The Onset of Convection in a Two-Layered Porous Medium with Anisotropic Permeability

We consider convection in a horizontal porous layer of uniform thickness which is heated from below and which is composed of two anisotropic sublayers with principal axes lying in the three coordinate directions. The aim is to determine criteria for the onset of convection by finding the critical Rayleigh number, wavenumber and roll orientation relative to the coordinate axes. The full set of nondimensional parameters has at least six members even when the sublayers are considered to be thermally isotropic, and therefore, we select some special cases in order to illuminate the type of qualitative behaviour which may be expected. One such case is where the anisotropic sublayers are identical except that one sublayer is rotated by an angle of 90∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90^\circ $$\end{document} to the other. In this situation, the most unstable roll is found to lie at an angle of ±45∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm \,45^\circ $$\end{document} to the principal axes. It is also found that fluid particles exhibit a mean longitudinal flow as they circulate about the vortex axis. This drift along the vortex is balanced by an equal and opposite drift in the two neighbouring vortices. Convection in each sublayer is shown to be two dimensional even though the full flow field is three dimensional.