Effects of Nonuniform Thermal Gradient and Adiabatic Boundaries on Convection in Porous Media

Abstract

The effects of different combinations of thermally insulated boundaries and nonuniform thermal gradient caused by either sudden heating or cooling at the boundaries or by distributed heat sources on convective stability in a fluid saturated porous medium are investigated using linear theory by considering the Brinkman model. In the case of sudden heating or cooling, solutions are obtained using single-term Galerkin expansion and attention is focused on the situation where the critical Rayleigh number is less than that for uniform temperature gradient and the convection is not maintained. Numerical values are obtained for various basic temperature profiles and some general conclusions about their destabilizing effects are presented. In particular, it is shown that the results of viscous fluid (σ = 0) and the usual Darcy porous medium (σ → ∞) emerge from our analysis as special cases. In the case of convection caused by heat source, since the effect of heat source is not brought out by the single-term Galerkin expansion, the critical internal Rayleigh number is determined using higher order expansion by specifying the external Rayleigh number. It is shown that, for values of σ2 ≥ 2.45 × 105, the different combinations of bounding surfaces give almost the same Rayleigh number and an explanation, following Lapwood, for this surprising behavior is given. It is found that the heat source’s effect on convection decreases for wave numbers up to the value 2.2 and drops suddenly around the critical value of 2.4 and then increases up to 2.5.