Linear instability of the Darcy-Hadley flow in an inclined porous layer

The buoyant flow in a saturated porous layer inclined to the horizontal is studied under the assumption that the plane impermeable boundaries are subject to linear temperature distributions up the layer. The basic solution is stationary and such that the temperature gradient is inclined to the boundary walls. Two parameters govern the thermal boundary conditions: the Rayleigh number, associated with the component of the basic temperature gradient orthogonal to the boundaries, the Hadley-Rayleigh number, associated with the component of the basic temperature gradient parallel to the boundaries. The linear stability of the basic solution with respect to the longitudinal normal modes is studied by employing two different numerical methods: a collocation method of weighted residuals, and a Runge–Kutta solver. Different regimes are considered: the upward–cooling condition, the upward–heating condition, and the buoyancy–balanced condition. The latter regime implies a vanishing velocity distribution and a vertical temperature gradient in the basic state. In the upward–cooling regime, for a fixed Hadley–Rayleigh number, the increasing inclination to the horizontal leads to a destabilising effect. When the inclination exceeds a threshold angle that depends on the Hadley–Rayleigh number, the basic solution becomes unstable for every Rayleigh number. The reverse holds true in the upward–heating regime, where the increasing inclination to the horizontal stabilises the basic flow. The general oblique normal modes are finally considered. It is shown that the longitudinal modes are selected at the onset of convection, except for the case of the Darcy–Bénard limiting case where all the oblique modes are equivalent.