## Electronic Theses and Dissertations

1992

Doctoral Thesis

Ph.D.

#### Department

Mathematics and Statistics

Mathematics.

#### Rights

info:eu-repo/semantics/openAccess

#### Abstract

This thesis deals with the study of certain flow and stability problems in porous media using some non-Darcian models. It is divided into three parts, which are essentially independent of each other. The first part deals with the theoretical study related with the solution of the Brinkman equation. An exact Cartesian-tensor form solution of the Brinkman equation is found, which facilitates the study of various boundary value problems. The problems of the flow past a porous sphere and creeping flow past a porous spherical shell are considered to illustrate the point. Then, a theoretical look is taken at the problem of steady convection in porous media, again using the generalized Brinkman equation. A variational formulation is introduced to define a weak solution and existence, uniqueness, and regularity of the weak solution are discussed. In the second part, two stability problems in porous media are examined. First, a theoretical investigation for the onset of Rayleigh-Benard convection is a porous layer, using the Brinkman equation with anisotropic permeability, is presented. The critical Rayleigh numbers, within the framework of linear theory, for both rigid and free boundaries, are calculated. The effect of considering anisotropy in the Brinkman equation is brought out. Next, an energy method is used to study the nonlinear stability of a rotating porous layer. Here the Brinkman-Boussinesq model is employed and a generalized energy functional is used to determine the energy stability bound. A comparison is also made with the linear instability bound. In the third part, another non-Darcian model is used to study the problem of unsteady flow of a power-law fluid in a porous medium. A mathematical analysis for two typical initial-boundary value problems, which correspond to well-test cases in the oil industry, is presented.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1992 .Q558. Source: Dissertation Abstracts International, Volume: 54-05, Section: B, page: 2541. Thesis (Ph.D.)--University of Windsor (Canada), 1992.

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