Magnetohydrodynamic unsteady free convection flow past vertical porous plates with suction and oscillating boundaries

Project Abstract

<p> </p><p>In this dissertation, the problems of Magnetohydrodynamic unsteady free convection flow<br>past vertical porous plates with suction and oscillating boundaries are studied. The linear<br>and nonlinear partial differential equations governing the flow problems and boundary<br>conditions were transformed into dimensionless form, and the perturbation techniques<br>applied in getting analytical solutions for the velocity, temperature, the skin friction<br>coefficient and Nusselt number. It was observed that an increase in the values of thermal<br>Grashof number, Eckert number and heat source increases velocity profile, while an<br>increase in Darcy term retards the velocity profile. An increase in heat source and Grashof<br>number, also increases the Heat transfer coefficient. The effects of various parameters on<br>the flow fields have been presented with the help of graphs and tables.<br>xii</p><p>&nbsp;</p> <br><p></p>

Project Overview

<p> INTRODUCTION<br>1.1 Background of the study<br>As understanding of the natural world has grown, human civilization and communities<br>have consistently been established at locations that feature a viable source of fluid flowing.<br>Throughout history, people have continuously attempted to manipulate the natural fluid<br>flow, in order to effect an improvement in such areas as agricultural stability, living<br>environment, and transportation.<br>The Magnetohydrodynamic (MHD) channel flow, was first described theoretically by<br>Hartmann (1937), who considered plane Poiseuille flow with a transverse magnetic field.<br>Since then, the study of MHD has been an active area of research because of its geophysical<br>and astrophysical applications. Ahmed and Batin (2013), investigated the effects of<br>conduction-radiation and porosity of the porous medium on laminar convective heat transfer<br>flow of an incompressible, viscous, electrically conducting fluid over an impulsively started<br>vertical plate embedded in a porous medium in presence of transverse magnetic field.<br>Modern technologies have emerged, and we have become increasingly reliant on the<br>fundamental principles of fluid flow. Humanity has come to depend upon the development<br>and design of modern transport, such as cars, ships and air-crafts, which are rooted in an<br>essential understanding and knowledge of fluid flows and this knowledge area, is an integral<br>area for solving aerodynamic problems. The area also provides a plethora of engineering<br>problems concerning energy conservation and transmission. Time past methodological<br>engineering, and even biomedical studies, have proven the universally accepted tenant that<br>understanding fluid flow is critical to the development of applied knowledge.<br>The effect of radiation, chemical reaction and variable viscosity on hydromagnetic heat and<br>mass transfer in the presence of magnetic field are studied by Seddeek and Almushigeh<br>1<br>(2010). Ahmed et al. (2012), considered MHD mixed convection and mass transfer from an<br>infinite vertical porous plate with chemical reaction in presence of a heat source. Uwanta<br>and Isah (2012) studied the boundary layer fluid flow in a channel with heat source, soret<br>effects and slip condition.<br>1.2 Statement of the Problems<br>Fluid flow is still a growing area, due to its wide application in technology, engineering,<br>science and medicine. This wide interest makes it necessary to undertake research, where<br>the governing equations are a set of unsteady, coupled and invariably the equations are also<br>nonlinear, at the same time, many of the phenomena that fluid flow shows for oscillating<br>boundary conditions.<br>Balasubramanyam et al. (2010) analysed the combined effects of magnetic field and viscous<br>dissipation on convective heat and mass transfer flow through a porous medium in a vertical<br>channel in the presence of heat generating sources. However, they had neither considered<br>the effect of unsteady state nor coupled within the governing equation or presence of Darcy<br>term.<br>Therefore, there is need for further research to address the above mention problems. The<br>governing equations of an adapted model are modified. Three problems of heat transfer<br>were formulated, both are unsteady and coupled. The first problem is linear with fixed<br>plates, while the second problem is non linear with infinity boundary and the third problem<br>is non linear with fixed boundary. The problems considered are:<br>(i) Magnetohydrodynamic unsteady free convection flow past vertical porous plates<br>with heat deposition.<br>(ii) Magnetohydrodynamic unsteady free convection flow past an infinite vertical porous<br>plates with heat deposition.<br>2<br>(iii) Darcy Forcheimer Magnetohydrodynamic unsteady free convection flow past vertical<br>porous plates with heat deposition.