Linear instability to longitudinal rolls of the Darcy-Hadley flow in a weakly heterogeneous porous medium

The aim of this contribution is to investigate the effects of a weak vertical heterogeneity of the porous medium on the stability of the Darcy-Hadley flow. We will assume that the permeability and the thermal conductivity of the porous medium undergo a weak linear change in the vertical direction. The stability analysis will be carried out by considering linear perturbations of the basic state in the form of longitudinal rolls, under the hypothesis that the weak heterogeneity does not alter the result that these modes are the most unstable. The neutral stability conditions and the critical values of the vertical Darcy-Rayleigh number will be obtained on varying the horizontal Darcy-Rayleigh number as well as the parameters of heterogeneity relative both to the permeability and to the conductivity. The solution of the linear disturbance equations will be obtained by employing Galerkin’s method of weighted residuals.