NUMERICAL STUDIES ON UNSTEADY MIXED CONVECTION …THESIS CERTIFICATE This is to certify that the thesis entitled \NUMERICAL STUDIES ON UNSTEADY MIXED CONVECTION FLOWS" submitted by

$Page 1: NUMERICAL STUDIES ON UNSTEADY MIXED CONVECTION …THESIS CERTIFICATE This is to certify that the thesis entitled \NUMERICAL STUDIES ON UNSTEADY MIXED CONVECTION FLOWS" submitted by$
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


3.3 MATHEMATICAL FORMULATION . . . . . . . . . . . . . . . . . . . . . 65


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


4.3 MATHEMATICAL FORMULATION OF THE PROBLEM . . . . . . . . 93


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


vii

5.3 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128


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


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)



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


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



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



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

.


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



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


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


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.


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)


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


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]


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


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



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


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)


Varga’s algorithm

Wn = −EnWn+1 + Jn, 1 ≤ n ≤ N


74


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∞


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]


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



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


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


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-


77




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


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


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.


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


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)


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


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)



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)


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)


Varga’s algorithm

Wm,n = −EnWm,n+1 + Jn, 1 ≤ 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)



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


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-





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


(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


(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


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.


u(0, x, r) = ui(x, r), v(0, x, r) = vi(x, r), T (0, x, r) = Ti(x, r) (5.4)

128


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∞.


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)



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



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)


Varga’s algorithm as follows:

Wn = −EnWn+1 + Jn, 1 ≤ 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)


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).


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)




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).


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)




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).


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

Documents

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