ON CONVECTION AND FLOW IN POROUS MEDIA WITH CROSS-DIFFUSION Ahmed A. Khidir

206 Pages·2012·1.18 MB·English

ON CONVECTION AND FLOW IN POROUS MEDIA
WITH CROSS-DIFFUSION
A THESIS SUBMITTED IN FULFILMENT OF THE ACADEMIC
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
By
Ahmed A. Khidir
School of Mathematics, Statistics and Computer Science
College of Agriculture, Engineering and Science
University of KwaZulu-Natal
August 2012
Contents
Abstract v
Declaration vii
Dedication viii
Acknowledgements ix
1 Introduction 1
1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The cross-diffusion effect . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Analytical and numerical studies . . . . . . . . . . . . . . . . . . . . 16
1.4 Method of quasilinearization . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 The SLM technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Comparison of the QLM and SLM . . . . . . . . . . . . . . . . . . . . 32
i
1.7 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.8 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Natural convection from a vertical plate immersed in a power-law
fluid saturated non-Darcy porous medium with viscous dissipation
and Soret effects 40
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.1 Aiding buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.2 Opposing buoyancy . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3 Soret effect on the natural convection from a vertical plate in a
thermally stratified porous medium saturated with non-Newtonian
liquid 65
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
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3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4 Thermal radiation and viscous dissipation effects on mixed convec-
tion from a vertical plate in a non-Darcy porous medium 95
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5 On cross-diffusion effects on flow over a vertical surface using a lin-
earisation method 118
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6 Cross-diffusion, viscous dissipation and radiation effects on an ex-
ponentially stretching surface in porous media 137
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6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7 Conclusion 165
References 170
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Abstract
In this thesis we studied convection and cross-diffusion effects in porous media.
Fluid flow in different flow geometries was investigated and the equations for mo-
mentum, heat and mass transfer transformed into a system of ordinary differential
equations using suitable dimensionless variables. The equations were solved using a
recent successive linearization method. The accuracy, validity and convergence of the
solutions obtained using this method were tested by comparing the calculated results
with those in the published literature, and results obtained using other numerical
methods such as the Runge-Kutta and shooting methods, the inbuilt Matlab bvp4c
numerical routine and a local non-similarity method.
We investigated the effects of different fluid and physical parameters. These
include the Soret, Dufour, magnetic field, viscous dissipation and thermal radiation
parameters on the fluid properties and heat and mass transfer characteristics.
The study sought to (i) investigate cross-diffusion effects on momentum, heat and
mass transport from a vertical flat plate immersed in a non-Darcy porous medium
saturated with a non-Newtonian power-law fluid with viscous dissipation and thermal
radiation effects, (ii) study cross-diffusion effects on vertical an exponentially stretch-
ing surface in porous medium and (iii) apply a recent hybrid linearization-spectral
technique to solve the highly nonlinear and coupled governing equations. We further
sought to show that this method is accurate, efficient and robust by comparing it
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with established methods in the literature.
In this study the non-Newtonian behaviour of the fluid is characterized using the
Ostwald-de Waele power-law model. Cross-diffusion effects arise in a broad range
of fluid flow situations in many areas of science and engineering. We showed that
cross-diffusion has a significant effect on heat and mass-transfer processes and cannot
be neglected.
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Declaration
The work described in this dissertation was carried out under the supervision and
direction of Professor P. Sibanda, School of Mathematics, Statistics and Computer
Science, University of KwaZulu-Natal, Pietermaritzburg, South Africa.
I, Ahmed A. Khidir, declare that this thesis under title
On Convection and Flow in Porous Media with Cross-Diffusion
is my own work and it has not been previously submitted to this or any other aca-
demic institution for a degree or diploma, or any other qualification. All published
and unpublished material used in this thesis has been given full acknowledgement.
Signatures:
....................................................
Mr. Ahmed A. Khidir (Student)
.......................................................
Prof. Precious Sibanda (Supervisor)
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Dedication
To
my mother . . . . . . Hawaa
my father . . . . . . Abdalmgid
my wife . . . . . . Aalya
and my son . . . . . . Adel
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Acknowledgements
My first and foremost gratitude goes to my PhD supervisor, Professor P. Sibanda for
his support, advice and for the effort given in these three years in the development of
my skills and the writting of this thesis. Without his help and support, this project
would never have been possible.
Also, I would like to thank Dr. Faiz G. Awad for many interesting discussions,
support, continuous and invaluable suggestions throughout my work. Thanks are also
due to Dr. M. Narayana who gave many useful comments. The three years in the
school has been fruitful, thanks to all the colleagues and members of the school. I feel
fortunate to have made so many good Sudanese friends in Pietermaritzburg (South
Africa), many thanks to all for supporting me and my family. Also, I wish to thank
the Sudan government for providing the required financial support. I would like to
extend my deepest gratitude to my father, brothers, sisters and friends. They always
have provided unwavering love and encouragement. Thank you for believing in me.
Lastly, but certainly not least, thanks to my family, my wife Aalya and my son
Adel for their love, tolerance and enduring support. Without their help and encour-
agement, I would have been finished well before I got started. I thank you from the
bottom of my heart, I dedicate this thesis with love, to you and to our new family
member.
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