Buoyant Darcy flow driven by a horizontal temperature gradient: a linear stability analysis

The linear stability of a fluid saturated porous layer bounded by two parallel impermeable plane walls is investigated. The lower wall is subject to a uniform heat flux, while the upper wall is subject to a linearly varying temperature in a horizontal direction. Two parameters govern the onset of convection in the porous layer: the vertical Darcy–Rayleigh number, and the horizontal Darcy–Rayleigh number. The objective of this study is to obtain the onset conditions for the instability of the basic parallel flow in the layer. The governing balance equations are written in a dimensionless form and solved on assuming oblique roll disturbances, arbitrarily oriented in the horizontal plane. Mathematically, this leads to a system of two ordinary differential equations to be solved as an eigenvalue problem. The solution, carried out numerically, provides the neutral stability condition. The numerical solution is performed by employing a procedure based on the sixth-order Runge–Kutta method and on the shooting method for satisfying the boundary conditions at the upper boundary wall.