Compressibility effect on Darcy porous convection*

Hydrodynamic stability problems have been widely analyzed and in this respect there are several notable results regarding Newtonian and incompressible fluids. It is commonly acknowledged that variations in temperature during non-isothermal processes induce changes in the properties of the fluid, such as density. Analyzing the complete effects of the density variations is so intricate that the use of certain approximations becomes indispensable. In this regard,  the majority of studies examining the stability of fundamental steady-state motions in both clear fluids and fluid-saturated porous media, employ a well-established hypothesis known as the Boussinesq approximation, which assumes density to be a constant function in all terms of the equations except in the body force term due to gravity, where the density ρ depends on temperature T, but not on the pressure p. However, this assumption is an approximation of the real phenomenon, since perfectly incompressible fluids do not exist in nature. This is the reason why the investigation of compressibility effects in hydrodynamic stability problems is worthy of consideration. In particular, a more realistic constitutive equation for the fluid density is the following ρ(p,T)=ρ₀[1-α(T-T₀)+β(p-p₀)), where ρ₀ is the reference density in correspondence of a temperature T₀ and pressure p₀, α is the thermal expansion coefficient and β is the compressibility factor. This fluids are called slightly compressible.

The main goal of the present talk is to describe the compressibility effect on the onset of convection in porous media, via instability analysis. We address the qualitative analysis of the solutions of the initial boundary value problem modelling slightly compressible convective currents in a Darcy's porous medium. The critical Rayleigh-Darcy number for the onset of convection is determined as a function of a dimensionless parameter  (which is proportional to the compressibility factor) proving that it enhances the onset of convective motions.