Anisotropic Inertia Effect

Abstract

Anisotropic porous media occur everywhere, although the anisotropy will not, in general, be regular. However, frequently the anisotropy is approximately transversely isotropic. One may expect properties like the permeability and the thermal diffusivity to vary strongly with changes of direction in a porous material. With wood or layered rocks one may expect strong permeability variation along the grain lines as opposed to directions orthogonal to these lines. An understanding of fluid flow in an anisotropic porous medium is highly desirable. In this chapter we consider the problem of double diffusive convection in a layer of Darcy porous material with a single temperature, but the emphasis is on assessing the effect of an anisotropic inertia coefficient. There is strong evidence to show that, in general, the inertia coefficient may be very anisotropic, especially in a microfluidic setting. Indeed, thermal convection in porous media is an area which is attracting much attention when the material properties are anisotropic. Since many real porous materials display strong anisotropy this focus of attention is natural. We describe a linear instability analysis together with a complimentary nonlinear energy stability analysis for the thermosolutal convection problem in a saturated porous medium allowing the inertia coefficient in Darcy’s equation to be an anisotropic tensor. Numerical calculations are presented which specialize to the case where the inertia term is composed of constants and is diagonal, but this still allows for the very important case of horizontal anisotropy in the inertia coefficient.