FINITE ELEMENT MODELING TO FLOW, HEAT AND MASS TRANSFER IN POROUS MEDIUM

Abstract

Since last few decades, the study of fluid flow has made a remarkable progress but still there are many physical phenomena which are yet to unveil. Flow, heat and mass transfer in porous media is one of such interesting research area due to its numerous applications e.g. transport of water in living plants and trees, geophysical flows, recovery of petroleum resource, storage of nuclear waste materials, insulation of buildings, grain storage and also in various engineering fields like irrigation engineering, chemical engineering, aeronautics, meteorology, biomedicine etc. Most of the studies in porous media done so far are based on darcy model. However recent research and further insight shows that this model is valid only for slow flows through porous media with low permeability but at higher flow rates or in highly porous media, 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 the Darcy-Forchheimer model. The consideration of inertial effects is becoming indispensable, specially in biomedical and geological engineering, to simulate meticulously the blood vessel blockage with deposits in the cardiovascular system and also to adequately evaluate production performance in well performance engineering. Provoked by this fact, darcy-Forchheimer model is used here in this thesis to study the flow, heat and mass transfer through porous medium in different cases. The purpose of the present study is to work on some probems of the flow heat and mass transfer of three type (Third grade fluid, Micropolar fluid and Biofluid) of fluids in the laminar boundary layer through different porous geometries. In the above studies, the problem ultimately reduces to the solution of a boundary value problem whose solution is always a hard nut to crack due to the nonlinear nature. So Finite Element Method (FEM) is used to solve the coupled system of nonlinear differential equations with corresponding boundary conditions. Instead of using the existing software's we have used our C++ codes as the existing codes had certain limitations when dealing with non linear system of equations. The Entire thesis comprises nine chapters. The first chapter is introductory in nature and deals with the fundamental concepts of flow, heat and mass transfer in porous media and basic idea regarding the third grade fluid, micropolar fluid and biofluid. It also contains some descriptions about the basic theoretical concepts used in subsequent chapters e.g. boundary layer, heat transfer, convection process, mass transfer etc. The rest of the eight chapters are divided into three parts, depending on the nature of the fluid. The first part consists of two chapters on third grade fluid. The second part consists of four chapters relating to micropolar fluid problems. The third and the last part consist of two chapters on biofluids. A brief description of the chapters is given as follows: The inadequacy of the classical Navier Stokes theory to describe rheological complex fluids such as important industrial fluids including intumescent paints, lubricants, greases, hydrocarbons, gels, complex polyalloys, polymer solution and certain oils, has led to the development of several theories of non-Newtonian fluids. Amongst the many models, used to describe the non-Newtonian behavior (e.g. shear thickening and shear thinning), exhibited by certain polymeric solutions and also geophysical materials, the fluids of grade n have received special attention. The third grade fluid (a subclass of grade n fluid) obeys a non-linear constitutive law in which the stress is expressed in terms of the pressure and of some tensors of Rivlin - Ericksen type. Seeing the innumerable applicability of this field, we have presented in second chapter ii , a mathematical model to simulate the steady laminar flow of an incompressible, third grade, non-newtonian fluid past an infinite porous plate embedded in a Darcy-Forchheimer porous medium. The special case of second order viscoelastic effects are also studied. The model is solved with appropriate boundary conditions using FEM. The study has shown that velocities are generally decreased with an increase in the forchheimer parameter, but enhances with the rise in permeability and third grade fluid parameter. Such type of flow scenario find applications in polymer extrusion processes, petroleum recovery processes, and other important industrial rheological systems. In the third chapter, a numerical solution is presented for the natural convective heat transfer of an incompressible, third grade, non-Newtonian fluid flowing past an infinite porous plate in a porous medium. A number of special cases of the general transformed model have also been discussed. Our investigation indicates that a rise in third grade viscoelasticity, thermal conductivity and permeability (Darcian effect) boosts velocities and/or temperatures, whereas an increase in Forchheimer parameter depresses velocity and also temperature. The heat transfer in such flow scenario is important in chemical engineering processes involving polyacrylamide and high-molecular weight polyethylene oxide. A new stage in the evaluation of fluid dynamic theory has been in progress because of the increasing importance in the processing industries and elsewhere of materials whose flow behavior cannot be characterized by usual Newtonian theory. Eringen formulated a new concept of theory of micropolar fluid, which was 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. iii Chapter four contains the problem of coupled heat and mass transfer in micropolar fluid flowing past a moving vertical surface embedded in a darcy-forchheimer porous medium with soret and Dufour effects. FEM is used to solve the transport equations. The results are shown graphically which indicates that increasing buoyancy parameter leads to an increase in linear velocity but decreases temperature functions, mass transfer function and angular velocity. It is interesting to note that an increase in Dufour number (reducing Soret number) lead to an increase in temperature function. Increasing Soret number (reducing Dufour number) increases the mass transfer function. The variation of skin friction and the heat transfer parameter with respect to Grashof number is also shown in the tabular form. Moreover, an increase in the surface parameter boosts the dimensionless microrotation profile. A numerical study of the buoyancy-induced convective flow and mass transfer