On Gill’s stability problem for non-Newtonian Darcy’s flow

Abstract

Gill’s stability problem is the analysis of the parallel buoyant flow in a vertical porous channel whose parallel walls are kept at different uniform temperatures. Gill’s classical paper [Journal of Fluid Mechanics, 35 (1969) 545–547] provides a rigorous proof that this flow is linearly stable. The aim of our study is to extend Gill’s analysis to the class of non-Newtonian viscous fluids modelled by Ostwald-de Waele power law. The main difference between Newtonian fluids and general power-law fluids is that the basic velocity profile is linear, in the Newtonian case, and nonlinear with an inflexion point at the mid-plane, in the non-Newtonian case. Despite the presence of the inflexion point, this study evidences a stable behaviour of the basic flow versus general normal mode perturbations: longitudinal, oblique and transverse rolls. Stability to longitudinal rolls is proved analytically, while the behaviour of transverse and oblique rolls is investigated numerically. The damping rates of perturbations, evaluated for oblique and transverse rolls, display increasing values as the Darcy–Rayleigh number increases. Numerical data thus suggest that linear stability holds for the whole class of power-law fluids.