The unsteady penetration of free convection flows caused by heating and cooling flat surfaces in a porous media

Abstract

Free convection flow is generated in a porous media adjacent to a vertical or horizontal flat surface which is suddenly heated and cooled, sinusoidally along its length. An approximate analytical solution, which is valid for small times and for any value of the Rayleigh number, Ra, is obtained by matching inner and outer expansions. A numerical solution is also obtained which matches the small time analytical solution to the ultimate steady-state solution when such a solution exists. In both configurations the flow pattern is that of a row of counter rotating cells situated close to the surface. When the surface is vertical and for Ra ⪆ 40, two recirculating regions develop at small times at the point of collision of two boundary layers which flow along the surface. However, for 40 ⪷ Ra ⪷ 150, the steady state solution, proposed by Bradean et al. [International Journal of Heat and Mass Transfer, 1996, 39, 2545–2557], using additional symmetrical conditions is unstable and at very large time the solution is periodic in time. In the situation in which the surface is horizontal the collision of convection boundary layers occurs without separation. As time increases, the height of the cellular flow penetration increases and then decreases to its steady-state value. The heat penetrates infinitely into the porous media and the steady-state is approached later in time as the distance from the surface increases.