Thermoconvective instabilities in an inclined porous channel heated from below

The thermoconvective instability in an inclined rectangular channel filled with a fluid saturated porous medium and heated from below with a uniform flux is investigated. A stationary parallel buoyant flow with a linear temperature change in the transverse direction is considered. The linear stability to transverse and longitudinal roll disturbances of this basic state is examined. The thermoconvective instability onset of transverse rolls occurs when the Darcy-Rayleigh number exceeds a critical value, that increases with the inclination angle. The critical Darcy-Rayleigh number is discontinuous at the inclination angle 23.4749° above the horizontal. It is shown that, when the inclination angle exceeds 31.3618°, either the transverse rolls are stable or a second discontinuous transition to a higher branch of instability occurs. The longitudinal rolls may be unstable for every inclination except for the vertical. A stability diagram is sketched displaying the geometrical conditions (inclination angle, aspect ratio) such that either the transverse rolls or the longitudinal rolls is the preferred mode of instability.