z)− T∞ = (Tw − T∞)θ(η, t∗), Tw − T∞ = (T0 − T∞)(x
L),
C(t, x, z)− C∞ = (Cw − C∞)φ(η, t∗), Cw − C∞ = (C0 − C∞)(x
L),
Gr1 = g∗β cos α∗(T0 − T∞)L3
ν2, ReL = Ω sin α∗
L2
ν, λ1 =
Gr1
Re2L
,
Gr2 = g∗β∗ cos α∗(C0 − C∞)L3
ν2, λ2 =
Gr2
Re2L
,
R(t∗) = 1 + εt∗2, α1 =Ω1
Ω, N =
λ2
λ1
, P r =ν
α, Sc =
ν
D, (4.8)
and for the prescribed heat flux (PHF) case:
η = (Ω sin α∗/ν)12 z, ve = Ω2x sin α∗R(t∗),
t∗ = (Ω sin α∗)t, u(t, x, z) = −2−1Ωx sin α∗R(t∗)Fη(η, t∗),
v(t, x, z) = Ωx sin α∗R(t∗)G(η, t∗),
w(t, x, z) = (νΩ sin α∗)12 R(t∗)F (η, t∗),
T (t, x, z)− T∞ = (Ω sin α∗/ν)−12 (
qw
k)Θ(η, t∗), qw = q0(
x
L),
C(t, x, z)− C∞ = (Ω sin α∗/ν)−12 (
mw
ρD)Φ(η, t∗), mw = m0(
x
L),
Gr∗1 =g∗β cos α∗q0L
4
kν2, λ∗1 =
Gr∗1
Re52L
, α1 =Ω1
Ω, N∗ =
λ∗2λ∗1
,
Gr∗2 =g∗β∗ cos α∗(m0)L
4
ρDν2, λ∗2 =
Gr∗2
Re52L
, P r =ν
α, Sc =
ν
D, (4.9)
95
to equations (4.1)-(4.5), we find that equations (4.1) is identically satisfied, and equations
(4.2)-(4.5) for the prescribed wall temperature (PWT) case reduce to
fηηη −R(t∗)ffηη + 2−1R(t∗)fη2 − 2R(t∗)[g2 − (1− α1)
2]
−2R(t∗)−1λ1(θ + Nφ)−R(t∗)−1(dR
dt∗)fη − fηt∗ = 0, (4.10)
gηη −R(t∗)[fgη − gfη]−R(t∗)−1(dR
dt∗)[g − (1− α1)]− gt∗ = 0, (4.11)
Pr−1θηη −R(t∗)(fθη − fηθ
2)− θt∗ = 0, (4.12)
Sc−1φηη −R(t∗)[fφη − fηφ
2]− φt∗ = 0, (4.13)
and for prescribed heat flux (PHF) case, the equations corresponding to equations (4.10)-
(4.13) are given by
Fηηη −R(t∗)FFηη + 2−1R(t∗)Fη2 − 2R(t∗)[G2 − (1− α1)
2]
−2R(t∗)−1λ1∗(Θ + N∗Φ)−R(t∗)−1(
dR
dt∗)Fη − Fηt∗ = 0, (4.14)
Gηη −R(t∗)[FGη −GFη]−R(t∗)−1(dR
dt∗)[G− (1− α1)]−Gt∗ = 0, (4.15)
Pr−1Θηη −R(t∗)(FΘη − FηΘ
2)−Θt∗ = 0, (4.16)
Sc−1Φηη −R(t∗)[FΦη − FηΦ
2]− Φt∗ = 0. (4.17)
The boundary conditions for the PWT case are given by
f(0, t∗) = 0 = fη(0, t∗), g(0, t∗) = α1, θ(0, t∗) = φ(0, t∗) = 1,
fη(∞, t∗) = 0, g(∞, t∗) = 1− α1, θ(∞, t∗) = φ(∞, t∗) = 0. (4.18)
The boundary conditions for the PHF case are expressed by
F (0, t∗) = 0 = Fη(0, t∗), G(0, t∗) = α1, Θη(0, t
∗) = Φη(0, t∗) = −1,
Fη(∞, t∗) = 0, G(∞, t∗) = 1− α1, Θ(∞, t∗) = Φ(∞, t∗) = 0. (4.19)
96
Here η is the similarity variable; f, F are the dimensionless stream functions for
the PWT and PHF cases respectively; f ′ and g are, respectively, the dimensionless velocity
along x-and y -directions for the PWT case; F ′ and G are the corresponding velocities for
the PHF case, respectively; θ and φ are the dimensionless temperature and concentration
for the PWT case; Θ and Φ are the dimensionless temperature and concentration for
the PHF case; ReL is the Reynolds number ; Gr1, Gr2, Gr∗1 and Gr∗2 are the Grashof
numbers; λ1, λ2, λ∗1 and λ∗2 are the buoyancy parameters; N , N∗ are the ratio of Grashof
numbers for PWT and PHF case, respectively; α1 is the ratio of angular velocity of the
cone to the composite angular velocity; Pr and Sc are the Prandtl and Schmidt numbers,
respectively;
It may be remarked that, α1 = 0 implies that the cone is stationary and the fluid
is rotating, α1 = 1 represents the case where the cone is rotating in an ambient fluid, and
for α1 = 0.5, the cone and the free stream fluid are rotating with equal angular velocity
in the same direction. Thus, for the case α1 < 0.5, Ω1 < Ω2 and for α1 > 0.5, Ω1 > Ω2.
Further for α1 < 0, the cone and the free stream fluid are rotating in the opposite
direction. The ratio of Grashof numbers denoted by the parameter N(PWT case) or N∗
(PHF case) measures the relative importance of thermal and mass diffusion in inducing
the buoyancy forces which drive the flow. N or N∗ = 0 for no species diffusion, infinite for
the thermal diffusion, positive for the case when the buoyancy forces due to temperature
and concentration difference act in the same direction and negative when they act in the
opposite direction.
We have assumed that the flow is steady at time t∗ = 0 and becomes unsteady
for t∗ > 0 due to the time-dependent angular velocities (R(t∗) = 1 + εt∗2, ε ≶ 0) of the
cone and the free stream fluid. Hence, the initial conditions (i.e., conditions at t∗ = 0) are
given by the steady state equations obtained from equations (4.10) - (4.17) by substituting
R(t∗) = 1, dRdt∗ = fηt∗ = gt∗ = θt∗ = φt∗ = 0 and Fηt∗ = Gt∗ = Θt∗ = Φt∗ = 0 when t∗ = 0
as
fηηη − ffηη + 2−1fη2 − 2[g2 − (1− α1)
2]− 2λ1(θ + Nφ) = 0 (4.20)
gηη − [fgη − gfη] = 0 (4.21)
97
Pr−1θηη − (fθη − fηθ
2) = 0 (4.22)
Sc−1φηη − [fφη − fηφ
2] = 0, (4.23)
and for prescribed heat flux (PHF) case, the equations corresponding to equations (4.20)-
(4.23) are given by
Fηηη − FFηη + 2−1Fη2 − [G2 − (1− α1)
2]− 2λ1∗(Θ + N∗Φ) = 0 (4.24)
Gηη − [FGη −GFη] = 0 (4.25)
Pr−1Θηη − (FΘη − FηΘ
2) = 0 (4.26)
Sc−1Φηη − [FΦη − FηΦ
2] = 0. (4.27)
The corresponding boundary conditions for PWT and PHF cases are obtained from (4.18)
and (4.19), respectively, when t∗ = 0 as :
f(0, 0) = 0 = fη(0, 0), g(0, 0) = α1, θ(0, 0) = φ(0, 0) = 1,
fη(∞, 0) = 0, g(∞, 0) = 1− α1, θ(∞, 0) = φ(∞, 0) = 0. (4.28)
F (0, 0) = 0 = Fη(0, 0), G(0, 0) = α1, Θη(0, 0) = Φη(0, 0) = −1,
Fη(∞, 0) = 0, G(∞, 0) = 1− α1, Θ(∞, 0) = Φ(∞, 0) = 0. (4.29)
It may be noted that the steady state equations (4.20)-(4.27) in the absence of
mass diffusion for α1 = 1 and N = 0 of PWT case are the same as those of Hering and
Grosh [42], and Himasekhar et al. [48]. Further, the unsteady equations (10)- (12) in the
absence of mass diffusion (i.e., without the species equation(13)) for α1 = 1 and N = 0
of the PWT case are the same for M = 0 case (i.e., in the absence of magnetic field) of
Takhar et al.[127] who investigated recently an unsteady mixed convection flow from a
rotating vertical cone with a magnetic field.
98
The quantities of physical interest are as follows [8, 99]:
The local surface skin friction coefficients in x- and y-directions for the PWT case are,
respectively, given by
Cfx =[2µ(∂u
∂z)]z=0
ρ[Ωx sin α∗]2= −Re
− 12
x R(t∗)fηη(0, t∗),
Cfy = − [2µ(∂v∂z
)]z=0
ρ[Ωx sin α∗]2= −2Re
− 12
x R(t∗)gη(0, t∗).
Thus,
Re12x Cfx = −R(t∗)fηη(0, t
∗),
2−1Re12x Cfy = −R(t∗)gη(0, t
∗), (4.30)
where Rex = Ωx2 sin α∗/ν. Similarly, the local surface skin friction coefficients in x- and
y- directions for PHF case are, respectively, given by
Re12x Cfx = −R(t∗)Fηη(0, t
∗),
2−1Re12x Cfy = −R(t∗)Gη(0, t
∗). (4.31)
The local Nusselt number and local Sherwood number for the PWT case can be expressed
as
Re− 1
2x Nux = −θη(0, t
∗),
Re− 1
2x Shx = −φη(0, t
∗), (4.32)
where Nux = − [x(∂T∂z
)]z=0
Tw − T∞and Shx = − [x(∂C
∂z)]z=0
Cw − C∞.
Similarly, the local Nusselt and Sherwood numbers for the PHF case are, respectively,
given in the form
Re− 1
2x Nux =
1
Θ(0, t∗),
Re− 1
2x Shx =
1
Φ(0, t∗). (4.33)
99
4.4 METHOD OF SOLUTION
The set of dimensionless equations (4.10) - (4.13) under the boundary conditions (4.18) for
PWT case (Eqs (4.14)- (4.17) under the boundary conditions (4.19) for PHF case) with
the initial conditions obtained from the corresponding steady state equations (4.20)-(4.27)
has been solved numerically using an implicit finite difference scheme in combination with
the quasilinearization technique. The method has been described in Section 2.4 of Chapter
2.
