Analytical study of Brinkman–Bénard convection in a bidisperse porous medium: Linear and weakly nonlinear study

Abstract

Linear and weakly nonlinear stability analyses of Brinkman–Bénard convection of a Newtonian fluid saturating a bidisperse porous medium (BDPM) are made. Local-thermal-non-equilibrium (LTNE) is assumed between the fluid and the porous spheres (micro-pores) that make up the macro porous medium. Two coupled linear momentum equations are considered one each for the macro- and micro-pores. Results of mono-disperse porous medium (MDPM) with solid spheres are recovered as a limiting case of the present study. Further, in the case of both types of porous media considered the results of Darcy–Bénard and Brinkman–Bénard convection are extracted under suitable limiting procedures. To keep the work analytical, we reduce the intractable hexa-modal Lorenz model with four quadratic nonlinearities into the tractable mono-modal Stuart–Landau equation (SLE) with cubic and quintic nonlinearities. Subcritical instability is discounted in the study thereby suggesting that cubic SLE and cubic–quintic SLE both expound similar results qualitatively. The concept of a BDPM is shown to be meaningful only when the pores are not large, and when they are very small, then the MDPM assumption applies. Similar observation can be made when the ratio of permeabilities is large. The presence of micro-pores does not alter the size of the convective cell significantly at the onset. The present study reiterates the findings of several earlier works.