APPLICATION OF FINITE ELEMENT METHOD TO SOME PROBLEMS OF MICROPOLAR FLUIDS

Abstract

In recent years, the study of microcontinuum fluid mechanics has made remarkable progress. Several theories describing fluids consisting of molecules whose lengths are not negligible when compared with the characteristic length of the geometry, have been formulated. Eringen, A.Cemal put forward the theory of micro fluids which has been further extended to micropolar fluids that exhibit certain microscopic effects arising from the local structure and micromotions of the fluid elements. The equations governing the problem of micropolar fluids ultimately reduce to highly nonlinear coupled differential equations. To find their analytical solution is a difficult task and many a times it appears not even possible to achieve the same. However, The numerical results may be obtained to a good degree of accuracy with the help of powerful computational techniques and give important clues about the real problem of interest. The thesis comprising of eight chapters, deals with the finite element solution of some problems of flow and heat transfer in micropolar fluid through certain geometries. Both forced and free ii convection problems, with heat sources or sinks present in some cases, as well as suction and injection applied on the boundaries have been considered. The first chapter is introductory and contain a brief description of the micropolar fluid model followed by a short survey of the literature. The governing equations of conservation of mass, momentum, angular momentum and energy for an incompressible micropolar fluid have been laid down. The subsequent contents of the thesis form the main contribution of the author. The, chapter-wise details are as follows:• In chapter 2, the problem of mixed convective flow of micropolar fluid between two co-axial porous circular cylinders has been analysed when the outer cylinder moves parallel to itself with a constant velocity. The temperature on the cylinders varies linearly along the walls. The coupled differential equations have been solved by finite element method to obtain the velocity and temperature distribution. The numerical results have been displayed graphically for various values of the parameters involved. It is observed that micropolar fluid may be used as a coolant. The skin friction, couple stress and heat transfer rate on the boundaries have also been discussed. In chapter 3, the mixed convective flow of a micropolar fluid over a stretching sheet has been examined. This study may be applicable to polymer technology involving the stretching of plastic sheets. The governing differential equations are non-linear and their finite element solution has been obtained. The numerical results are shown through graphs. The wall shear stress, couple stress and the rate of heat transfer from the wall are calculated. Chapter 4 deals with the problem of non-steady plane stagnation point flow of a micropolar fluid with hard blowing. The technique of hard blowing is usually employed to reduce wall shear and heat transfer on a body. Due to hard blowing with high velocities normal to the body surface at the stagnation point, the boundary layer is blown off the body to form a shear layer. The wall temperature and free stream temperature both are assumed constant. The velocity of the potential flow is chosen to vary inversely as a linear function of time. The results are shown through graphs. It is concluded that the heat transfer on the wall is zero, as expected, The. chapter 5 is devoted to the study of micropolar fluid flow from an enclosed rotating disc with suction and injection. Finite element method is used to solve highly non-linear governing differential equations. The values of dimensionless radial distance at which there is no-recirculation for the cases of net radial out flow and net radial inflow fpr several values of the micropolar parameter and suction/injection parameter, have been tabulated. The numerical results for radial and axial velocities and microrotation functions for different values of various parameters involved have been obtained. The behaviour of velocity components have been investigated in detail in the regions of recirculation and no-recirculation for the cases of radial outflow and inflow and shown graphically. The finite element solution of combined convective non-steady three dimensional micropolar fluid flow at a stagnation point is the subject matter of the cilapter 6. The flow is considered in the vicinity of the forward stagnation point of a blunt-nosed body. The velocity of potential flow is chosen to vary inversely as a linear function of time. The velocity function, microrotation function, temperature, skin friction coefficient, couple stress coefficient and heat transfer coefficient have been obtained numerically by varying four parameters iv viz. the micropolar parameter, the parameter characterising the surface around the stagnation point, the degree of acceleration or deceleration of the potential flow and the Grashof number characterising the free convection. From the analysis of results, it may be inferred that the presence of micropolar additives thoroughly influences the characteristic features of the flow and heat transfer. The chapter 7 contains the problem of unsteady boundary layer flow of a micropolar fluid at a 2-dimensional stagnation point on a moving wall. Unsteadiness occurs due to time dependence of free stream velocity and wall temperature. The partial differential equations governing the flow have been solved numerically using finite element method as well as Keller-Box method. It is found that Keller box method too is an efficient tool for solving parabolic partial differential equations. The velocity distribution has been illustrated for several positive and negative values of the wall velocity. The skin friction, couple stress and transfer rate, are found to be strongly dependent on the coupling parameter and time, however, the effect of variation in the microrotation parameter is visible appreciably in case of couple stress only. The concluding chapter 8 is devoted to the study of the free convection heat transfer in a micropolar fluid confined between a long vertical wavy wall and a parallel flat wall. Analysis of fluid flow over a wavy wall is of interest because of its physical applications such as transpiration cooling of re-entry vehicles, rocket booster and„ film vaporization in combustion chambers etc. The equations governing the fluid flow and heat transfer have been solved subject to the relevant boundary conditions by assuming that the solution consists of two parts viz. a mean part and a perturbed one. To obtain the perturbed part of V the solution, use has been made of the long-wave approximation. The sets of differential equations have been solved by FEM. The zeroth-order, the first order and the total solution of the problem have been evaluated numerically for several sets of values of the various parameters entering the problem and are depicted graphically.