Flow past a suddenly heated vertical plate in a porous medium

Abstract

An analysis is presented for the steady free convection flow about a semi-infinite vertical flat plate that is embedded in a saturated porous medium at high Rayleigh numbers. Similarity solutions are obtained for a class of problems where the wall temperature varies as a power of the distance from the leading edge of the plate. The existence and uniqueness of the solutions are considered. The approach to this steady-state solution is also considered by investigating the temporal development of the flow when the temperature of the plate is impulsively increased from that of the surroundings. A numerical solution is presented that matches the small and large time solutions. For some temperature distributions on the plate it is found that the velocity achieves its maximum value within the boundary layer. For these the disturbance from the leading edge of the plate travels fastest within the boundary layer. An asymptotic solution valid at large times is presented and the approach of the numerical solution to this asymptotic solution is illustrated. For the situation in which the plate is impulsively heated to a constant temperature an analysis is presented for the early stages of the departure from the one-dimensional solution.