The inclined Wooding problem

A constant-thickness thermal boundary layer is formed by the presence of uniform suction into a constant temperature hot surface bounding a porous medium—the Wooding problem. When the hot surface lies below the porous medium, the system is susceptible to thermoconvective instability. The present paper is concerned with how the classical linear stability analysis is modified by inclining the heated surface. Analysis is confined to disturbances in the form of transverse rolls because the equivalent analysis for longitudinal rolls may be described analytically in terms of that for the horizontal layer. The linear stability analysis is made difficult by the absence of a second bounding surface, and the method of multiple shooting is needed in order to obviate the consequence of having a stiff system of disturbance equations. Therefore, the computational domain is split into 5 or 10 subdomains. It is found that all modes of instability travel up the heated surface unless the surface is horizontal. The system is found to be linearly stable for all inclinations above 31. 85473 ^∘, a value which is remarkably close to that for the inclined Darcy-Bénard problem.