Unstable buoyant flow in an inclined porous layer with an internal heat source

The buoyant flow with zero average velocity, namely free convection, in an inclined porous layer is studied. The heating is supplied by an internal volumetric source with a uniform distribution. The boundaries are either isothermal at the same temperature, or the lower one adiabatic and the upper one isothermal. The stability to small-amplitude perturbations is analysed for three-dimensional normal modes. It is proved that the longitudinal rolls, viz. normal modes with wave vector perpendicular to the basic flow, are the most unstable modes. It is also shown that neutrally stable transverse modes may grow in time if the inclination angle of the layer to the horizontal is smaller than a threshold value. The threshold angle depends on the imposed boundary conditions, isothermal/isothermal or adiabatic/isothermal. When the threshold angle is approached from below, the neutral stability curves assume a closed-loop shape, they gradually shrink their size and eventually collapse to a point.