Density maximum and finite Darcy–Prandtl number outlooks on Gill's stability problem subject to a lack of thermal equilibrium

Abstract

The Gill stability problem encompasses the investigation of stability of natural convection flow in a vertical porous layer governed by Darcy's law under a local thermal equilibrium (LTE) perspective and was proved analytically by Gill [Gill, J. Fluid Mech. 35, 545–547 (1969)] that the flow is always stable. The present study deals with the simultaneous influence of the Darcy–Prandtl number and the density maximum property on Gill's stability problem subject to a lack of thermal equilibrium. The density variation with fluid temperature is assumed to be pure quadratic, and it is established that the linear stability of the basic state for a three-dimensional problem can be Squire-transformed. It is observed that Gill's proof of linear stability cannot be extended to the present model and hence we approached numerically by evaluating the growth rate of normal mode perturbations. The neutral stability curves are obtained, and the critical parameters at the onset of instability are determined. Even though the isolation presence of time-dependent velocity term and the density maximum property evidence the basic flow to be stable for all infinitesimal perturbations, their simultaneous occurrence induces instability under certain parametric conditions. The finite range of values of the scaled interphase heat transfer coefficient within which the flow is stable is found to increase with increasing Darcy–Prandtl number but vanishes with increasing porosity-modified conductivity ratio. Moreover, the basic state becomes stable when the Darcy–Prandtl number is larger than 7.08. The results of LTE model are delineated as a particular case from the present study.