Gill's problem in a sandwiched porous slab

TL;DR

This paper extends Gill's stability problem to a three-layer porous slab with high thermal conductivity external layers and a low conductivity core, revealing flow instability conditions related to the Rayleigh number.

Contribution

It introduces a three-layer model with infinite thermal conductivity ratio, providing new insights into flow instability in layered porous structures.

Findings

Flow instability can occur at high Rayleigh numbers.

Instability conditions match previous homogeneous layer predictions in a limiting case.

Layered configuration influences the onset of buoyant flow instability.

Abstract

The classical Gill's stability problem for the stationary and parallel buoyant flow in a vertical porous slab with impermeable and isothermal boundaries kept at different temperatures is reconsidered in a different perspective. A three-layer slab is studied instead of a homogeneous slab as in Gill's problem. The three layers have a symmetric configuration where the two external layers have a high thermal conductivity, while the core layer has a much lower conductivity. A simplified model is set up where the thermal conductivity ratio between the external layers and the internal core is assumed as infinite. It is shown that a flow instability in the sandwiched porous slab may arise with a sufficiently large Rayleigh number. It is also demonstrated that this instability coincides with that predicted in a previous analysis for a homogeneous porous layer with permeable boundaries, by considering the limiting case where the permeability of the external layers is much larger than that of the core layer.