## Electronic Theses and Dissertations

1999

Dissertation

Ph.D.

#### Department

Mathematics and Statistics

Kaloni, P. N.,

Mathematics.

CC BY-NC-ND 4.0

#### Abstract

In this thesis, we study several stability problems related to mono-diffusive (thermal) convection, or double-diffusive (thermal-solutal) convection, in a porous medium and in a viscous fluid, induced by inclined temperature gradients with mass flow, or through-flow, or with surface tension. The thesis begins with a brief outline of porous media, the Darcy model, and a brief exposition of linear and nonlinear stability methods. It also briefly discusses the compound matrix method and the Chebyshev tau method for solving eigenvalue problems. Two chapters deal with the convection problems in a porous medium induced by inclined temperature gradients. Problems of horizontal mass flow or vertical through-flow are discussed, and the effect of including solutal gradient and mass flow is analyzed. In each case, an energy functional with coupling parameter(s) is chosen to establish the differential inequality, which gives the sufficient condition for the nonlinear stability of the basic steady solution. The associated variational problem is formulated from this condition. The compound matrix method with secant method and golden search routine is employed to numerically solve the resulting eigenvalue problem. Numerical results and comparisons are presented to show that a lower-order Galerkin approximation may not be accurate enough to predict the correct results, and that there may be a wide difference between the stability bounds computed by the linear stability theory and the energy stability theory. The numerical solution of the inclined temperature gradient convection problem in a viscous fluid is also presented. The discussion includes the effect of horizontal mass flow. Both energy and linear stability methods are employed and the compound matrix method with secant method and golden section search routine is used to solve the eigenvalue problems. This results in an eighth-order differential equation. The effect of mass flow-rate, Prandtl number, and horizontal temperature gradient on critical Rayleigh number are discussed. A comparison between the energy and linear stability results is also reported. The linear stability problem of convection induced by inclined temperature gradient and surface tension in a viscous flow is also studied. The resulting eigenvalue problem, in this case, turns out to be complex and thus prohibits the use of the compound matrix method. The problem is solved by implementing the Chebyshev tau-QZ method. Numerical results and comparisons are presented to demonstrate that the implementation of the approach and algorithm proposed are correct and powerful. Moreover, this algorithm can also be used to numerically verify whether the principle of exchange of stability is valid or not.Dept. of Economics, Mathematics, and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1999 .Q25. Source: Dissertation Abstracts International, Volume: 61-09, Section: B, page: 4765. Adviser: P. N. Kaloni. Thesis (Ph.D.)--University of Windsor (Canada), 1999.

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