Unstable Convection in a Vertical Double–Layer Porous Slab

Abstract

A convective stability analysis of the flow in a vertical fluid-saturated porous slab made of two layers with different thermophysical properties is presented. The external boundaries are isothermal with one of them impermeable while the other is open to an external fluid reservoir. This study is a development of previous investigations on the onset of thermal instability in a vertical heterogeneous porous slab where the heterogeneity may be either continuous or piecewise as determined by a multilayer structure. The aim of this paper is investigating whether a two-layer structure of the porous slab may lead to the onset of cellular convection patterns. The linear stability analysis is carried out under the assumption that one porous layer has a thermal conductivity much higher than the other layer. This assumption may be justified for the model of a heat transfer enhancement system involving a saturated metal foam. A flow model for the natural convection based on Darcy’s momentum transfer in a porous medium is adopted. The buoyancy-induced basic flow state is evaluated analytically. Small-amplitude two-dimensional perturbations of the basic state are introduced, thus leading to a linear set of governing equations for the disturbances. A normal mode analysis allows one to formulate the stability eigenvalue problem. The numerical solution of the stability eigenvalue problem provides the onset conditions for the thermal instability. Moreover, the results evidence that the permeability ratio of the two layers is a key parameter for the critical conditions of the instability.