Further thoughts on buoyancy-induced instability of a variable viscosity fluid saturating a porous slab

Abstract

The stability of buoyant flow in a vertical porous layer bounded by impermeable-isothermal boundaries is studied insofar as the variability of fluid viscosity with temperature is concerned. The flow is governed by modified Darcy's law with two different forms of viscosity–temperature relationships namely, quadratic and exponential. The modal analysis is carried out with a velocity–temperature formulation of the governing equations for the perturbations. The temperature-dependence of the viscosity forbids the energy analysis of Gill [“A proof that convection in a porous vertical slab is stable,” J. Fluid Mech. 35, 545–547 (1969)] in embarking upon any definite conclusion on the stability of fluid flow even under the limit of an infinite Prandtl–Darcy number, and consequently, the stability eigenvalue problem is solved numerically. Types of temperature-dependent viscosity laws are found to demonstrate conflicting behavior on the stability characteristics of the base flow. The results show that the base flow is linearly stable if the viscosity varies with temperature quadratically. On the contrary, instability emerges for an exponential type of variation in viscosity beyond a certain range of values of the corresponding viscosity parameter depending on the Prandtl–Darcy number. It is established that an increase in the value of the viscosity parameter is to decrease the critical Darcy–Rayleigh number markedly and thereby destabilizes the fluid flow.