ELEMENT FREE GALERKIN METHOD FOR FLUID FLOW, HEAT AND MASS TRANSFER IN POROUS MEDIUM

Abstract

The study of fluid flow and heat transfer over continuous flat surface has been of great interest. The boundary layer control in aerodynamics, the cooling of an infinite metallic plate in a cooling bath, the boundary layer along a liquid film in condensation process and polymer sheet or filament extruded continuously from a die are few examples of practical applications of a continuous flat surface. Such analysis in boundary layer flow arises in the design of thrust bearing and radial diffusers, transpiration cooling, drag reduction, and thermal recovery of oil, etc. Now a days, a lot of research is going on fluid flow through porous media due to its numerous applications e.g. transport of water in living plants and trees, geophysical flows, recovery of petroleum resources, storage of nuclear waste materials and also in various engineering fields like irrigation engineering, chemical engineering, aeronautics, meteorology, biomedicine, etc. Porous media is also useful to alter flow kinematics in a way so as to achieve a better control on the rate of cooling. The studies in porous media are based on Darcy model which is valid only for slow flows through porous media with low permeability, but at higher flow rates or with high permeability, inertial effects become significant. These effects can be accounted for, through the addition of a velocity squared term in the momentum equation, resulting in a new model, popularly known as Darcy-Forchheimer model. The consideration of inertial effects is becoming indispensable, especially in biomedical and geological engineering, to simulate meticulously the blood vessel blockage with deposits in the cardiovascular system, etc. Present thesis is divided into nine chapters. Chapter 1 is introductory in nature and deals with the fundamental concepts of fluid flow, heat and mass transfer in porous medium and basic theoretical concepts used in subsequent chapters e.g. boundary layer, micropolar fluid, Magnetohydrodynamics (MHD), etc. Chapter 2 contains a brief description of numerical techniques namely finite element method (FEM) and element free Galerkin method (EFGM), which are used to solve the mathematical models of the problems discussed in the following chapters. The last seven chapters contain a discussion of the problems solved in this thesis work. In chapter 3, numerical solution is presented for heat transfer through an incompressible, viscous fluid flowing over an unsteady stretching surface embedded in a porous medium. A variable magnetic field is applied transversely to the direction of the flow. The time dependent nonlinear differential equations governing the problem have been transformed by a similarity transformation into a system of non-linear ordinary differential equations, which are solved numerically by EFGM. This system of equations is non-linear, therefore an iterative method maintaining an accuracy of 0.0001 has been used to solve it. EFGM study has been made by choosing appropriate penalty parameter, weight functions and scaling parameter. An excellent validation of the present numerical results has been achieved with the earlier steady state results of Grubkha and Bobba (1985) and Chen (1998) for local Nusselt number. Finally, the influence of unsteadiness parameter, hydromagnetic parameter, porosity parameter and Prandtl number on the velocity and temperature profiles is presented graphically. The problem is also solved with FEM, which shows that EFGM results are in well agreement with FEM results. In Chapter 4, the effect of viscous dissipation and heat source/sink on unsteady boundary layer flow and heat transfer past a permeable stretching surface embedded in a porous medium in the presence of magnetic field is studied. If the fluid is very viscous, considerable heat can be produced even though at relatively low speed, e.g. in the extrusion of a plastic. Thus, the heat transfer results may alter appreciably due to viscous dissipation. 11 Recently, a new model for viscous dissipation in a porous medium was proposed by Al-Hadhrami et al. (2003), which is probably adequate for most practical purposes. Therefore, in this chapter, the viscous dissipation effect is modeled in according to Al-Hadhrami et al. (2003). The effect of suction parameter, unsteadiness parameter, Eckert number, porosity parameter and heat source/sink parameter on the velocity and temperature profiles are shown graphically. The impact of physical parameters on skin friction coefficient and wall temperature gradient is shown in tabulated form. It is found that the skin friction decreases numerically with the increase in unsteadiness parameter, porosity parameter and suction parameter. The rate of heat transfer increases with the increase in unsteadiness parameter, suction parameters and heat sink parameter. Thus, a fast cooling of the plate can be achieved by implementing these parameters. Chapter 5 deals with the study of thermal radiation effect on unsteady flow and heat transfer characteristics of viscous fluid with variable fluid properties over a permeable stretching sheet placed in a porous medium in the presence of magnetic field. The thermophysical properties of fluid such as viscosity and thermal conductivity may change with temperature. To predict fluid flow and heat transfer rate accurately, it is therefore more realistic to take into account the variation of physical properties with temperature. Numerical results are carried out for important parameters namely unsteady parameter, buoyancy parameter, temperature-dependent fluid viscosity parameter, thermal conductivity parameter and radiation parameter associated with the fluid flow and heat transfer phenomena. It is observed that heat transfer rate decreases with an increase in the temperature-dependent fluid viscosity and thermal conductivity parameters while it increases with the increase in buoyancy and radiation parameters. The limiting case of our results is in excellent agreement with the earlier steady state result of El-Aziz (2009). Chapter 6 contains the study of the Magnetohydrodynamics (MHD) buoyancy-induced stagnation point flow, heat and mass transfer of an electrically conducting micropolar fluid through porous medium on a vertical non-linear stretching surface with heat source. The theory of micropolar fluid was formulated by Eringen, which is capable of describing the non-Newtonian fluids consisting of dumb-bell molecules or short rigid cylindrical elements like mucus, light and heavy chemicals, fluid suspensions and Synovial fluid in human joints. The presence of dust or smoke suspended in a gas may also be modeled using this theory. The element free Galerkin solution is illustrated and the effect of non-linear stretching rate parameter, material parameter and buoyancy parameter on linear velocity, angular velocity, temperature and mass transfer functions is reported graphically. In chapter 7, the study of fully-developed, transient free convection flow and heat transfer along a semi-infinite vertical permeable moving plate through porous medium is examined including the effect of viscous heating. The Darcy-Forchheimer model, which includes the effects of boundary and inertia forces is employed. A one dimensional spatial and transient model has been derived and solved using EFGM. 3-D graphs of velocity and temperature are also plotted to provide a better perspective of the flow field evolution with respect to time. The parameters governing the problem are the Prandtl number, Darcy number, Forchhimer number, Grashof number, Eckert number and plate velocity. The velocity and temperature profiles are presented for different parameters. The Nusselt numbers at the plates are also evaluated. In chapter 8, MHD flow of micropolar fluid past a semi-infinite vertical moving plate in a Darcy-Forchheimer porous medium with heat absorption effect is examined. Profiles for velocity, microrotation and temperature are presented for a wide range of plate velocity, material parameter, Darcy number, Forchhimer number, magnetic field parameter and heat absorption parameter. The skin friction and Nusselt number at the plates are also calculated. iv In chapter 9, the effects of thermo-diffusion (Soret effect) and diffuso-thermal gradient (Dufour effect) on the unsteady incompressible fluid flow with heat and mass transfer past a semi-infinite vertical moving plate in a Darcy-Forchheimer porous medium. The governing system of coupled, nonlinear partial differential equations is solved numerically using the element free Galerkin method. The computations indicate that the increase in chemical rate parameter decreases velocity, temperature and concentration value. Temperatures are increased substantially with the decrease in Soret number and increase in Dufour number. Concentration values are conversely enhanced with the increase in Soret number and a concurrent decrease in Dufour number. Finally, the numerical values of skin friction coefficient, rate of heat transfer and rate of mass transfer are also presented in tabular form