Buoyancy-induced flow adjacent to a periodically heated and cooled horizontal surface in porous media

Abstract

Analytical and numerical solutions are presented for the problem of steady, natural convection from a sinusoidally heated and cooled horizontal surface which is embedded in a fluid saturated porous media which is maintained at a constant temperature T ∞. The plate is assumed to have a harmonic temperature variation, and at large distances from the plate we investigate two different boundary conditions, namely, when we enforce either a constant temperature T ∞ or an adiabatic boundary condition. For small values of the Rayleigh number an analytical solution has been obtained provided that the temperature at infinity is constant but depends on the Rayleigh number or there is zero heat flux at infinity. On specifying an arbitrary temperature T ∞ at infinity then no analytical solution could be obtained and the numerical solution procedure was not convergent. However, we were able to obtain an analytical solution in the situation when the constant temperature T ∞ is enforced at a finite distance, say y = d, from the plate and, as d → ∞, the solution approaches that obtained by enforcing the adiabatic condition at infinity. When the constant temperature T ∞ is enforced at the station y = d we found that the numerical solution is dependent on the value of d, but when the zero heat flux condition is enforced at y = d we obtained that d = 4 π is sufficiently large for the solution to be independent of the value of d. At very small values of Ra there is very good agreement between the analytical and numerical solutions and when Ra is very large the boundary-layer scalings for the Nusselt number and the mean fluid velocity along the plate are confirmed by the numerical calculations.