For the case of prescribed wall temperature(PWT) case, linearizing the equations
(4.10) - (4.13) under the boundary condition (4.18) using the quasilinearization, we get
the following equations are obtained
h(i+1)ηη + X i
1h(i+1)η + X i
2h(i+1) + X i
3h(i+1)t∗ + X i
4g(i+1) + X i
5θ(i+1) + X i
6φ(i+1) = X i
7(4.34)
g(i+1)ηη + Y i
1 g(i+1)η + Y i
2 g(i+1) + Y i3 g
(i+1)t∗ + Y i
4 h(i+1) = Y i5 (4.35)
θ(i+1)ηη + Zi
1θ(i+1)η + Zi
2θ(i+1) + Zi
3θ(i+1)t∗ + Zi
4h(i+1) = Zi
5 (4.36)
φ(i+1)ηη + M i
1φ(i+1)η + M i
2φ(i+1) + M i
3φ(i+1)t∗ + M i
4h(i+1) = M i
5 (4.37)
where h = fη and the corresponding boundary conditions for equations (4.18) become
hi+1 = 0, gi+1 = α1, θi+1 = 1 = φi+1 at η = 0
hi+1 = 0, gi+1 = 1− α1, θi+1 = 0 = φi+1 at η = η∞ (4.38)
The coefficient functions with iterative index i are known and the functions with iterative
index i + 1 are to be determined. The coefficients in equations (4.34) - (4.37) are given
by
X i1 = −R(t∗)f
X i2 = R(t∗)h−R−1∂R
∂t∗
100
X i3 = −1
X i4 = −4R(t∗)g
X i5 = −2R−1λ1
X i6 = −2R−1λ1N
X i7 = 2R(t∗)
[h2
4− g2 − (1− α1)
2
]
Y i1 = −R(t∗)f
Y i2 = R(t∗)h−R−1∂R
∂t∗
Y i3 = −1
Y i4 = R(t∗)g
Y i5 = R(t∗)gh−R−1(1− α1)
∂R
∂t∗
Zi1 = −PrR(t∗)f
Zi2 = Prh/2R(t∗)
Zi3 = −Pr
Zi4 = PrR(t∗)θ/2
Zi5 = PrR(t∗)hθ/2
M i1 = −Sc R(t∗) f
M i2 = Sc R(t∗) h/2
M i3 = −Sc
101
M i4 = Sc R(t∗) φ/2
M i5 = Sc R(t∗) hφ/2
The linearized system (4.34)-(4.37) in (i + 1)th iterates with prescribed boundary
conditions (4.38) can be solved using an implicit finite difference scheme. Central differ-
ence formula is used in η− direction and backward difference formula in t∗− direction
with constant step size 4η and 4t∗ in η and t∗− directions, respectively, to approximate
the unknowns and their derivatives.
The boundary η →∞ is replaced by a finite boundary η = η∞. Now the interval
[0; η∞] is divided into N equal subintervals of height 4η. In accordance with the above
mesh system , the difference approximations for the unknowns and their derivatives are
written as
hηη =(hm,n+1 − 2hm,n + hm,n−1)
(4η)2,
hη =(hm,n+1 − hm,n−1)
2(4η), (4.39)
ht∗ =(hm,n − hm−1,n)
(4t∗), (4.40)
Similar expressions can be written for g, θ and φ. The finite difference method
in combination with quasilinearization technique was outlined by Inouye and Tate[56] to
solve the two point boundary value problems. Thus, applying the finite difference scheme,
we get a system of equations which can be written in the matrix form as
AnWm,n−1 + BnWm,n + CnWm,n+1 = Dn, 2 ≤ n ≤ N for fixed m. (4.41)
where the vectors and coefficient matrices are given by
Wm,n =
h
g
θ
φ
m,n
, Dn =
d1
d2
d3
d4
n
, An =
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
n
,
102
Bn =
b11 b12 b13 b14
b21 b22 b23 b24
b31 b32 b33 b34
b41 b42 b43 b44
n
, Cn =
c11 c12 c13 c14
c21 c22 c23 c24
c31 c32 c33 c34
c41 c42 c43 c44
n
The elements of matrices An, Bn and Cn are
a11 = 1−X14η2
b11 = −2 + X2(4η)2 + X3(4η)2/4t∗ c11 = 1 + X14η2
a12 = 0 b12 = X4(4η)2 c12 = 0
a13 = 0 b13 = X5(4η)2 c13 = 0
a14 = 0 b14 = X6(4η)2 c14 = 0
a21 = 0 b21 = Y3(4η)2 c21 = 0
a22 = 1− Y14η2
b22 = −2 + Y2(4η)2 + Y3(4η)2/4t∗ c22 = 1− Y14η2
a23 = 0 = a24 b23 = 0 = b24 c23 = 0 = c24
a31 = 0 b31 = Z3(4η)2 c31 = 0
a32 = 0 = a34 b32 = 0 = b34 c32 = 0 = c34
a33 = 1− Z14η2
b33 = −2 + Z2(4η)2 + Z3(4η)2/4t∗ c33 = 1 + Z14η2
a41 = 0 b41 = M3(4η)2 c41 = 0
a42 = 0 = a43 b42 = 0 = b43 c42 = 0 = c43
a44 = 1−M14η2
b44 = −2 + M2(4η)2 + M3(4η)2/4t∗ c44 = 1 + M14η2
and the elements of matrix Dn are
d1 =
[X7 + X3
hm,n−1
4t∗
](4η)2
d2 =
[Y5 + Y3
gm,n−1
4t∗
](4η)2
d3 =
[Z5 + Z3
θm,n−1
4t∗
](4η)2
d4 =
[M5 + M3
φm,n−1
4t∗
](4η)2
103
Wm,1 and Wm,N+1 can be obtained from boundary conditions at η = 0 and at
η = η∞:
Wm,1 =
h
g
θ
φ
m,η=0
=
0
α1
1
1
Wm,N+1 =
h
g
θ
φ
m,η=η∞
=
0
1− α1
0
0
(4.42)
The equations (4.41) together with the boundary conditions (4.42) can be solved by
Varga’s algorithm
Wm,n = −EnWm,n+1 + Jn, 1 ≤ n ≤ N
where En = (Bn − AnEn−1)−1Cn
Jn = (Bn − AnEn−1)−1(Dn − AnJn−1), 2 ≤ n ≤ N
E1 = EN+1 =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
and J1 =
0
α1
1
1
, JN+1 =
0
1− α1
0
0
The linear system (4.41)is solved for t∗ > 0 using the previous values of t∗ and for t∗ = 0,
the computations are initiated using the following initial profiles.
f =(1− (1 + η)e−η)
η∞; h = fη =
η
η∞e−η
g = α1e−η + (1− α1)
η
η∞; θ = e−η = φ (4.43)
104
The algorithm for the solution of (4.41) is as follows:
1. Using the initial profiles(4.43), the matrix elements and vectors are evaluated.
2. Using the values of the vectors and matrix elements obtained from the step(1), the
system (4.41) is solved by the Varga algorithm[137], at the point t∗ = 0.
3. Using calculated values at t∗ = 0 from steps(1) and (2) the vectors and matrix
elements are again evaluated and then the system (4.41) is solved at a point t∗ =
4t∗ marching in t∗- direction. Values of t∗ are used to calculate values of t∗ +4t∗.
4. The step(3) is repeated until a prefixed value of t∗ is reached. In the above steps
the solutions are assumed to be converged when
Max|(hη)
i+1w − (hη)
iw|, |(gη)
i+1w − (gη)
iw|, |θη)
i+1w − (θη)
iw|, |(φη)
i+1w − (φη)
iw|
< 10−4
Similarly for the case of constant heat flux(CHF) case, linearizing the equations (4.14)
- (4.17) under the boundary condition (4.19) using the quasilinearization, we get the
following equations are obtained
H(i+1)ηη +X i
1H(i+1)η +X i
2H(i+1)+X i
3H(i+1)t∗ +X i
4G(i+1)+X i
5Θ(i+1)+X i
6Φ(i+1) = X i
7(4.44)
G(i+1)ηη + Y i
1 G(i+1)η + Y i
2 G(i+1) + Y i3 G
(i+1)t∗ + Y i
4 H(i+1) = Y i5 (4.45)
Θ(i+1)ηη + Zi
1Θ(i+1)η + Zi
2Θ(i+1) + Zi
3Θ(i+1)t∗ + Zi
4H(i+1) = Zi
5 (4.46)
Φ(i+1)ηη + M i
1Φ(i+1)η + M i
2Φ(i+1) + M i
3Φ(i+1)t∗ + M i
4H(i+1) = M i
5 (4.47)
where H = Fη and the corresponding boundary become
H i+1 = 0, Gi+1 = α1, Θi+1η = −1 = Φi+1
η at η = 0
H i+1 = 0, Gi+1 = 1− α1, Θi+1 = 0 = Φi+1 at η = η∞ (4.48)
The coefficient functions with iterative index i are known and the functions with iterative
index i + 1 are to be determined. The coefficients in equations (4.44) - (4.47) are given
105
by
X i1 = −R(t∗)F Y i
1 = −R(t∗)F
X i2 = R(t∗)H −R−1 ∂R
∂t∗ Y i2 = R(t∗)H −R−1 ∂R
∂t∗
X i3 = −1 Y i
3 = −1
X i4 = −4R(t∗)G Y i
4 = R(t∗)G
X i5 = −2R−1λ∗1 Y i
5 = R(t∗)GH −R−1(1− α1)∂R∂t∗
X i6 = −2R−1λ∗1N
∗
X i7 = 2R(t∗)
[H2
4−G2 − (1− α1)
2]
Zi1 = −PrR(t∗)F M i
1 = −Sc R(t∗) F
Zi2 = PrH/2R(t∗) M i
2 = Sc R(t∗) H/2
Zi3 = −Pr M i
3 = −Sc
Zi4 = PrR(t∗)Θ/2 M i
4 = Sc R(t∗) Φ/2
Zi5 = PrR(t∗)HΘ/2 M i
5 = Sc R(t∗) HΦ/2
The linearized system (4.44)-(4.47) in (i + 1)th iterates with prescribed boundary condi-
tions (4.48) can be solved using an implicit finite difference scheme. Central difference
formula is used in η− direction and backward difference formula in t∗− direction with
constant step size 4η and 4t∗ in η and t∗− directions, respectively, to approximate the
unknowns and their derivatives, as
Hηη =(Hm,n+1 − 2Hm,n + Hm,n−1)
(4η)2,
Hη =(Hm,n+1 −Hm,n−1)
2(4η), (4.49)
Ht∗ =(Hm,n −Hm−1,n)
(4t∗), (4.50)
with the similar expression for G, Θ, Φ etc. Finally, we obtain the linear system in matrix
form:
AnWm,n−1 + BnWm,n + CnWm,n+1 = Dn, 2 ≤ n ≤ N (4.51)
106
where the vectors and coefficient matrices are given by
Wm,n =
H
G
Θ
Φ
m,n
, Dn =
d1
d2
d3
d4
n
, An =
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
n
,
Bn =
b11 b12 b13 b14
b21 b22 b23 b24
b31 b32 b33 b34
b41 b42 b43 b44
n
, Cn =
c11 c12 c13 c14
c21 c22 c23 c24
c31 c32 c33 c34
c41 c42 c43 c44
n
The elements of matrices An, Bn and Cn are
a11 = 1−X14η2
b11 = −2 + X2(4η)2 + X3(4η)2/4t∗ c11 = 1 + X14η2
a12 = 0 b12 = X4(4η)2 c12 = 0
a13 = 0 b13 = X5(4η)2 c13 = 0
a14 = 0 b14 = X6(4η)2 c14 = 0
a21 = 0 b21 = Y3(4η)2 c21 = 0
a22 = 1− Y14η2
b22 = −2 + Y2(4η)2 + Y3(4η)2/4t∗ c22 = 1− Y14η2
a23 = 0 = a24 b23 = 0 = b24 c23 = 0 = c24
a31 = 0 b31 = Z3(4η)2 c31 = 0
a32 = 0 = a34 b32 = 0 = b34 c32 = 0 = c34
a33 = 1− Z14η2
b33 = −2 + Z2(4η)2 + Z3(4η)2/4t∗ c33 = 1 + Z14η2
a41 = 0 b41 = M3(4η)2 c41 = 0
a42 = 0 = a43 b42 = 0 = b43 c42 = 0 = c43
a44 = 1−M14η2
b44 = −2 + M2(4η)2 + M3(4η)2/4t∗ c44 = 1 + M14η2
and the elements of Dn are
d1 =[X7 + X3
hm,n−1
4t∗
](4η)2
d2 =[Y5 + Y3
gm,n−1
4t∗
](4η)2
107
d3 =[Z5 + Z3
θm,n−1
4t∗
](4η)2
d4 =[M5 + M3
φm,n−1
4t∗
](4η)2
Wm,1 and Wm,N+1 can be obtained from boundary conditions at η = 0 and at
η = η∞, respectively. Following the same procedure the equations (4.51) together with
the boundary conditions can be solved by Varga’s algorithm. The linear system (4.51)is
solved for t∗ > 0 using the previous values of t∗ and for t∗ = 0, the computations are
initiated using the following initial profiles.
