Magnetohydrodynamic Unsteady Free Convection Flow Past Vertical Porous Plates with Suction and Oscillating Boundaries

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 – Magnetohydrodynamic Unsteady Free Convection Flow Past Vertical Porous Plates with Suction and Oscillating Boundaries – 

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ABSTRACT  

In this dissertation, the problems of Magnetohydrodynamic unsteady free convection flow past vertical porous plates with suction and oscillating boundaries are studied. The linear and nonlinear partial differential equations governing the flow problems and boundary conditions were transformed into dimensionless form, and the perturbation techniques applied in getting analytical solutions for the velocity, temperature, the skin friction coefficient and Nusselt number.

It was observed that an increase in the values of thermal Grashof number, Eckert number and heat source increases velocity profile, while an increase in Darcy term retards the velocity profile. An increase in heat source and Grashof number, also increases the Heat transfer coefficient. The effects of various parameters on the flow fields have been presented with the help of graphs and tables. 

TABLE OF CONTENTS

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
CERTIFICATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
TABLE OF CONTENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
DIMENSIONLESS NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
GREEK SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

CHAPTER ONE
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background of the study . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Statement of the Problems . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Aim and Objectives of the Study . . . . . . . . . . . . . . . . . . . . 3
1.4 Significance of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Definitions of Basic Concepts . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Structure of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 5

CHAPTER TWO
LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Some Related Literature Review . . . . . . . . . . . . . . . . . . . . 6

CHAPTER THREE
METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Regular perturbation expansions . . . . . . . . . . . . . . . . . . . . 10
3.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.1 Magnetohydrodynamic Unsteady Free Convection Flow Past
Vertical Porous Plates with Heat Deposition . . . . . . . . . . 12
3.3.2 Magnetohydrodynamic Unsteady Free Convection Flow Past
an Infinite Vertical Porous Plates with Heat Deposition . . . . 17
3.3.3 Darcy Forchcheimer Magnetohydrodynamic Unsteady Free
Convection Flow Past Vertical Porous Plates with Heat Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

CHAPTER FOUR
RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Magnetohydrodynamic Unsteady Free Convection Flow Past Vertical Porous Plates with Heat Deposition . . . . 50
4.2.1 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Temperature field . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.3 Skin friction coefficient and Nusselt number . . . . . . . . . . 56
4.3 Magnetohydrodynamic Unsteady Free Convection Flow Past an Infinite Vertical Porous Plates with Heat Deposition . . . . . . . . . . . 57
4.3.1 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 Temperature field . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.3 Skin friction coefficient and Heat transfer coefficient . . . . . 61
4.4 Darcy Forchcheimer Magnetohydrodynamic Unsteady Free Convection Flow Past Vertical Porous Plates with Heat Deposition . . . . . . 61
4.4.1 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.2 Temperature field . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.3 Skin friction coefficient and Heat transfer coefficient . . . . . 67

CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS . . . . . . . . . . . . . . . . . . 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.1 Magnetohydrodynamic Unsteady Free Convection Flow Past
Vertical Porous Plates with Heat Deposition . . . . . . . . . . 70
5.3.2 Magnetohydrodynamic Unsteady Free Convection Flow Past
an Infinite Vertical Porous Plates with Heat Deposition . . . . 70
5.3.3 Darcy Forchcheimer Magnetohydrodynamic Unsteady Free
Convection Flow Past Vertical Porous Plates with Heat Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.6 Limitations of the Study . . . . . . . . . . . . . . . . . . . . . . . . . 72

REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

INTRODUCTION  

As understanding of the natural world has grown, human civilization and communities have consistently been established at locations that feature a viable source of fluid flowing.

Throughout history, people have continuously attempted to manipulate the natural fluid flow, in order to effect an improvement in such areas as agricultural stability, living environment, and transportation.

The Magnetohydrodynamic (MHD) channel flow, was first described theoretically by Hartmann (1937), who considered plane Poiseuille flow with a transverse magnetic field.

Since then, the study of MHD has been an active area of research because of its geophysical and astrophysical applications.

Ahmed and Batin (2013), investigated the effects of conduction-radiation and porosity of the porous medium on laminar convective heat transfer flow of an incompressible, viscous, electrically conducting fluid over an impulsively started vertical plate embedded in a porous medium in presence of transverse magnetic field.

Modern technologies have emerged, and we have become increasingly reliant on the fundamental principles of fluid flow.

Humanity has come to depend upon the development and design of modern transport, such as cars, ships and air-crafts, which are rooted in an essential understanding and knowledge of fluid flows and this knowledge area, is an integral area for solving aerodynamic problems.

The area also provides a plethora of engineering problems concerning energy conservation and transmission. Time past methodological engineering, and even biomedical studies, have proven the universally accepted tenant that understanding fluid flow is critical to the development of applied knowledge.

The effect of radiation, chemical reaction and variable viscosity on hydromagnetic heat and mass transfer in the presence of magnetic field are studied by Seddeek and Almushigeh 1 (2010). Ahmed et al. (2012), considered MHD mixed convection and mass transfer from an infinite vertical porous plate with chemical reaction in presence of a heat source.

Uwanta and Isah (2012) studied the boundary layer fluid flow in a channel with heat source, soret effects and slip condition. 

REFERENCES

Abdel-Naby, M. A. (2003). Finite difference solution of radiation effects on mhd unsteady
free convection flow over vertical plate with variable surface temperature. Journal of
Applied Mathematics, 2:65 – 86.

Ahmed, N., Sarma, D., and Deka, H. (2012). Mixed convection and mass transfer from an
infinite vertical porous plate with chemical reaction in presence of heat source. Applied
Mathematical Sciences, 6(21):1011 – 1020.

Ahmed, S. (2007). Effects of unsteady free convective mhd flow through a porous medium
bounded by an infinite vertical porous plate. Bull. Cal. Math. Soc., 99(5):511 – 522.

Ahmed, S. (2010). Influence of chemical reaction on transient mhd free convective flow
over a vertical plate in slip flow regime. Emirates Journal for Engineering Research,
15(1):25 – 34.

Ahmed, S. and Ahmed, N. (2004). Two dimensional mhd oscillatory flow along a uniformly
moving infinite vertical porous plate bounded by porous medium. International Journal
of Pure and Applied Maths, 35:1309 – 1319.

Ahmed, S. and Batin, A. (2013). Convective laminar radiating flow over an accelerated vertical plate embedded in a porous medium with an external magnetic field. International
Journal of Engineering and Technology, 3(1):66 – 72.

Ali, H. N. (1993). Introduction to Perturbation Techniques. John Wiley and Sons, INC.,
wiley classics library edition.

Anand, R. J. and Shivaiah, S. (2011). Chemical reaction effects on an unsteady mhd flow
past a semi infinite vertical porous plate with viscous dissipation. Journal of Applied
Mathematics and Mechanics, 32(8):1065 – 1078.

CSN Team.

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