Natural convection in shallow porous cavities near the density maximum: The conduction and intermediate regimes

We investigate two-dimensional natural convection in a shallow rectangular cavity filled with a porous medium which is saturated with a fluid which has a density maximum. One sidewall of the cavity is maintained at a temperature below the density maximum and the other sidewall is maintained at a temperature above the density maximum. The top and bottom boundaries are insulating. Attention is paid to the case when the aspect ratio, A, defined as the ratio of the height of the cavity to its width, is asymptotically small. The domain is divided into two end zone regions near the hot and cold walls, a density maximum turning zone region, and two central core regions in between the end-zone regions and the density maximum region. Asymptotic analysis is performed in the conduction regime when the Rayleigh number Ra ∼ O(1) and in the intermediate regime when Ra ∼ O(1/A), and heat transfer correlations are presented. Numerical solutions to the full governing solutions are obtained and these solutions are compared with the asymptotic results.