Onset of Convection in a Porous Box with Partly Conducting and Partly Penetrative Sidewalls

The onset of thermal convection in a 2D porous box is investigated analytically. The lateral walls are partly heat conducting and partly penetrative. The top and bottom are impermeable and perfect heat conductors. The linear stability problem is solved only for the symmetric configuration of equal conditions at each sidewall. The problem is degenerate when the parameters of semi-conduction and semi-penetration coincide. The degenerate problem has one symmetric and one antisymmetric eigenfunctions, and the cell width varies with minimum cell width in the middle. Our primary model for the partly penetrative wall is a thin and highly permeable layer near a closed wall. We also study a secondary model of a partly penetrative wall, with a thin layer of small permeability near a hydrostatic reservoir.