The Wooding problem revisited

The threshold conditions to convective instability in a semi-infinite porous layer saturated by a fluid are determined. The classical setup for this problem in geothermal fluid dynamics was originally modeled by Wooding in 1960. Its formulation is here reconsidered to allow for a finite heat transfer coefficient across the boundary, parametrized through the Biot number. The temperature boundary condition considered by Wooding is here recovered as the limit of an infinite Biot number. The linear stability analysis of the stationary boundary layer, which establishes in the porous medium, when a boundary steady suction occurs is carried out. Two different versions of the Rayleigh number are considered, namely, a temperature-difference-based version and a heat-flux-based version. While the former is the classical Rayleigh number for flow in porous media, the latter is a variant definition, which displays a finite limit at neutral stability in both the opposite limiting cases of an infinite or a zero Biot number.