F =(1− (1 + η)e−η)
η∞; H = Fη =
η
η∞e−η
G = α1e−η + (1− α1)
η
η∞; Θ =
(0.5− η
η∞+
(η√2η∞
)2)
η∞ = Φ (4.52)
4.5 RESULTS AND DISCUSSION
The nonlinear coupled partial differential equations (4.10)-(4.13) with boundary condi-
tions (4.18) for constant wall temperature case [Equations (4.14)-(4.17) for constant heat
flux case with boundary conditions (4.19)] have been solved numerically using an implicit
finite difference scheme in combination with quasilinearization technique and its detailed
description is presented in the previous section. To ensure the convergence of the numer-
ical solution to the true solution, the step size ∆η and the edge of the boundary layer
η∞ have been optimized. The results presented here are independent of ∆η and η∞ at
least up to the fourth decimal place. The computations have been carried out for various
values of Pr(0.7 ≤ Pr ≤ 7.0), λ1(0 ≤ λ1 ≤ 5), α1(−0.25 ≤ α1 ≤ 1.0), Sc(0.22 ≤ Sc ≤2.57), N(−0.5 ≤ N ≤ 1.0), λ∗1(0 ≤ λ∗1 ≤ 5) and N∗(−0.5 ≤ N∗ ≤ 1.0). The edge of the
boundary layer η∞ is taken between 4 and 6 depending on the values of parameters. The
results have been obtained for both increasing (R(t∗) = 1 + εt∗2, ε = 0.2, 0 ≤ t∗ ≤ 2) and
decreasing (R(t∗) = 1 + εt∗2, ε = −0.15, 0 ≤ t∗ ≤ 2) angular velocity of the cone and the
free stream fluid. In order to validate our method, we have compared steady state results
of skin-friction and heat transfer coefficients (−fηη(0, 0),−gη(0, 0),−θη(0, 0)) for PWT
case with those of Hering and Grosh [42] and Himasekhar et al. [48]. The results are
108
found in excellent agreement. We have also compared the skin friction and heat transfer
coefficients (−fηη(0, t∗),−gη(0, t
∗),−θη(0, t∗)) for PWT case in the absence of mass diffu-
sion (N = 0) with the results of Takhar et al. [127] for M = 0, who studied recently the
unsteady mixed convection flow from a rotating vertical cone with a magnetic field and
found them in excellent agreement. Some of the comparisons are shown in Figure 4.1.
0 1 20.5
1
2
3
t*
Re x1/
2 Cfx
, 2
−1R
e x1/2 C
fy,
Re x−1
/2 N
ux
Rex1/2 C
fx
2−1Rex1/2 C
fy
Rex−1/2 Nu
x
Present results
* Takhar et al. [125]
R(t*) = 1+ ε t*2, ε = 0.2
Figure 4.1: Comparison of the results (CfxRe1/2x , 2−1CfyRe
1/2x , Re
−1/2x Nux).
The results for the prescribed wall temperature (PWT) case are presented in
Figures 4.2-4.17 and for the prescribed heat flux (PHF) case in Figures 4.18-4.21.
Case(i): Prescribed Wall Temperature.
For the prescribed wall temperature(PWT) case, the effects of the buoyancy pa-
rameter λ1 and the parameter α1,( which is the ratio of the angular velocity of the cone
to the composite angular velocity) on the velocity , temperature and concentration pro-
files (−fη, g, θ, φ) are displayed in Figures 4.2-4.5 for R(t∗) = 1 + εt∗2, ε = 0.2, t∗ =
1, N = 1, Sc = 0.94 and Pr = 0.7. Also, the effects of λ1 and α1 on the local skin
friction coefficients (Re1/2x Cfx, 2
−1Re1/2x Cfy) and the local Nusselt and Sherwood numbers
(Re−1/2x Nux, Re
−1/2x Shx) are presented in Figures 4.6-4.9.
109
0 2 4 6−0.2
0
0.4
0.8
1.2
−fη
α1 = 0.0 ; λ
1 = 1
α1 = 0.2 ; λ
1 = 1
α1 = 0.5 ; λ
1 = 1
α1 = 1.0 ; λ
1 = 1
α1 = 1.0 ; λ
1 = 3
α1 = 0.0 ; λ
1 = 3
α1 = −0.2 ; λ
1 = 1
η
Figure 4.2: Effects of λ1 and α1 on velocity profiles (−fη) for PWT case when
t∗ = 1, N = 1, P r = 0.7 and Sc = 0.94
0 2 4 6−0.2
0
0.4
0.8
1.2
η
g
α1 = −0.2; λ
1 = 1
α1 = 0.0; λ
1 = 1
α1 = 0.2; λ
1 = 1
α1 = 0.5; λ
1 = 1
α1 = 0.7; λ
1 = 1
α1 = 1.0; λ
1 = 1
α1 = 0.0; λ
1 = 3
α1 = 0.1; λ
1 = 3
Figure 4.3: Effects of λ1 and α1 on velocity profiles (g) for PWT case when
t∗ = 1, N = 1, P r = 0.7 and Sc = 0.94
110
0 2 4 6 8
0.5
1
η
θ
α1 = 0.0; λ
1 = 1
α1 = 0.5; λ
1 = 1
α1 = 1.0; λ
1 = 1
α1 = 0.0; λ
1 = 3
α1 = 1.0; λ
1 = 3
Figure 4.4: Effects of λ1 and α1 on temperature profiles (θ) for PWT case when
t∗ = 1, N = 1, P r = 0.7 and Sc = 0.94
0 2 4 6 8
0.5
1
η
φ
α1 = 0.0; λ
1 = 1
α1 = 0.5; λ
1 = 1
α1 = 1.0; λ
1 = 1
α1 = 0.0; λ
1 = 3
α1 = 1.0; λ
1 = 3
Figure 4.5: Effects of λ1 and α1 on concentration profiles (φ) for PWT case when
t∗ = 1, N = 1, P r = 0.7 and Sc = 0.94
111
0 1 2−4
0
3
6
α1 = 0.0; λ
1 = 1.0
α1 = 0.2; λ
1 = 1.0
α1 = 0.5; λ
1 = 1.0
α1 = 0.7; λ
1 = 1.0
α1 = 0.5; λ
1 = 3.0
α1 = 0.7; λ
1 = 3.0
α1 = 0.0; λ
1 = 3.0
α1 = −0.2; λ
1 = 1.0
t*
Re x1/
2 Cfx
Figure 4.6: Effects of λ1 and α1 on skin friction coefficients (CfxRe1/2x ) for PWT case
when ε = 0.2, N = 1, P r = 0.7 and Sc = 0.94
0 1 2−2
−1
0
1
2
α1 = 0.0; λ
1 = 1.0
α1 = 0.2; λ
1 = 1.0
α1 = 0.5; λ
1 = 1.0
α1 = 0.7; λ
1 = 1.0
α1 = 1.0; λ
1 = 1.0
α1 = 0.2; λ
1 = 1.0
α1 = 0.7; λ
1 = 1.0
α1 = −0.2; λ
1 = 3.0
t*
0.5
Re x1/
2 Cfy
Figure 4.7: Effects of λ1 and α1 on skin friction coefficients (2−1CfyRe1/2x ) for PWT case
when ε = 0.2, N = 1, P r = 0.7 and Sc = 0.94
112
0 1 2−0.2
0
0.4
0.8
α1 = 0.0; λ
1 = 1.0
α1 = 0.25; λ
1 = 1.0
α1 = 0.5; λ
1 = 1.0
α1 = 0.75; λ
1 = 1.0
α1 = 0.25; λ
1 = 3.0
α1 = 0.75; λ
1 = 3.0
α1 = −0.2; λ
1 = 1.0
t*
Re x−1
/2 N
ux
Figure 4.8: Effects of λ1 and α1 on Re−1/2x Nux for PWT case when
s = 0.5, N = λ1 = 1, P r = 0.7 and Sc = 0.94
0 1 2−0.25
0
0.5
1
α1 = 0.0; λ
1 = 1.0
α1 = 0.25; λ
1 = 1.0
α1 = 0.5; λ
1 = 1.0
α1 = 0.75; λ
1 = 1.0
α1 = 0.25; λ
1 = 3.0
α1 = 0.75; λ
1 = 3.0
α1 = −0.2; λ
1 = 1.0
t*
Re x−1
/2 S
hx
Figure 4.9: Effects of λ1 and α1 on Re−1/2x Shx for PWT case when
s = 0.5, N = 1, λ1 = 3, P r = 0.7 and Sc = 0.94
113
It is observed from Figures 4.2 and 4.3 that the ratio of angular velocities α1
strongly affects the velocity profiles (−fη, g). For α1 = 0.5, the cone and the fluid are
rotating with equal angular velocity in the same direction and the non-zero velocities in
tangential and azimuthal directions (−fη, g) for α1 = 0.5 are only due to the positive buoy-
ancy parameter λ1 = 1, which acts like a favourable pressure gradient. When α1 > 0.5,
the fluid is being dragged by the rotating cone and due to the combined effects of buoyancy
force and rotation, the tangential velocity (−fη) increases its magnitude but the azimuthal
velocity (g) decreases its magnitude within the boundary layer. On the other hand, when
α1 < 0.5, the cone is dragged by the fluid and the combined effects of buoyancy force and
rotation parameter is just the opposite. Since the positive buoyancy force (λ1 > 0) implies
favourable pressure gradient, the fluid gets accelerated which results in thinner momen-
tum, thermal and concentration boundary layers. Consequently, the velocity, temperature
and concentration gradients are increased (see Figures 4.2-4.5). Hence, the local skin fric-
tion coefficients (Re1/2x Cfx, 2
−1Re1/2x Cfy) and the local Nusselt and Sherwood numbers
(Re−1/2x Nux, Re
−1/2x Shx) are also increased at any time t∗ as shown in Figures 4.6-4.9.
For example, α1 = 0.75 at time t∗ = 1, Figures 4.6-4.9 show that the percentage increase
in Re1/2x Cfx, 2
−1Re1/2x Cfy, Re
−1/2x Nux and Re
−1/2x Shx due to the increase in λ1 from 1
to 3 are about 111%, 28%, 33% and 27%, respectively. The effect of the time variation
is found to be more pronounced on the skin friction coefficients (Re1/2x Cfx, 2
−1Re1/2x Cfy)
than on the Nusselt and Sherwood numbers (Re−1/2x Nux, Re
−1/2x Shx) because the change
in the angular velocity with time strongly affect the velocity components. To be more
specific for λ1 = 1, α1 = 1, N = 1, the values of Re1/2x Cfx and 2−1Re
1/2x Cfy increase by
about 20% and 103%, respectively, when the time t∗ increases from 0 to 2. On the other
hand for the same data, Re−1/2x Nux and Re
−1/2x Shx increase approximately by 3% and
4%, respectively, for the increase of t∗ from 0 to 2.
Figures 4.10-4.13 display the effect of the ratio of the buoyancy forces N (which
measures the relative importance of the species and thermal diffusion) on the local skin
friction coefficients (Re1/2x Cfx, 2
−1Re1/2x Cfy) and the local Nusselt and Sherwood numbers
(Re−1/2x Nux, Re
−1/2x Shx) for increasing and decreasing angular velocities (R(t∗) = 1 +
εt∗2, ε = 0.2 and ε = −0.15) when λ1 = 1, α1 = 0.75, P r = 0.7 and Sc = 0.94.
