Vortex Instability in Buoyancy-Induced Flow over Inclined Heated Surfaces in Porous Media

A linear stability analysis is performed for the study of the onset of vortex instability in free convective flow over an inclined heated surface in a porous medium. The undisturbed state is assumed to be the steady two-dimensional buoyancy-induced boundary layer flow which is characterized by a non-linear temperature profile. By a scaling argument, it is shown that the length scales of disturbances are smaller than those for the undisturbed boundary layer flow, thus, confirming the so-called “bottling effects” whereby the disturbances are confined within the boundary layer. By neglecting the lowest order terms in the three-dimensional disturbances equations, the simplified equations are solved based on the local similarity approximations, wherein the disturbances are assumed to have a weak dependence in the streamwise direction. The resulting eigenvalue problem is solved numerically. The critical parameter and the critical wave number of disturbances at the onset of vortex instability are computed for different prescribed wall temperature distribution of the inclined surface. It is found that the larger the inclination angle with respect to the vertical, the more susceptible is the flow for the vortex mode of disturbances; and in the limit of zero inclination angle (i.e., a vertical heated plate) the flow is stable for this form of disturbances.