Compressibility effect on Darcy porous convection

Abstract

Perfectly incompressible materials do not exist in nature but are a useful approximation of several media which can be deformed in non-isothermal processes but undergo very small volume variation. In this paper the linear analysis of the Darcy-Bénard problem is performed in the class of extended-quasi-thermal-incompressible fluids, introducing a factor β which describes the compressibility of the fluid and plays an essential role in the instability results. In particular, in the Oberbeck-Boussinesq approximation, a more realistic constitutive equation for the fluid density is employed in order to obtain more thermodynamic consistent instability results. Via linear instability analysis of the conduction solution, the critical Rayleigh-Darcy number for the onset of convection is determined as a function of a dimensionless parameter β proportional to the compressibility factor β, proving that β enhances the onset of convective motions.

On the Use and Misuse of the Oberbeck–Boussinesq Approximation

“…Such discussions on the origin and on the range of applicability of the Oberbeck-Boussinesq approximation stem from pioneering papers such as those by Spiegel and Veronis [4], by Gray and Giorgini [5] and by Hills and Roberts [6]. On the other hand, further papers offer interesting perspectives on possible extended forms of the Oberbeck-Boussinesq approximation [7][8][9][10][11][12].…”