114
0 1 21
2
3
Re1/
2x
Cfx
N = −0.5; ε = 0.2 N = 0.0; ε = 0.2 N = 0.5; ε = 0.2 N = 1.0; ε = 0.2 N = −0.5; ε = −0.15N = 0.0; ε = −0.15N = 0.5; ε = −0.15N = 1.0; ε = −0.15
t*
Figure 4.10: Effect of N on skin friction coefficient (CfxRe1/2x ) for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75, P r = 0.7 and Sc = 0.94
0 10 2
0.4
0.8
1.2
t*
0.5
Re1/
2x
Cfy
N = −0.5; ε = 0.2N = 0.0; ε = 0.2N = 0.5; ε = 0.2N = 1.0; ε = 0.2N = −0.5; ε = −0.15N = 0.0; ε = −0.15N = 0.5; ε = −0.15N = 1.0; ε = −0.15
Figure 4.11: Effect of N on skin friction coefficient (CfyRe1/2x ) for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75, P r = 0.7 and Sc = 0.94
115
0 1 2
0.52
0.6
0.68
t*
Re−1
/2x
Nu
x
N = −0.5;ε = 0.2N = 0.0; ε = 0.2N = 0.5; ε = 0.2N = 1.0; ε = 0.2N = −0.5; ε = −0.15N = 0.0; ε = −0.15N = 0.5; ε = −0.15N = 1.0; ε = −0.15
Figure 4.12: Effect of N on heat transfer coefficient (NuRe−1/2x ) for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75, P r = 0.7 and Sc = 0.94
0 1 2
0.6
0.7
0.8
Re−1
/2x
Sh
x
N = −0.5; ε = −0.15N = 0.0; ε = −0.15N = 0.5; ε = −0.15N = 1.0; ε = −0.15N = −0.0; ε = 0.2N = 0.0; ε = 0.2N = 0.5; ε = 0.2N = 1.0; ε = 0.2
t*
Figure 4.13: Effect of N on mass transfer coefficient (ShRe−1/2x ) for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75, P r = 0.7 and Sc = 0.94
116
Due to the increase in N , the velocity gradient in the primary flow as well as in
the secondary flow i.e., skin friction coefficients (Re1/2x Cfx, 2
−1Re1/2x Cfy) and the Nusselt
and Sherwood numbers (Re−1/2x Nux, Re
−1/2x Shx) increase for both increasing (ε > 0) and
decreasing (ε < 0) angular velocity cases at any time t∗ . For example, the percentage
increase in Re1/2x Cfx, 2
−1Re1/2x Cfy, Re
−1/2x Nux and Re
−1/2x Shx due to the increase in N
from −0.5 to 1 are about 100%, 25%, 29% and 28%, respectively, for the increasing angular
velocity case (ε = 0.2) at time t∗ = 1.0. It is observed from Figures 4.10 and 4.11 that for
a fixed N , the skin friction coefficients increase with time t∗ for an increasing angular ve-
locity but for decreasing angular velocity, the trend is reverse. For example, for increasing
angular velocity with N = 0.5 and ε = 0.2, Re1/2x Cfx and 2−1Re
1/2x Cfy increase approxi-
mately by 25% and 107%, respectively, as the time t∗ increases from 0 to 2. But for a de-
creasing angular velocity with N = 0.5 and ε = −0.15, Re1/2x Cfx and 2−1Re
1/2x Cfy decrease
approximately by 10% and 78%, respectively, as the time t∗ increases from 0 to 2. Since
an increase in the angular velocity with time directly affects the tangential and azimuthal
velocity components, the skin friction coefficients are significantly affected. However, the
effect of an increase in the angular velocity on the energy and species equations are rather
indirect. Hence, the local Nusselt and Sherwood numbers (Re−1/2x Nux, Re
−1/2x Shx) are
weakly affected. In fact, the changes in the values of Re−1/2x Nux and Re
−1/2x Shx for an
increasing/decreasing angular velocity are within 3% as the time t∗ increases from 0 to 2,
which are displayed in Figures 4.12 and 4.13.
The effects of Pr and Sc on the temperature and the concentration profiles (θ, φ) for
increasing angular velocity (R(t∗) = 1+εt∗2, ε = 0.2) when λ1 = 1, N = 1.0 and α1 = 0.75
are presented in Figures 4.14 and 4.15. Also,the effects of Pr and Sc on the local Nus-
selt and Sherwood numbers for the same data are shown in Figures 4.16 and 4.17. The
temperature and concentration profiles in Figures 4.14 and 4.15 reveal the fact that the
thermal boundary layer thickness increases rapidly with decreasing Pr and the concentra-
tion boundary layer thickness increases significantly with decreasing Sc. Hence in Figures
4.16 and 4.17, Re−1/2x Nux increases with Pr and Re
−1/2x Shx increases with Sc. For exam-
ple, Sc = 0.22, Re−1/2x Nux increases by about 132% as Pr increases from 0.7 to 7.0 and
for Pr = 0.7, Re−1/2x Shx increases by about 165% as Sc increases from 0.22 to 2.57.
117
0 1 2 3 4
0.5
1
η
θ
Pr = 0.7; Sc = 0.94Pr = 0.7; Sc = 2.57Pr = 7.0; Sc = 0.94Pr = 7.0; Sc = 2.57
Figure 4.14: Effects of Pr and Sc on θ for CWT case when
λ1 = 1, α1 = 0.25, s = 0.75 and N = 1
0 1 2 3 4
0.5
1
η
φ
Sc = 0.94; Pr = 0.7Sc = 2.57; Pr = 0.7Sc = 0.94; Pr = 7.0Sc = 2.57; Pr = 7.0
Figure 4.15: Effects of Pr and Sc on φ for CWT case when
λ1 = 1, α1 = 0.25, s = 0.75 and N = 1
118
0 1 2
0.6
1
1.6
Pr = 0.7; Sc = 0.22Pr = 0.7; Sc = 2.57Pr = 3.0; Sc = 0.22Pr = 3.0; Sc = 2.57Pr = 7.0; Sc = 0.22Pr = 7.0; Sc = 2.57
t*
Re x−1
/2 N
ux
Figure 4.16: Effects of Pr and Sc on Re−1/2x Nux for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75 and N = 1
0 1 2
0.4
0.7
1
Pr = 0.7; Sc = 0.22Pr = 0.7; Sc = 0.94Pr = 0.7; Sc = 2.57Pr = 7.0; Sc = 0.22Pr = 7.0; Sc = 0.94Pr = 7.0; Sc = 2.57
t*
Re x−1
/2 S
hx
Figure 4.17: Effects of Pr and Sc on Re−1/2x Shx for PWT case when
λ1 = 1, α1 = 0.25, s = 0.75 and N = 1
119
It may be remarked that the skin friction coefficients (Re1/2x Cfx, 2
−1Re1/2x Cfy)
are less affected by Pr and Sc as compared to the Nusselt and Sherwood numbers
(Re−1/2x Nux, Re
−1/2x Shx). In particular, for an increasing angular velocity (R(t∗) = 1 +
εt∗2, ε = 0.2) at t∗ = 1, when Pr changes from 0.7 to 7.0 the variations in the values
of Re1/2x Cfx and 2−1Re
1/2x Cfy are within 18% and the variations are within 8% for the
change of Sc from 0.22 to 2.57.
Case(ii): Prescribed Heat Flux.
For the prescribed heat flux (PHF) case, the effects of buoyancy parameter λ∗1 and
Prandtl number Pr on temperature profile (Θ) for increasing angular velocity (R(t∗) =
1 + εt∗2, ε = 0.2) with Sc = 0.94, N = 1 and α1 = 0.75 are displayed in Figure 4.18.
Further, the effects of buoyancy parameter λ∗1 and Schmidt number Sc on concentration
profiles (Φ) for increasing angular velocity (R(t∗) = 1+εt∗2, ε = 0.2) with Pr = 0.7, N = 1
and α1 = 0.75 are displayed in Figure 4.19. Figures 4.18 and 4.19 shows that unlike the
PWT case where temperature and concentration profiles (θ, φ) vary from 1 on the wall to
0 at the edge of the boundary layer η∞, in the PHF case temperature and concentration
profiles (Θ, Φ) on the wall are different from 1. This behavior is to be expected as the
boundary conditions Θη(η, t∗) = Φη(η, t∗) = −1 on the wall (i.e.,Θη(0, t∗) = Φη(0, t
∗) =
−1 ) are being imposed and all temperature and concentration profiles Θ(η) and Φ(η)
are equally inclined to the vertical axis at η = 0 as shown in Figures 4.18 and 4.19. It is
observed from Figure 4.18 that the thermal boundary layer thickness reduces significantly
with the increase of Prandtl number Pr. The physical reason is that the higher Prandtl
number fluid has a lower thermal conductivity which results in thinner thermal boundary
layer and similarly in Figure 4.19 the concentration boundary layer thickness reduces
significantly with the increase of Schmidt number Sc. Figures 4.20 and 4.21 display the
effects of λ∗1 and Pr on surface skin friction coefficients (Re1/2Cfx, 2−1Re1/2Cfy). The local
skin friction coefficients (Re1/2Cfx,2−1Re1/2Cfy) are increased at any time t∗ as shown in
Figures 4.20 and 4.21. Since the structure of the equations in both the cases (PWT and
PHF) are almost similar, it is natural to expect the effects of α1, λ1∗, N∗ and Sc are to
be similar in the present PHF case as in the PWT case and the detailed discussion the
results are therefore not presented here.
120
0 1 2 3
0.5
1
1.5
η
Θ
λ*1 = 1.0; Pr = 0.7
λ*1= 1.0; Pr = 7.0
λ*1 = 5.0; Pr = 0.7
λ*1 = 5.0; Pr = 7.0
Figure 4.18: Effects of λ1∗ and Pr on Θ for PHF case when
α1 = 0.5, s = 0.75, N∗ = 1 and Sc = 0.94
0 1.5 2.5
0.5
1
1.5
η
Φ
λ*1 = 1; Sc = 0.22
λ*1 = 1; Sc = 2.57
λ*1 = 5; Sc = 0.22
λ*1 = 5; Sc = 2.57
Figure 4.19: Effects of λ1∗ and Sc on Φ for PHF case when
α1 = 0.5, s = 0.75, N∗ = 1 and Pr = 0.7
121
0 1 21.5
4
6
t*
Re x1/
2 Cf x
λ*1 = 1.0; Pr = 7.0; ε =0.2
λ*1 = 1.0; Pr = 7.0; ε =−0.15
λ*1 = 3.0; Pr = 0.7; ε =0.2
λ*1 = 3.0; Pr = 0.7; ε =−0.15
λ*1 = 1.0; Pr = 0.7; ε =0.2
λ*1 = 1.0; Pr = 0.7; ε =−0.15
λ*1 = 3.0; Pr = 7.0; ε =0.2
λ*1 = 3.0; Pr = 7.0; ε =−0.15
Figure 4.20: Effects of Pr and λ∗1 on Re1/2x Cfx for PHF case when
α1 = 0.5, Sc = 0.94 and N∗ = 1
0 1 2
1
1.6
t*
Re x1/
2 Cf y
λ*1 = 1.0; Pr = 0.7; ε = 0.2
λ*1 = 1.0; Pr = 0.7; ε =−0.1
λ*1 = 1.0; Pr = 7.0; ε = 0.2
λ*1 = 1.0; Pr = 7.0; ε =−0.15
λ*1 = 3.0; Pr = 0.7; ε = 0.2
λ*1 = 3.0; Pr = 0.7; ε =−0.15
λ*1 = 3.0; Pr = 7.0; ε = 0.2
λ*1 = 3.0; Pr = 7.0; ε =−0.15
Figure 4.21: Effects of Pr and λ∗1 on Re1/2x Cfy for PHF case when
α1 = 0.5, Sc = 0.94and N∗ = 1
122
4.6 CONCLUSIONS
Unsteady mixed convection flow on a rotating cone in a rotating fluid due
to the combined effects of thermal and mass diffusion has been studied numerically to
obtain semi-similar solutions for both PWT and PHF conditions. The results have been
obtained for increasing and decreasing angular velocity cases. Comparisons with previ-
ously published work on modified cases of the problem were performed and found to be
in good agreement. The results indicate that skin friction coefficients in x- and y- di-
rections change significantly with time but the change in Nusselt and Sherwood numbers
are comparatively very small. It was found that the buoyancy force (λ1 or λ1∗) enhances
the skin friction coefficients, Nusselt and Sherwood numbers. For a fixed buoyancy force,
the Nusselt and Sherwood numbers increase with Prandtl number but the skin friction
coefficients decrease. In fact, the increase in Prandtl number and Schmidt number causes
a significant reduction in the thickness of thermal and concentration boundary layers,
respectively. Due to the increase in the ratio of buoyancy forces (N), the skin friction
coefficients, Nusselt and Sherwood numbers increase.
