Convection in a fluid-saturated porous layer with non-uniform temperature gradient

Abstract

The effect of a non-uniform thermal gradient caused by either sudden heating or cooling at the boundaries or by distributed heat sources on convective heat transfer in a fluid saturated porous medium is investigated using the Brinkman model by means of linear stability analysis. The case of isothermal boundary conditions is examined by considering different combinations of bounding surfaces. In the case of sudden heating or cooling, analytical solutions are obtained using a single-term Galerkin expansion and attention is focussed on the situation where the critical Rayleigh number is less than that for a uniform thermal gradient and the convection is not maintained. Numerical results are obtained for various basic temperature profiles and some general conclusions about their destabilizing effects are presented. In the case of distributed heat sources it is shown, in general, that the effect of heat source on the Rayleigh number is second order and is not felt in the single-term Galerkin expansion. Hence the critical internal Rayleigh number is determined using the higher order expansion by specifying the external Rayleigh number. Both analytical and numerical solutions are obtained and it is shown that the numerical results obtained by a sixth-order approximation agree well to an error of 3.3% for the two-term expansion and closely for the three-term expansion. In particular, it is shown that for values of σ ² ≥ 2.45 × 10⁵ the different combinations of bounding surfaces give almost the same Rayleigh number and an explanation, following Lapwood, for this surprising behaviour is given.