Buoyancy Driven Convection in a Liquid Layer with Insulating Permeable Boundaries

In this paper, we investigate the onset of buoyancy driven thermal convection in a horizontal layer of fluid with thermally insulating permeable boundaries, using the classical linear stability analysis. It is proved that the principle of exchange of stabilities is valid. The eigenvalue problem is solved by using the Galerkin method. Results are obtained and discussed？ for a wide range of values of the boundary parameters characterizing the permeable nature of boundaries. Attention is focused on a situation where the value of the critical Rayleigh number is less than that for the case when one of the boundaries is rigid while the other one is free and the convection is not maintained in general. In the case, when permeability parameter of either one of the two boundaries varies inversely to that of the other, we discover that the critical Rayleigh number decreases and goes through a lowest minimum at a certain value of the permeability parameter and this situation pertains when the critical wave number is zero. In addition, existing results for various combinations of the boundary conditions namely, when both the bounding surfaces are either dynamically free or rigid and when either one of them is dynamically free while the other one is rigid, are obtained as limiting cases of the boundary parameters