Unstably stratified Darcy flow with impressed horizontal temperature gradient, viscous dissipation and asymmetric thermal boundary conditions

The free convection flow in a horizontal porous layer with an adiabatic bottom boundary and a top boundary with a stationary and non-uniform temperature distribution is investigated. The top boundary temperature distribution is assumed to have a constant gradient and the effect of viscous dissipation is taken into account. A basic parallel buoyant flow develops in the horizontal direction where the top boundary temperature changes. The governing parameters are the Gebhart number and the horizontal Rayleigh number associated with the gradient of the prescribed boundary temperature distribution. In fact, the system experiences a more and more intense effect of the frictional heating as the Gebhart number increases. A linear stability analysis of the basic buoyant flow is carried out. Oblique roll disturbances in any arbitrary horizontal direction are studied and the critical values of the horizontal Rayleigh number are evaluated numerically. It is shown that, for realistic values of the Gebhart number, the longitudinal rolls are the most unstable. Moreover, it is proved that the viscous dissipation yields a destabilising effect.