123
CHAPTER 5
UNSTEADY MIXED CONVECTION FROM A
MOVING VERTICAL SLENDER CYLINDER
5.1 INTRODUCTION
Unsteady mixed convection flows do not necessarily admit similarity solutions in many
practical situations. During the last two decades, a wide range of problems has appeared
that demand detailed analysis of unsteady mixed convection flows which necessitate taking
non-similarity into account. The unsteadiness and non-similarity in such flows may be
due to the free stream velocity or due to the curvature of the body or due to the surface
mass transfer or even possibly due to all these effects. Due to mathematical difficulties
involved in obtaining non-similar solutions for such problems, most investigators have
confined their studies either to steady non-similar flows or to unsteady semi-similar or
self-similar flows [109, 138]. In contrast, different approximate and numerical methods
have also been used to obtain non-similar solutions of such problems. A review of the
non-similarity solution methods along with citations of some relevant publications is given
in an earlier study by Dewey and Gross[28].
One of the often used concept in the solution of non-similarity boundary layers
is the principle of local similarity. In this method, the boundary layer equations suitably
transformed and divested of non-similar terms, and are applied locally at discrete stream-
wise locations. The resulting equations can be treated as ordinary differential equations
and can be readily solved by standard techniques. However, this approach cannot be jus-
tified convincingly because there is no positive way to estimate the effect of non-similar
terms on the final results. Hence the local similarity method is of uncertain accuracy
and this is not suitable for all problems. Sparrow et al.[122, 123] have developed the
local non-similarity method, which retains all the non-similar terms in the conservation
124
equations. Another asymptotic method had been developed by Kao and Elrod[57, 58]
to improve the accuracy of the local non-similarity method. The local non-similarity
method has since been employed successfully by a number of investigators in a variety
of boundary layer flow analysis (Chen and Mucoglu[22]; Minkowycz and Sparrow [85];
Hasan and Eichhorn[40]). Moffatt and Duffy[87] have provided an analysis about the
limitations of the local non-similarity solutions and presented examples where local non-
similarity method breaks down. Terrill[133] and Smith and Clutter[116] have used the
differential-difference method to solve non-similar flows. Nath[93] solved a class of non-
similar boundary layer flow problems using an approximate method based on a series
expansion of derivatives of stream function. The finite difference scheme serves well in
many cases to find the solution of non-similar flows. Marvin and Sheaffer[80], and Venkat-
achala and Nath[140] have used finite difference scheme to compute the non-similar flows.
Flows over cylinder are usually consider to be two dimensional as long as
the body radius is large compared to the boundary layer thickness. On the other hand
for slender cylinder, the radius of the cylinder may be of the same order as that of the
boundary layer thickness. Therefore, the flow may be consider as axisymmetric instead
of two dimensional. The flow past a slender body of revolution (here the term “slender”
means that the body radius is small and hence boundary layer thickness is not negligible
compared to the local radius of the body) has received a considerable attention, because
the use of a slender body reduces the drag and even produces sufficient lift to support the
body in certain situations. The flow nature on slender body is much characterized by its
two surface curvatures, viz., the longitudinal one in the meridian plane and the transverse
one in a plane normal to the axis of symmetry. The former is a quantity that is associated
with any curved surface which causes centrifugal force in the flow. In the usual treatment
of boundary layer analysis longitudinal curvature is assumed to be very small compared
to unity. Therefore, the effect of the longitudinal curvature in the boundary layer is
negligible. In such cases, attention is paid only to the transverse curvature which affects
the boundary layer and the effect is similar to that of a favourable pressure gradient. In
such a case, the governing equations contain the transverse curvature term which strongly
influences the velocity and temperature fields and correspondingly the skin friction and
125
heat transfer rate at the wall. Among the earlier studies, the magnitude of the transverse
curvature effect has been investigated for isothermal laminar flows by Stewartson[124]
and Cebeci[11]. The results show that the local skin friction can be altered by an order
of magnitude due to an appropriate change in the ratio of boundary layer thickness to
cylinder radius. It is therefore evident that the calculations of momentum and heat
transfer on slender cylinders should consider the transverse curvature effect, especially in
applications such as wire and fiber drawing, where accurate predictions are required and
thick boundary layers can exist on slender or near-slender bodies.
Mixed convection flow over a slender vertical cylinder due to the thermal
diffusion has been considered by Chen and Mucoglu[22] and Mucoglu and Chen[91] for the
constant wall temperature and constant heat flux conditions, respectively. They solved
the partial differential equations approximately using the local non-similarity method.
Subsequently Bui and Cebeci[11], Lee et.al.[71], Wang and Kleinstreuer[144] and most
recently Takhar et.al[130] have solved this problem using an implicit finite difference
scheme. All the above studies pertain to steady flows. In many practical problems, the
flow could be unsteady due to the velocity of the moving slender cylinder which varies
with time or due to the impulsive changes in the cylinder velocity or due to the free stream
velocity which varies with time. There are several transport processes with surface mass
transfer i.e., injection(or suction) in industry where the buoyancy force arises from thermal
diffusion caused by the temperature gradient such as polymer fiber coating or the coating
of wires etc. In these applications, the careful control of the yarn quenching temperature or
the heating and cooling temperature has a strong bearing on the final product quality[151].
When the ratio of the boundary layer thickness to the radius of the cylinder becomes larger
than one, the curvature effect leads to an increase of the heat transfer coefficient compared
to that characterizing the flat plate situation[123]. Therefore, it is interesting as well as
useful to investigate the combined effects of transverse curvature, viscous dissipation and
thermal diffusion on a continuously moving vertical slender cylinder where the cylinder
velocity and free stream velocity vary arbitrarily with time.
126
5.2 STATEMENT OF THE PROBLEM
The objective of the present investigation is to obtain a non-similar solution for the
unsteady mixed convection flow along a slender vertical heated cylinder which is mov-
ing in the same direction as that of free stream velocity. The unsteadiness is intro-
duced by the time dependent velocity of the slender cylinder as well as that of the free
stream. The effects of transverse curvature, viscous dissipation and surface mass trans-
fer(injection/suction) are also included in the analysis. The governing boundary layer
equations along with the boundary conditions are first cast into a dimensionless form by
a non-similar transformation and the resulting system of nonlinear coupled partial dif-
ferential equation is then solved by an implicit finite difference scheme in combination
with the quasilinearization technique[56, 9]. It appears from the brief literature survey
presented in the previous section that the present analysis is more general than those
of previous investigations. Particular cases of the present results have been compared
with those of Chen and Mucoglu[22] and Takhar et al.[130] and they are found to be in
excellent agreement.
-
6
︷ ︸︸ ︷
6
±M
-
6
?
6 6
ª
2R
x, u
r, v
g∗
6
or
x
uw
Tw
o
u∞, T∞
Figure 5.1: Physical model and coordinate system .
127
5.3 ANALYSIS
We consider the unsteady laminar mixed convection flow along a heated vertical slender
cylinder with injection and suction. The blowing rate is assumed to be small and it does
not affect the inviscid flow at the edge of the boundary layer. It is assumed that the
injected fluid processes the same physical properties as the boundary layer fluid and has
a static temperature equal to the wall temperature. The flow is taken to be axisymmetric
and Figure 5.1 shows the co-ordinate system and the physical model. The Boussinesq
approximation is invoked for the fluid properties to relate density changes to temperature
changes. Under the above assumptions the governing boundary layer equations can be
expressed as [8, 99, 130]:
∂(ru)
∂x+
∂(rv)
∂r= 0 (5.1)
∂u
∂t+ u
∂u
∂x+ v
∂u
∂r=
∂ue
∂t+ (ν/r)
∂
∂r
(r∂u
∂r
)+ g∗β(T − T∞) (5.2)
∂T
∂t+ u
∂T
∂x+ v
∂T
∂r=
ν
Pr
1
r
∂
∂r
(r∂T
∂r
)+
µ
ρCp
(∂u
∂r
)2
(5.3)
where u, v and T represents the axial and radial velocities and the temperature,
and t represents time. Here x and r are axial and radial co-ordinates and u, v are the
velocity component of the axial and radial directions, respectively; t is the time, g∗ is
the acceleration due to gravity; α and ν are thermal diffusivity and kinematic viscosity,
respectively; K is the thermal conductivity; Pr(= µCp/K) is the Prandtl number; T is the
temperature in the boundary layer; β is the volumetric co-efficient of thermal expansion.
The initial conditions are
u(0, x, r) = ui(x, r), v(0, x, r) = vi(x, r), T (0, x, r) = Ti(x, r) (5.4)
128
and the boundary conditions are given by
u(t, x, R) = uw(t) = uw,0φ(t∗), v(t, x, R) = vw, T (t, x, R) = Tw
u(t, x,∞) = ue(t) = u∞φ(t∗), T (t, x,∞) = T∞. (5.5)
where the subscripts w and ∞ denote conditions on the wall r = R and at infinity.
The equation of continuity (5.1) is clearly satisfied by a stream function ψ(x, r, t) defined
as
u =ψr
r, v = −ψx
r
Applying the following transformations :
ξ =
(4
R
)(νx
u∞
) 12
, η =
(νx
u∞
)− 12[r2 −R2
4R
], t∗ =
ν
R2t,
r2
R2= [1 + ξη] , ψ(x, r, t) = R(νu∞x)
12 φ(t∗)f(ξ, η, t∗), φ(t∗) = 1 + εt∗2,
u =1
2u∞φfη, v =
1
2rRφ
(νu∞x
) 12
(ηfη − f − ξ
∂f
∂ξ
),
θ(ξ, η, t∗) =T − T∞Tw − T∞
, Ec =u∞2
4cp(Tw − T∞), λ1 =
Grx
Re2x
,
P r =ν
α, Grx = gβx3(Tw0 − T∞)/ν2, Rex =
u∞x
ν, (5.6)
to equations (5.1) and (5.3), we find that equation (5.1) is identically satisfied,
and equations (5.2) and (5.3) reduce to
[(1 + ξη)Fη]η + φfFη + 8λ1φ−1θ + (
ξ2
4)[φ−1φt∗(2− F )− Ft∗ ] = φξ[FFξ − Fηfξ](5.7)
Pr−1[(1 + ξη)θη]η + φfθη + Ec(1 + ξη)φ2Fη2 − ξ2
4θt∗ = φξ[Fθξ − θηfξ] (5.8)
129
Here η is the similarity variable; ξ is the transverse curvature, f and fη are the dimension-
less stream functions and velocity components, respectively; θ and t∗ is the dimensionless
temperature and time, respectively; Rex is the Reynolds number; Grx is the Grashof
number; λ1 is the buoyancy parameter. The vanishing of λ1 corresponds to the case of
forced convection. φ(t∗) is the function of t∗ with first order continuous derivative. Cp is
a specific heat at constant pressure and ε is constant. The parameter Ec is the viscouss
dissipation parameter known as Eckert number. The boundary conditions for equations
(5.7) and (5.8) are expressed by
F (ξ, 0, t∗) = α2, θ(ξ, 0, t∗) = 1 for 0 ≤ t∗, ξ ≤ 1
F (ξ,∞, t∗) = 2, θ(ξ,∞, t∗) = 0 for 0 ≤ t∗, ξ ≤ 1 (5.9)
where α2 = 2
(uw
ue
)and f(ξ, η, t∗) =
∫ η
0
Fdx + fw;
fw =
(−Rvwξ
4νφ
)=
(Aξ
φ
), A =
(−Rvw
4ν
)= constant.
