On a free convection problem over a vertical flat surface in a porous medium

The problem of the free convection boundary-layer flow over a semi-infinite vertical flat surface in a porous medium is considered, in which the surface temperature has a constant value T-1 at the leading edge, where T-1 is above the ambient temperature, and takes a value T-2 at a given distance L along the surface, varying linearly between these two values and remaining constant afterwards. Numerical solutions of the boundary-layer equations are obtained as well as solutions valid for both small and large distance along the surface. Results are presented for the three cases, when the temperature T-2 is greater, equal or less than the ambient temperature T-infinity. In the first case, T-2 > T-infinity, a boundary-layer flow develops along the surface starting with a flow associated with the temperature difference T-1 - T-infinity at the leading edge and approaching a flow associated with the temperature difference T-2 - T-infinity at large distances. In the second case, T-2 = T-infinity, the convective flow set up on the initial part of the surface drives a wall jet in the region where the surface temperature is the same as ambient. In the final case, T-2 < T-infinity, a singularity develops in the numerical solution at the point where the surface temperature becomes T-infinity. The nature of this singularity is discussed.