| dc.description.abstract | This research work is a study of fluid flow problems by the Boundary (Finite) Element Method. The widely used Finite Element Method has the advantages of its flexibility in choosing the shape and size of the elements, the arbitrariness of the mesh structure as well as the freedom of choice in approximating the formulae; but it leads to the requirement of very large core (memory) when applied to problems in fluid dynamics. The Boundary Element Method makes use of the existing analytical solution while approximating the given equations, and the rest of the approximations are the same as in the Finite Element Method. Moreover, it reduces the dimensionality of the problem by one. Its simplicity and elegance to implement on modern computers show its superiority over other numerical methods like the Finite Difference and the Finite Element Method. This thesis is constituted of five chapters. Chapter I presents a fairly extensive literature survey and gives the introduction to this method. Superiority over the other numerical methods is brought out to justify the reason for choosing the Boundary Element Method.
Chapter II is concerned with molecular diffusion in oscillating viscous flows in a pipe. The effect of flow oscillations on the axial diffusion of a solute in a pipe is analyzed by the Boundary Element Method. The equation governing the fluid flow is solved analytically and using these results the nonlinear concentration equation coupled with fluid flow is solved by the Boundary Element Method, using the fundamental solution of the heat equation for a general initial distribution. The nonlinear terms are solved by an iterative method. Since the fundamental solution itself contains the time variable, there is no necessity to have a separate iteration for the time variable. The elements of the boundary and the domain are generated as in the Finite Element Method and these integral equations are solved with linear variations. The flow variables like velocities and skin friction are calculated; backflow and distribution of concentration are also discussed. It also presents the error analysis and the method of implementation.
In Chapter III, viscous flow in a square cavity, that is, a flow induced by a steady motion of one of the walls in its own plane and the temperature distribution of the cavity, is considered. The problem is coupled with the heat equation and solved by the Modified Biharmonic Boundary Element Method. The effective numerical scheme is presented where the analytical solution is obtained in the neighborhood of the singular point. While applying the Boundary Element approximation, we use two Green’s functions and for the bilinear equation after converting the system of equations by a separable method. The error due to the singular point is minimized and the flow variables like velocity and pressure are calculated; and to show its efficiency the results are compared with the results available in the literature. By solving for different mesh and node points, eddy formation is confirmed, the center of the eddy is determined and the vorticity and the stream function at the center of the eddy are presented.
In Chapter IV, we apply the Boundary Element Method to biofluid mechanics problems, namely, the steady and unsteady viscous flows in channels of variable cross-section by using the direct Boundary Element Method. New types of boundary conditions on vorticity are described, and an upwind Boundary Element Scheme is presented which increases the efficiency of the computational scheme. Using this method, extensive analysis is done to calculate the flow separation and the reattachment in the stenosis region. The unknown stream function, vorticity, shear stress, and temperature distribution are discussed in detail. For both steady and unsteady flows, the behaviour of velocity and shear stress are analyzed in the upstream of the transition region. The separation and the reattachment results are compared. Four types of stenosis are considered for the above study.
In the final chapter, in order to show the efficiency of the Boundary Element Method, external flow, namely, two-dimensional flow around a cylinder is considered. To make the method effective, a complex boundary element is considered to analyze the flow. After explaining the advantages and presenting the error analysis for this method, some flow variables of interest are discussed in detail. The advantage of this method is that the approximation given by the Complex Boundary Element Method is an analytic function in the domain of interest, and the integrals involved in the boundary element are solved exactly without using numerical approximations. This leads to minimization of the error.
Thus, this thesis clearly demonstrates the efficacy of the Boundary Element Method, which had hitherto been mainly used in solving problems in solid mechanics, for solving various fluid flow problems. Since this method utilizes the existing analytical solution before approximating the boundary integrals, it drastically reduces the computer time and the required memory.
The books and original papers referred to in the text of the thesis are enlisted at the end of each chapter. The symbols used are defined as and when they occur in the text. Also, for easy reference, they are listed at the end of the thesis. Papers based on the work reported in the thesis will be communicated for publication shortly. | |