The surface mass transfer parameter A > 0 or A < 0 according to whether
there is a suction or injection.We have assumed that the flow is steady at time t∗ = 0
and becomes unsteady for t∗ > 0 due to the time-dependent free stream velocity (ue(t) =
u∞φ(t∗)) and the cylinder velocity (uw(t) = uw,0φ(t∗)), where φ(t∗) = 1 + εt∗2; ε ≶ 0.
Hence, the initial conditions (i.e., conditions at t∗ = 0) are given by the steady state
equations obtained from equations (5.7) and (5.8) by substituting φ(t∗) = 1, dφdt∗ = Ft∗ =
θt∗ = 0 when t∗ = 0 as:
[(1 + ξη)Fη]η + fFη + 8λ1θ = ξ[FFξ − Fηfξ] (5.10)
Pr−1[(1 + ξη)θη]η + fθη + Ec(1 + ξη)Fη2 = ξ[Fθξ − θηfξ] (5.11)
130
The corresponding boundary conditions are obtained from (5.9) when t∗ = 0 as:
F (ξ, 0) = α2, θ(ξ, 0) = 1 for 0 ≤ ξ ≤ 1
F (ξ,∞) = 2, θ(ξ,∞) = 0 for 0 ≤ ξ ≤ 1 (5.12)
It may be noted that the steady state equations in the absence of viscous dissi-
pation for α2 = 1 are the same as those of Chen and Mucoglu[22], and Takhar et al.[130]
with N = 0. The quantities of physical interest are as follows [8, 99]:
The local surface skin friction coefficient is given by
Cf =2τw
ρu2∞= 2−1Rex
− 12 φ(t∗)(Fη)w.
Thus,
Rex
12 Cf = 2−1φ(t∗)(Fη)w
The local Nusselt number can be expressed as
Re− 1
2x Nu = −2−1(θη)w, (5.13)
where Nu = −[x(∂T
∂r)]w
Tw − T∞.
5.4 METHOD OF SOLUTION
The set of dimensionless equations (5.7) and (5.8) under the boundary conditions (5.9)
with the initial conditions obtained from the corresponding steady state equations (5.10)
and (5.11) has been solved numerically using an implicit finite difference scheme in combi-
nation with the quasilinearization technique[56]. Since the method is described in detail
in Section 2.4.1 of Chapter2, it is not repeated here. The initial conditions necessary
for the computation of the unsteady case are obtained by solving steady state equations
(5.10) and (5.11) under the boundary conditions (5.12) using the same method.
131
Applying the quasilinearization technique to the non-linear coupled partial differ-
ential equations (5.7) and (5.8), we get
X i1F
i+1ηη + X i
2Fi+1η + X i
3Fi+1 + X i
4Fi+1ξ + X i
5Fi+1t∗ + X i
6θi+1 = X i
7 (5.14)
Y i1 θi+1
ηη + Y i2 θi+1
η + Y i3 θi+1
ξ + Y i4 θi+1
t∗ + Y i5 F i+1
η + Y i6 F i+1 = Y i
7 (5.15)
The coefficient functions with iterative index i are known and the functions with iterative
index i + 1 are to be determined. The boundary conditions become
F i+1 = α2, θi+1 = 1 at η = 0
F i+1 = 2, θi+1 = 0 at η = η∞ (5.16)
where η∞ is the edge of the boundary layer. The coefficients in equations (5.14) and (5.15)
are given by
X i1 = (1 + ξη) Y i
1 = (1+ξη)Pr
X i2 = ξ + φf + φ ξ fξ Y i
2 = ξPr
+ φf + φ ξ fξ
X i3 = − ξ2
4φφt∗ − φ ξ fξ Y i
3 = −φ ξ F
X i4 = −φ ξ F Y i
4 = − ξ2
4
X i5 = − ξ2
4Y i
5 = 2Ec(1 + ξη)φ2Fη
X i6 = 8λ1
φY i
6 = −φ ξ θξ
X i7 = −
[φ ξFFξ + ξ2
2φφt∗
]Y i
7 = φ[Ec(1 + ξη)φFη
2 − ξFθξ
]
The implicit finite difference scheme has been applied to equations(5.14) and
(5.15). Central difference scheme is used in η- direction and backward difference scheme is
implemented in ξ and t∗- directions with constant step-sizes4η,4ξ and4t∗, respectively.
According to the above mesh system, the difference approximations for the derivative of
unknown can be written as
F = Fm,p,n
Fη = (Fm,p,n+1 − Fm,p,n−1)/24η
132
Fξ = (Fm,p,n − Fm,p−1,n)/4ξ
Ft∗ = (Fm,p,n − Fm−1,p,n)/4t∗
Fηη = (Fm,p,n+1 − 2Fm,p,n + Fm,p,n−1)/(4η)2
with Fm,p,n = F (t∗m, ξp, ηn)
t∗m = (m− 1)4t∗; m = 1, 2, ...., M ; t∗M = 1
ξp = (p− 1)4ξ; p = 1, 2, ...., p; ξp = 2
ηn = (n− 1)4η; n = 1, 2, ...., N ; ηN = η∞ (5.17)
Similar expressions can be written for θ. The subscripts p,m and n denote particular
location corresponding to ξ, t∗ and η, respectively.
Introducing (5.17) in the equations (5.14) and (5.15), we get the system of linear
equations in matrix form as
AnWn−1 + BnWn + CnWn+1 = Dn, 2 ≤ n ≤ N (5.18)
where the vectors and coefficient matrices are given by
Wn =
F
θ
m,p,n
, Dn =
d1
d2
m,p,n
, An =
a11 a12
a21 a22
m,p,n
,
Bn =
b11 b12
b21 b22
m,p,n
, Cn =
c11 c12
c21 c22
m,p,n
133
The elements of the matrices An, Bn, Cn and Dn are
a11 = X1 −X24η2
a12 = 0
a21 = −Y54η2
a22 = Y1 − Y24η2
b11 = −2X1 + X3(4η)2 + X4(4η)2
4ξ+ X5
(4η)2
4t∗ b12 = X6(4η)2
b21 = Y6(4η)2 b22 = −2Y1 + Y3(4η)2
4ξ+ Y4
(4η)2
4t∗
c11 = X1 + X24η2
c12 = 0
c21 = Y54η2
c22 = Y1 + Y24η2
(5.19)
d1 = [X7+X4
4ξFm,p−1,n+
X5
4t∗Fm−1,p,n]4η2, d2 = [Y7+
Y3
4ξθm,p−1,n+
Y4
4t∗θm−1,p,n]4η2.(5.20)
Wm,p,1 and Wm,p,N+1 can be obtained from boundary conditions at η = 0 and at η = η∞:
Wm,p,1 =
F
θ
m,p,η=0
=
α2
1
Wm,p,N+1 =
F
θ
m,p,η=η∞
=
2
0
(5.21)
The equations (5.18) together with the boundary conditions (5.21) can be solved by
Varga’s algorithm as follows:
Wn = −EnWn+1 + Jn, 1 ≤ n ≤ N
where En = (Bn − AnEn−1)−1Cn
Jn = (Bn − AnEn−1)−1(Dn − AnJn−1), 2 ≤ n ≤ N
E1 = EN+1 =
0 0
0 0
and J1 =
α2
1
, JN+1 =
2
0
The equations (5.18) together with the boundary conditions (5.21) can be solved for any
t∗ > 0, ξ > 0 provided that the values of the dependent variables for t∗ = 0, ξ > 0 and
ξ = 0, t∗ > 0 are known.
134
As mentioned earlier, the solution for the system of equations(5.18) requires so-
lutions at time t∗ = 0 and ξ > 0 and they are given by steady state equations (5.10) and
(5.11). Applying the qusailinearization technique with combination of an implicit finite
difference scheme to the equations (5.10) and (5.11), we get
X i1F
i+1ηη + X i
2Fi+1η + X i
3Fi+1 + X i
4Fi+1ξ + X i
5θi+1 = X i
6 (5.22)
Y i1 θi+1
ηη + Y i2 θi+1
η + Y i3 θi+1
ξ + Y i4 F i+1
η + Y i5 F i+1 = Y i
6 (5.23)
The coefficient functions with iterative index i are known and the functions with iterative
index i+1 are to be determined. The boundary conditions for equations (5.22) and (5.23)
are given by (5.16). The coefficients in equations (5.22) and (5.23) are :
X i1 = (1 + ξη) Y i
1 = (1+ξη)Pr
X i2 = ξ + f + ξ fξ Y i
2 = ξPr
+ f + ξ fξ
X i3 = − ξ Fξ Y i
3 = − ξ F
X i4 = ξ F Y i
4 = Y i4 = 2Ec(1 + ξη)Fη
X i5 = 8λ1 Y i
5 = −ξ θξ
X i6 = −ξFFξ Y i
7 =[Ec(1 + ξη)Fη
2 − ξFθξ
]
Applying finite difference scheme for derivatives in equations (5.22) and (5.23), the re-
sulting linear system of equations leads to the matrix form as in equation (5.18).
The elements of the matrices An, Bn, Cn and Dn are
a11 = X1 −X24η2
a12 = 0
a21 = −Y44η2
a22 = Y1 − Y24η2
b11 = −2X1 + X3(4η)2 + X4(4η)2
4ξb12 = X5(4η)2
b21 = Y5(4η)2 b22 = −2Y1 + Y3(4η)2
4ξ
c11 = X1 + X24η2
c12 = 0
c21 = Y44η2
c22 = Y1 + Y24η2
(5.24)
d1 = X6(4η)2 + X4(4η)2
4ξF1,p−1,n, d2 = Y6(4η)2 + Y3
(4η)2
4ξθ1,p−1,n. (5.25)
135
Similarly, the initial conditions at ξ = 0 and t∗ ≥ 0 in the (η, t∗) plane can be
obtain putting ξ = 0 in (5.7) and (5.8) and after applying the quasilinearization technique
to the resulting equations, we get,
X i1F
i+1ηη + X i
2Fi+1η + X i
3θi+1 = X i
4 (5.26)
Y i1 θi+1
ηη + Y i2 θi+1
η + Y i3 F i+1
η = Y i4 (5.27)
The coefficient functions with iterative index i are known and the functions with iterative
index i+1 are to be determined. The boundary conditions for equations (5.26) and (5.27)
are given by (5.16). The coefficients in equations (5.26) and (5.27) are :
X i1 = 1 Y i
1 = 1
X i2 = φf Y i
2 = Pr φf
X i3 = 8λ1
φY i
3 = 2Pr Ec φ2 Fη
X i4 = 0 Y i
4 = Pr Ec φ2 F 2η
Applying the implicit finite difference scheme for derivatives in equations (5.26)
and (5.27), the resulting linear system of equations leads to the matrix form as in equation
(5.18).
