Convection in porous media and Legendre, Chebyshev Galerkin methods

Abstract

The subject of thermal convection in fluid and porous media is investigated, coupled with the development of efficient spectral finite element methods to improve on the more commonly used techniques for these types of problems. Convection induced by the selective absorption of radiation in a porous medium is investigated in the first four chapters. For the Darcy and Brinkman models for fluid flow the thresholds of the linear and nonlinear theories are shown to be extremely close, demonstrating that the linear theory is accurate enough to predict the onset of convective motion. The exploration of a quadratically modelled internal heat source is discussed next. It is shown that the linear and nonlinear thresholds are close unless the quadratic term becomes dominant over the linear term. Developing a double- diffusive model yields a critical parameter for which no oscillatory convection occurs when it is exceeded. This is an unobserved phenomenon in the present literature. Thermal convection in a linearly viscous fluid in a finite box is also explored. It is demonstrated that the linear and nonlinear thresholds do not coincide, which contradicts results by Georgescu & Mansutti [25]. Legendre and Chebyshev polynomial based spectral methods are also developed for the evaluation of eigenvalues and eigenfunctions inherent in stability analysis in porous media, drawing on the experience of the implementation of the well established techniques in the previous work. These generate sparse matrices, where the standard homogeneous boundary conditions for both porous and fluid media problems are contained within the method.