The Wooding problem revisited

TL;DR

This paper revisits the classical problem of convective instability in porous layers, incorporating imperfect boundary heat transfer via the Biot number, and analyzes stability under different Rayleigh number definitions.

Contribution

It extends Wooding's 1960 model by including boundary heat transfer imperfections and compares two Rayleigh number formulations in stability analysis.

Findings

The classical boundary condition is recovered as the limit of infinite Biot number.

Stability thresholds differ between temperature-difference and heat-flux Rayleigh numbers.

The heat-flux-based Rayleigh number remains finite at neutral stability for all Biot numbers.

Abstract

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 modelled by Wooding in 1960. Its formulation is here reconsidered to allow for an imperfect heat transfer across the boundary, parametrised 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 of a zero Biot number.