On the stability of parallel flow in a vertical porous layer with annular cross-section

TL;DR

This paper proves that buoyant parallel flow in a vertical porous annular layer with impermeable boundaries is linearly stable, extending Gill's theorem, and explores how partial permeability can induce instability.

Contribution

It extends Gill's theorem to annular geometries and demonstrates the stability of the flow under impermeable conditions through numerical analysis.

Findings

Flow is asymptotically stable with impermeable boundaries.

Partial permeability can induce axisymmetric instability.

Numerical solutions confirm the negative growth rate of perturbations.

Abstract

The linear stability of buoyant parallel flow in a vertical porous layer with an annular cross-section is investigated. The vertical cylindrical boundaries are kept at different uniform temperatures and they are assumed to be impermeable. The emergence of linear instability by convection cells is excluded on the basis of a numerical solution of the linearised governing equations. This result extends to the annular geometry the well-known Gill's theorem regarding the impossibility of convective instability in a vertical porous plane slab whose boundaries are impermeable and isothermal with different temperatures. The extension of Gill's theorem to the annular domain is approached numerically by evaluating the growth rate of normal mode perturbations and showing that its sign is negative, which means asymptotic stability of the basic flow. A concurring argument supporting the absence of linear instability arises from the investigation of cases where the impermeability condition at the vertical boundaries is relaxed and a partial permeability is modelled through Robin boundary conditions for the pressure. With partially permeable boundaries, an instability emerges which takes the form of axisymmetric normal modes.