of a micropolar, chemically-reacting fluid over a vertical stretching surface embedded in a porous medium has been presented in chapter five. The rate of chemical reaction is assumed to be constant throughout the fluid. Finite element solution is developed and the effect of Grashof number, Reynolds number, Darcy number, Forchheimer number, Schmidt number and chemical reaction number on velocity, temperature, micro-rotation and mass transfer functions is presented graphically. Our computation indicates that linear velocity decreases, temperatures increase, micro-rotation increases and mass transfer function decreases with a rise in chemical reaction parameter. Moreover increasing Schmidt number reduces mass transfer function both in the reactive and non-reactive flow cases, although mass transfer function values are always higher for any Sc value in the non-reactive case (i.e. Chemical reaction parameter = 0). iv Chapter six contains the study of fully-developed, transient MHD (Magnetohydrodynamics) free convection heat and mass transfer of an electrically-conducting micropolar fluid between two vertical plates containing a porous medium with heat generation/absorption and asymmetric wall temperature and concentration. A one-dimensional spatial and transient model has been derived and solved using FEM. 3-D graphs of velocity and microrotation are also plotted to provide a better perspective of the flow field evolution with respect to time. Temperature profiles are shown to be reduced with rising thermal conductivity parameter but increased with rise in heat source parameter. The numerical investigation reveals that thermal conductivity parameter can also be efficiently used to regulate skin friction and heat transfer, which thus can be of potential importance in manufacturing processes, nuclear energy system control etc. Moreover, the results indicate that volumetric flow rate, total heat rate added to the fluid, and the total species rate added to the fluid is lower in case of micropolar fluids as compared to Newtonian fluids. Chapter seven deals with the study of two dimensional flow heat and mass transfer of a chemically reacting hydromagnetic micropolar fluid over a non linear stretching sheet with variable heat flux in a non-darcy porous medium. Numerical results regarding local Nusselt number are shown graphically with Magnetic number for variation in heat transfer exponent. In addition, the effect of Magnetic number on skin friction parameter U1(0) and the wall couple stress function g'(0) is examined in the tabular form. This study also analyzes the effect of velocity exponent, heat transfer exponent, material parameter, Magnetic Number, Darcy Number, Forchheimer Number, Prandtl number Pr, Schmidt Number Sc and Chemical reaction rate parameter on velocity, microrotation, temperature and concentration profiles. It may be emphasized that prior to the present study, in most of the earlier literature, the microinertia density and heat flux were assumed to be constant but here we had taken them as variable which makes this work more interesting and important. Bio-Fluid (fluids with bio-molecules) mechanics is a field that is gaining immense importance from last few decades due to its numerous applicability everywhere, e.g. blood flow (flow in the heart), air flow in humans (nose, lungs), heat transfer in humans (scalp cooling, thermoregulation) and developing of new medicines for various diseases etc. Motivated by the applicability of this field, in drug delivery and diffusion in the cardiovascular system, we investigate in Chapter eight, the pulsatile hydromagnetic flow and mass transfer of a newtonian biofluid through a porous channel. The physically important behaviour of blood vessel blockage are assumed by taking a non-darcian porous drag model incorporating a darcian linear impedance for low Reynolds numbers and a forchheimer quadratic drag for higher Reynolds number. Two dimensional velocity and concentration profiles are plotted against variation in Reynolds number, Darcy number, Forchheimer number, hydromagnetic number, steady pressure gradient parameter and also Schmidt number with time. Three dimensional profiles of velocity varying in space and time are also provided. The conduit considered, is rigid with a pulsatile pressure applied through an appropriate pressure gradient term. The present study is one of the leading analyses of fundamental pharmaco-hydrodynamic flows. The flow model has applications in the analysis of electrically conducting hematological fluids flowing through filtration media, diffusion of drug species in pharmaceutical hydromechanics and also in general fluid dynamics of pulsatile systems. Seeing the importance of study of coupled flow and heat transfer phenomena in biorheological flows, we examine in the last chapter, the transient pulsatile hydromagnetic vi flow and heat transfer in a non-newtonian biofluid through a darcy forchheimer porous channel. Here the flow is assumed to be free form entrance edge effect. Using a bi-viscosity fluid model, our numerical results shows that fluid velocity increases with an increase in Reynolds number, steady pressure gradient and rheological parameter 13 whereas it decreases with an increase in magnetic number and Forchheimer parameter. Temperature profile decreases with an increase in Prandtl number. Moreover, an increase in Eckert number leads to a gradual increase in temperature profile but its intensity rises considerably for higher Eckert number values, thus revealing the fact that viscous heating phenomena must be taken into consideration especially in bio-rheological fluids. Each chapter begins with a brief literature review pertaining to the concerned problem and the motivation behind the study. Emphasis had been given to the physical interpretation and the new findings. Moreover, the computational results are summarized at the end of each chapter. The thesis ends with an appendix describing the fundamental concepts of finite difference and finite element method and finally a list of complete References. Numerical results obtained using FEM are verified with the results obtained from Finite difference numerical technique. The convergence characteristic of finite element approximation is also investigated by refining the mesh and comparing the solution with those obtained by higher order elements. All the above mentioned studies are novel and to the best of the authors knowledge thus far have not been examined in the scientific literature. vii