The elements of the matrices An, Bn, Cn and Dn are
a11 = X1 −X24η2
a12 = 0
a21 = −Y34η2
a22 = Y1 − Y24η2
b11 = −2X1 b12 = X3(4η)2
b21 = 0 b22 = −2Y1
c11 = X1 + X24η2
c12 = 0
c21 = Y34η2
c22 = Y1 + Y24η2
(5.28)
d1 = X44η2, d2 = Y44η2 (5.29)
While computing the results in the plane (η, t∗), the results at t∗ = 0 provides
the initial values for computation. When t∗ = 0 and ξ = 0 the equations (5.7) and (5.8)
136
reduced to
Fηη + fFη + 8λ1θ = 0 (5.30)
θηη + Pr fθη + Pr EcFη2 = 0 (5.31)
with the same boundary conditions(5.9).
Applying the quasilinearization to these equations (5.30) and (5.31), we get
X i1F
i+1ηη + X i
2Fi+1η + X i
3θi+1 = X i
4 (5.32)
Y i1 θi+1
ηη + Y i2 θi+1
η + Y i3 F i+1
η = Y i4 (5.33)
The coefficient functions with iterative index i are known and the functions with iterative
index i+1 are to be determined. The boundary conditions for equations (5.32) and (5.33)
are given by (5.16). The coefficients in equations (5.32) and (5.33) are :
X i1 = 1 Y i
1 = 1
X i2 = f Y i
2 = Pr f
X i3 = 8λ1 Y i
3 = 2Pr Ec Fη
X i4 = 0 Y i
4 = Pr Ec F 2η
Applying finite difference scheme for derivatives in equations (5.32) and (5.33), the re-
sulting linear system of equations leads to the matrix form as in equation (5.18).
The elements of the matrices An, Bn, Cn and Dn are
a11 = X1 −X24η2
a12 = 0
a21 = −Y34η2
a22 = Y1 − Y24η2
b11 = −2X1 b12 = X3(4η)2
b21 = 0 b22 = −2Y1
c11 = X1 + X24η2
c12 = 0
c21 = Y34η2
c22 = Y1 + Y24η2
.
(5.34)
d1 = X44η2 d2 = Y44η2 (5.35)
137
To initiate the computations at ξ = 0 and t∗ = 0 i.e., the initial profiles for velocity
and temperature which satisfy the boundary conditions (5.21)are taken as
F =ηe−η
η∞θ = e−η (5.36)
The procedure for obtaining non-similar solutions described above can be summarized by
the following algorithm:
1. Using the initial profiles (5.36), the matrix elements(5.34) and vectors(5.35) are
evaluated. Using these values, the system (5.18) has been solved by Varga’s
algorithm[56] at ξ = 0 and t∗ = 0.
2. Using the calculated values at ξ = 0 and t∗ = 0 from step(1) in (5.28) and (5.29),
the system (5.18) has been solved by Varga’s algorithm for ξ = 0 and t∗ = 4t∗
marching in t∗− direction. To calculate the values at t∗+4t∗, the values at t∗ have
been used.
3. Using the calculated vales at ξ = 0 and t∗ = 0 from step(1) in (5.24) and (5.25), the
system (5.18) has been solved by Varga’s algorithm at ξ = 4ξ and t∗ = 0 marching
in ξ− direction. To calculate the values at ξ = ξ +4ξ, the values at ξ have been
used.
4. Using the calculated values from step(2) and step(3) at ξ > 0 and t∗ = 0 in (5.24)
and (5.25), the system (5.18) has been solved by Varga’s algorithm for ξ > 0 and
t∗ = 4t∗ marching in t∗− direction. To calculate the values at t∗ + 4t∗ for any
ξ > 0, the values at t∗ have been used. In the above steps, the solutions are assumed
to be converged when
Max |(Fη)
i+1w − (Fη)
iw|, |(θη)
i+1w − (θη)
iw|
< 10−4 (5.37)
5.5 RESULTS AND DISCUSSION
The computations have been carried out for various values of Pr(0.7 ≤ Pr ≤7.0), λ1(0 ≤ λ1 ≤ 3), α2(0 ≤ α2 ≤ 2), Ec(0 ≤ Ec ≤ 0.3) and A(−0.5 ≤ A ≤ 0.5).
138
The edge of the boundary layer η∞ is taken between 3 and 5 depending on the values of
parameters. The results have been obtained for both accelerating (φ(t∗) = 1 + εt∗2, ε >
0, 0 ≤ t∗ ≤ 1) and decelerating (φ(t∗) = 1 + εt∗2, ε < 0, 0 ≤ t∗ ≤ 1) free stream velocities
of the fluid. In order to validate our method, we have compared steady state results
of skin-friction and heat transfer coefficients (Fη(0, 0), θη(0, 0)) with those of Chen and
Mucoglu [22] and Takhar et al.[130]. The results are found in an excellent agreement
and comparisons are shown in Table 5.1 . The effects of buoyancy parameter λ1 on
Present Results Chen and Mucoglu. Takhar et al.
ξ λ1 fηη(ξ, 0) −θη(ξ, 0) fηη(ξ, 0) −θη(ξ, 0) fηη(ξ, 0) −θη(ξ, 0)
0 0 1.3282 0.5854 1.3282 0.5854 1.3281 0.5854
0 1 4.9664 0.8220 4.9666 0.8221 4.9663 0.8219
0 2 7.7122 0.9304 7.7126 0.9305 7.7119 0.9302
1 0 1.9169 0.8666 1.9172 0.8669 1.9167 0.8666
1 1 5.2580 1.0621 5.2584 1.0621 5.2578 1.0617
1 2 7.8871 1.1688 7.8871 1.1690 7.8863 1.1685
Table 5.1: Comparison of the steady state results (fηη(ξ, 0),−θη(ξ, 0)) with those of Chen
and Mucoglu[22], Takhar et al.[130].
velocity and temperature profiles (F, θ) for accelerating flow φ(t∗) = 1 + εt∗2, ε = 0.5,
when α2 = 1, Ec = 0.1, A = 0, P r = 0.7 and 7.0 are displayed in Figures 5.2-5.3. The
action of the buoyancy force shows the overshoot in the velocity profiles (F ) near the
wall for lower Prandtl number (Pr = 0.7) but for higher Prandtl number (Pr = 7.0) the
velocity overshoot in F is not observed as shown in Figure 5.2. The magnitude of the
overshoot increases with the buoyancy parameter λ1. The reason is that the buoyancy
force (λ1) effect is larger in a low Prandtl number fluid (Pr = 0.7, air) due to the lower
viscosity of the fluid which enhances the velocity as the assisting buoyancy force acts like
a favorable pressure gradient and the velocity overshoot occurs. For higher Prandtl num-
ber fluid (Pr = 7.0, water) the velocity overshoot is not present because higher Prandtl
number fluid implies more viscous fluid which makes it less sensitive to the buoyancy
parameter λ1. The effect of λ1 is comparatively less on the temperature (θ) as shown in
Figure 5.3.
139
0 1 2 31
2
3
3.5
η
F
λ1=0;t*=0;Pr=0.7
λ1=1;t*=0;Pr=0.7
λ1=2;t*=0;Pr=0.7
λ1=1;t*=1;Pr=0.7
λ1=2;t*=1;Pr=0.7
λ1=1;t*=1;Pr=7.0
λ1=3;t*=0;Pr=0.7
Figure 5.2: Effects of λ1 and Pr on F for φ(t∗) = 1 + εt2, ε = 0.5 when
Ec = 0.1, α2 = 1, ξ = 0.5, A = 0
0 1 2 3
0.5
1
η
θ
λ1=0; t*=0; Pr=0.7
λ1=1; t*=0; Pr=0.7
λ1=0; t*=1; Pr=0.7
λ1=1; t*=1; Pr=0.7
λ1=1; t*=1; Pr=7.0
Figure 5.3: Effects of λ1 and Pr on θ for φ(t∗) = 1 + εt2, ε = 0.5 when
Ec = 0.1, α2 = 1, ξ = 0.5, A = 0
140
The effects of surface curvature parameter ξ on velocity and temperature profiles
(F, θ) for accelerating flow φ(t∗) = 1 + εt∗2, ε = 0.5, when α2 = 1, Ec = 0.1, A = 0 and
Pr = 0.7 are displayed in Figure 5.4. Also, the effects of ξ and λ1 on the skin friction and
heat transfer coefficients (Re1/2x Cf , Re
−1/2x Nu) are presented in Figures 5.5 and 5.6. Due
to the increase in surface curvature parameter ξ, the steepness in velocity and temperature
profiles (F, θ) near the wall increases but the magnitude of the velocity overshoot is slightly
decreases. Even though the increase in ξ acts as a favourable pressure gradient, it’s effect is
small so that it does not cause the velocity profiles to have overshoots as in buoyancy aided
flow. Similar trends have been observed by Chen and Mucoglu[22] and Takhar et.al.[130]
for the steady state case. Further from Figures 5.5 and 5.6, it is observed that the skin
friction and heat transfer coefficients (Re1/2x Cf , Re
−1/2x Nu) increase with the increase of
buoyancy parameter(λ1). The physical reason is that the positive buoyancy force (λ1 > 0)
implies favourable pressure gradient and the fluid gets accelerated which results in thinner
momentum and thermal boundary layers. Consequently, the local skin friction (Re1/2x Cf )
and the local Nusselt number (Re−1/2x Nu) are also increase at any time (t∗) as shown
in Figures 5.5 and 5.6. For example for α2 = 1.0, P r = 0.7, Ec = 0.1, A = 0.0 at time
t∗ = 0.5, Figure 5.5 shows that the percentage increase in Re1/2x Cf for the increase of λ1
from 1 to 3 is approximately 121% and in Figure 5.6 shows that the percentage increase
in Re−1/2x Nu for the increase of λ1 from 1 to 3 is approximately 14%, respectively for the
same data.
The effects of wall velocity α2 and mass transfer parameter A on skin friction
and heat transfer coefficients for φ(t∗) = 1 + εt∗2, ε = 0.5, α2 = 1, P r = 0.7, Ec = 0.1
are presented in Figures 5.7 and 5.8. Results indicate that the skin friction coefficient
(Re1/2x Cf ) decreases but heat transfer rate at the wall increases for all values of A and
time t∗ due to the increase of wall velocity α2 . Actually, the increase of wall velocity (α2)
gives a slip over the surface as a result of which skin friction coefficient decreases. For
the sake of clarity, we also present some of the results quantitatively. For example, for
t∗ = 1.0, A = −0.5, the skin friction coefficient (Re1/2x Cf ) reduces approximately by 19%
(Figure 5.7) and the heat transfer coefficient (Re−1/2x Nu) increases approximately by 25%
(Figure 5.8) as the wall velocity α2 increases from 0 to 1.
141
0 1 2 31
1.5
2
2.5
η
F
ξ=0;t*=0ξ=1;t*=0ξ=0;t*=1ξ=1;t*=1
1 2 30
0.5
1
ηθ
ξ=0;t*=0ξ=1;t*=0ξ=0;t*=1ξ=1;t*=1
Figure 5.4: Effect of ξ on F and θ for φ(t∗) = 1 + εt2, ε = 0.5 when
λ1 = 1, P r = 0.7, Ec = 0.1, α2 = 1, A = 0.