<br>The governing equations are continuity equations, momentum equations and energy equations<br>which were solved using perturbation technique with the help of fitting boundaries.<br>The process of nondimensionalising the governing equations prior to the start of developing<br>the perturbation approximation will also be addressed.<br>1.3 Aim and Objectives of the Study<br>The main aim of this dissertation is to modify an existing work of Balasubramanyam<br>et al. (2010), by introducing conditions that do change with time, coupled with and in the<br>presence of modified Darcy term. With the following specific objectives of the study, to<br>solve:<br>(i) Magnetohydrodynamic unsteady free convection flow past vertical porous plates<br>with heat deposition.<br>(ii) Magnetohydrodynamic unsteady free convection flow past an infinite vertical porous<br>plates with heat deposition.<br>(iii) Darcy Forcheimer Magnetohydrodynamic unsteady free convection flow past vertical<br>porous plates with heat deposition.<br>1.4 Significance of the Study<br>This study, augments to the body of knowledge, based on the following rationales, which<br>were not taken care of in the work of Balasubramanyam et al. (2010):<br>(i) Unsteady state of the modified adapted work is to be considered.<br>(ii) The presence of coupled between momentum and energy equations of problems<br>(3.3.2) and (3.3.3) surface.<br>3<br>(iii) Inclusion of Darcy Forcheimer term in the problem formation.<br>1.5 Definitions of Basic Concepts<br>Eckert number (Ec): Is the ratio of kinetic energy to enthalpy change, Raisinghania<br>(2003).<br>Grashof number (Gr): It is the ratio of the product of the inertial force and the buoyant<br>force to the square of viscous force in the convection, Raisinghania (2003).<br>Hall effect: It is the production of a voltage difference across the electric conductor,<br>transverse to the electric current in the conductor and a magnetic field perpendicular to the<br>current, Raisinghania (2003).<br>Heat dissipation: It si that energy which is dissipated in a viscous liquid in motion of<br>account of the internal friction, Douglas and Gibilisco (2003).<br>Heat transfer: Is the transfer of heat energy from one body to another as a result of<br>temperature difference, Douglas and Gibilisco (2003).<br>Incompressible fluid: Is one that requires a large variation in pressure to produce some<br>appreciable variation in density, John et al. (2011).<br>Magnetohydrodynamic (MHD): Is an important branch in fluid dynamics, which is<br>concerned with the interaction of electrically conducting fluids and electromagnetic fluids,<br>Raisinghania (2003).<br>Nusselt number or Heat transfer coefficient (Nu): It is defined as ratio of convective<br>heat transfer to conductive heat transfer across the boundary, Raisinghania (2003).<br>Prandtl number (Pr): It is defined as the ratio of kinematic viscosity (v) to thermal<br>diffusivity (k) of a fluid, Raisinghania (2003).<br>Porous medium: Is a solid matrix containing holes either connected or non connected,<br>dispersed with in the medium in a regular or random manner provided such holes occur<br>4<br>frequently in the medium, Raisinghania (2003).<br>Porous parameter: Is defined as the ratio of Darcy resistance to viscous force, Borowski<br>and Borwein (2005).<br>Reynold’s number (Re): The number ensures dynamic similarity at corresponding points<br>near the boundaries where viscous effects are most important. Its reciprocal is called<br>Reynold’s number and is denoted by Re, John et al. (2011).<br>Skin friction: The dimensionless shear stress at the surface, John et al. (2011).<br>Steady flow: Is one in which the velocity, pressure and concentration may vary from point<br>to point but do not change with time, John et al. (2011).<br>Unsteady flow: Is one in which properties and conditions associated with the motion of<br>the fluid depend on the time so that the flow pattern varies with time, John et al. (2011).<br>Viscous dissipation: The heat generated by internal friction within the fluid element of<br>the fluid per unit time, John et al. (2011).<br>Viscosity: is the internal friction of a fluid which makes it resist flowing past a solid surface<br>or other layers of the fluid, John et al. (2011).<br>1.6 Structure of the Dissertation<br>The dissertation has been presented in five (5) chapters. Chapter one serves as the general<br>introduction to the research report. It provides background of the study, statement of the<br>problem, its significance, aim and objectives of the study as well as definitions of the basic<br>concept. Chapter two examines the related literature review, while Chapter three discusses<br>the methodology adapted for the research report, which includes model formation and<br>solutions to the mathematical models under study, Chapter four presents the results and<br>discussion to illustrate the flow characteristics for the velocity, temperature, skin friction<br>coefficient and Nusselt number. Finally Chapter five draws together the research project. <br></p>