Compressibility effect on Darcy porous convection

TL;DR

This study investigates how fluid compressibility influences the onset of Darcy-Bénard convection, revealing that increased compressibility lowers the critical Rayleigh-Darcy number and promotes convection.

Contribution

It introduces a compressibility parameter into the linear analysis of Darcy convection, providing more thermodynamically consistent instability results.

Findings

Compressibility parameter lowers the critical Rayleigh-Darcy number.

Enhanced compressibility promotes earlier onset of convection.

Analysis extends classical models to include realistic fluid compressibility effects.

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-B\'enard problem is performed in the class of extended-quasi-thermal-incompressible fluids, introducing a factor $\beta$ 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 $\widehat{\beta}$ proportional to the compressibility factor $\beta$, proving that $\widehat{\beta}$ enhances the onset of convective motions.