0 0.5 1
2
4
6
t*
(Re x)1/
2 Cf
λ1 = 0
1
2
3
ξ = 0
ξ = 1
Figure 5.5: Effects of λ1 and ξ on Re1/2x Cf for φ(t∗) = 1 + εt2, ε = 0.5 when
Pr = 0.7, Ec = 0.1, α2 = 1, A = 0.
142
0 0.5 10.45
0.5
0.6
0.7
(Re x)−1
/2N
u
t*
λ1 =1
2
3
λ1 =1
2
3
ξ=0
ξ=1
Figure 5.6: Effects of λ1 and ξ on Re−1/2x Nu for φ(t∗) = 1 + εt2, ε = 0.5 when
Pr = 0.7, Ec = 0.1, α2 = 1, A = 0.
0 0.5 12
2.5
3
3.5
4
t*
(Re x)1/
2 Cf
A =0; α2 =0
A =0; α2 =1
A =0.5; α2 =0
A =0.5; α2 =1
A =−−0.5; α2 =0
A =−−0.5; α2 =1
Figure 5.7: Effects of A and α2 on Re1/2x Cf for φ(t∗) = 1 + εt2, ε = 0.5 when
λ1 = 1, P r = 0.7, Ec = 0.1.
143
0 0.5 1
0.4
0.6
0.8
t*
(Re x)−1
/2N
uA = 0; α
2 = 0
A = 0; α2 = 1
A = 0.5; α2 = 0
A = 0.5; α2 = 1
A = −0.5; α2 = 0
A = −0.5; α2 = 1
Figure 5.8: Effects of A and α2 on Re1/2x Cf and Re
−1/2x Nu for φ(t∗) = 1 + εt2, ε = 0.5
when λ1 = 1, P r = 0.7, Ec = 0.1.
0 1 2 31
1.5
2
2.5
F
η
A=0.5; t*=0A=0.5; t*=1A=−0.5; t*=0A=−0.5; t*=1
0 1 2 3
1
2
η
F
α2=0; t*=0
α2=0; t*=1
α2=1; t*=0
α2=1; t*=1
α2=2; t*=0
α2=2; t*=1
Figure 5.9: Effects of A and α2 on F for φ(t∗) = 1 + εt2, ε = 0.5 when
λ1 = 1, P r = 0.7, ξ = 0.5, Ec = 0.1.
144
Figures 5.7 and 5.8 are also shows that for all time t∗, both Re1/2x Cf and Re
−1/2x Nu
increase with suction(A > 0) but decrease with the increase of injection (A < 0). In case of
injection, the fluid is carried away from the surface causing reduction in velocity gradient
as it tries to maintain the same velocity over a very small region near the surface and this
effect is reversed in the case of suction. In addition, Figure 5.9 displays the effect of α2
and A on velocity profiles (F ) for time t∗ = 0 and t∗ = 1. The graphs of the velocity and
temperature profiles (F ) versus η in Figure 5.9 show that the injection (A < 0) as well as
the wall velocity (α2) cause a decrease in the steepness of the profiles (F ) near the wall
but the steepness of the profiles (F ) increases with suction. For all the cases, the profiles
(F ) at a later time t∗ = 1.0 are comparatively less steeper near the wall than those at the
initial time t∗ = 0.
Figures 5.10 and 5.11 display the effects of Prandtl number for accelerating and
decelerating free-stream flows (φ(t∗) = 1 + εt∗2, ε = 0.5 and ε = −0.5) on the local
skin friction and heat transfer coefficients (Re1/2x Cf , Re
−1/2x Nu) where α2 = 1.0, λ1 = 1.0
and A = 0.0. It is found from Figure 5.10 that the skin friction coefficient decrease
with the increase of Prandtl number. Because the higher Prandtl number fluid means
more viscous fluid which increases the boundary layer thickness and consequently reduces
the shear stress. On the other hand, Figure 5.11 reveal that the surface heat transfer
rate increases significantly with Pr, as the higher Pr number fluid has a lower thermal
conductivity, which results the thinner thermal boundary layer and hence a higher heat
transfer rate at the wall. To be more specific for λ1 = 1, α2 = 1, A = 0, Ec = 0.1 and
t∗ = 0.5 as Pr increases from 0.7 to 7.0, Re1/2x Cf decreases by about 36% and Re
−1/2x Nu
increases by 114%, respectively. Thus, the heat transfer rate at the wall can be reduced
by using a low Prandtl number fluid such as air (Pr = 0.7). In case of accelerating
flow, Figures 5.10 and 5.11 show that both skin friction coefficient and heat transfer rate
increase with time t∗ and the effect of the time variations is found to be more pronounced
on the skin friction coefficient than on heat transfer rate. Because, the change in the
freestream velocity with time strongly affect the velocity component. For example, for
Pr = 7.0 the values of Re1/2x Cf and Re
−1/2x Nu increase by about 25% and 6%, respectively
when the time t∗ increases from 0 to 1.
145
0 0.5 11.2
1.6
2
2.4
t*
(Re x)1/
2 Cf
Pr = 0.7; ε=0.5Pr = 0.7; ε=−0.5Pr = 3.0; ε=0.5Pr = 3.0; ε=−0.5Pr = 7.0; ε=0.5Pr = 7.0; ε=−0.5
Figure 5.10: Effect of Pr on Re1/2x Cf for φ(t∗) = 1 + εt2, ε = 0.5 and ε = −0.5 when
λ1 = 1, Ec = 0.1, α2 = 1, A = 0, ξ = 0.5.
0 0.5 10.4
0.8
1.2
t*
(Re x)−1
/2N
u
Pr=0.7; ε=0.5Pr=0.7; ε=−0.5Pr=7.0; ε=0.5Pr=7.0; ε=−0.5Pr=3.0; ε=0.5Pr=3.0; ε=−0.5
Figure 5.11: Effect of Pr on Re−1/2x Nu for φ(t∗) = 1 + εt2, ε = 0.5 and ε = −0.5 when
λ1 = 1, Ec = 0.1, α2 = 1, A = 0, ξ = 0.5.
146
0 0.5 12.25
2.5
2.75
(Re x)1/
2 Cf
Ec =0.0 ; ε=0.5Ec =0.0 ; ε=−0.5Ec =0.1 ; ε=0.5Ec =0.1 ; ε=−0.5Ec =0.2 ; ε=0.5Ec =0.2 ; ε=−0.5Ec =0.3 ; ε=0.5Ec =0.3 ; ε=−0.5
t*
Figure 5.12: Effect of Ec on Re1/2x Cf for φ(t∗) = 1 + εt2, ε = 0.5 and ε = −0.5 when
λ1 = 1, P r = 0.7, α2 = 1, A = 0, ξ = 0.5.
0 0.5 10.35
0.45
0.55
0.65
t*
(Re x)−1
/2N
u
Ec=0.0 ; ε=0.5Ec=0.0 ; ε=−0.5Ec=0.1 ; ε=0.5Ec=0.1 ; ε=−0.5Ec=0.2 ; ε=0.5Ec=0.2 ; ε=−0.5
Figure 5.13: Effect of Ec on Re−1/2x Nu for φ(t∗) = 1 + εt2, ε = 0.5 and ε = −0.5 when
λ1 = 1, P r = 0.7, α2 = 1, A = 0, ξ = 0.5.
147
0 1 2 3
0.5
1
η
θ
Pr =0.7, t* = 0Pr =0.7, t* = 1Pr =7.0, t* = 0Pr =7.0, t* = 1
Figure 5.14: Effects of Pr on θ for φ(t∗) = 1 + εt2, ε = 0.5 when
λ1 = 1, A = 0, α2 = 1, ξ = 1.0.
0 1 2 3
0.5
1
η
θ
Ec=0.0; t*=0Ec=0.3; t*=0Ec=0.0; t*=1Ec=0.3; t*=1
Figure 5.15: Effects of Ec on θ for φ(t∗) = 1 + εt2, ε = 0.5 when
λ1 = 1, A = 0, α2 = 1, ξ = 1.0.
148
Also, the effects of Eckert number for accelerating and decelerating free-stream
flows (φ(t∗) = 1 + εt∗2, ε = 0.5 and ε = −0.5) on the local skin friction and heat transfer
coefficients (Re1/2x Cf , Re
−1/2x Nu) where α2 = 1.0, λ1 = 1.0 and A = 0.0, are displayed in
Figures 5.12 and 5.13 with the same data. It is observed from Figures 5.12 and 5.13 that
due to increase of the viscous dissipation parameter Ec, Re1/2x Cf increases but Re
−1/2x Nu
decreases and the effect is more pronounced on the heat transfer coefficient (Re−1/2x Nu).
In particular, it is found that at ξ = 1.0 and t∗ = 1.0 the percentage decrease of Re−1/2x Nu
for an increase in Ec from 0 to 0.5 is 100% as compared to 10% of Re1/2x Cf for the same
data. This behaviour is in support of the common fact that the viscous dissipation affects
the thermal boundary layer more than the momentum boundary layer.
Figures 5.14 and 5.15 display the effects of Pr and Ec on temperature profiles
(θ) for accelerating free-stream flows (φ(t∗) = 1 + εt∗2, ε = 0.5) where α2 = 1.0, λ1 = 1.0
and A = 0. It is observed from Figure 5.14 that the higher Pr number fluid has a lower
thermal conductivity, which results the thinner thermal boundary layer and hence a higher
heat transfer rate at the wall. It is observed from Figure 5.15 that the increase of the
viscous dissipation parameter Ec a higher thermal boundary layer thickness and hence a
lower heat transfer rate at the wall.
5.6 CONCLUSIONS
Non-similar solution of an unsteady mixed convection flow over a continuously
moving slender cylinder has been obtained for both accelerating and decelerating free-
stream velocities. Results indicate that the skin friction and heat transfer coefficients are
significantly affected by the time dependent free-stream velocity distributions. It is found
that the buoyancy force (λ1) enhances the skin friction coefficient and Nusselt number. In
the presence of the buoyancy force(λ1 > 0), the velocity profile exhibit velocity overshoot
for lower Prandtl number and the buoyancy parameter tends to increase its magnitude.
For a fixed buoyancy force, the Nusselt number increase with Prandtl number but the skin
friction coefficient decreases. In fact, the increase in Prandtl number causes a significant
reduction in the thickness of thermal boundary layer. As expected, both skin friction and
149
heat transfer coefficients increase with suction but decrease with the increase of injection.
In contrast, the effect of wall velocity is to decrease the skin friction but to increase
the heat transfer rate. Further, it is noted that the curvature parameter steepens both
the velocity and temperature profiles, but injection(A < 0) does the reverse. The heat
transfer rate is found to depend strongly on viscous dissipation, but the skin friction is
little affected by it.
150
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LIST OF PUBLICATIONS BASED ON THE RESEARCH WORK
I REFEREED JOURNALS
1. D. Anilkumar and S. Roy(2004) Self-similar solution of the unsteady mixed
convection flow in the stagnation point region of a rotating sphere. Heat and
Mass Transfer, 40, 487-493.
2. D. Anilkumar and S. Roy(2004) Unsteady mixed convection flow on a rotating
cone in a rotating fluid. Applied Mathematics and Computation, 155, 545 -
561.
3. S. Roy and D. Anilkumar(2004) Unsteady mixed convection from a rotating
cone in a rotating fluid due to the combined effects of thermal and mass diffusion.
International Journal of Heat and Mass Transfer, 47, 1673-1684.
4. S. Roy and D. Anilkumar. Unsteady mixed convection from a moving vertical
slender cylinder (Communicated).
II PRESENTATIONS IN CONFERENCES
1. D. Anilkumar and S. Roy. Self-similar solution to unsteady mixed convection
flow from a rotating cone in a rotating fluid. SIAM conference on Applied
Mathematics, Gainesville, Florida, March, 2004.
163