Compressibility Effect on Darcy Porous Convection
Giuseppe Arnone, Florinda Capone, Roberta De Luca, Giuliana Massa
Abstract
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 variations. 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 $$\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> 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 thermodynamically consistent instability results. The critical Rayleigh-Darcy number for the onset of convection is determined, via linear instability analysis of the conduction solution, as a function of a dimensionless parameter $$\widehat{\beta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>β</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> proportional to the compressibility factor $$\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> , proving that $$\widehat{\beta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>β</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> enhances the onset of convective motions. Article Highlights The onset of convection in fluid-saturated porous media is analyzed, taking into account fluid compressibility effect. The critical Rayleigh-Darcy number is determined in a closed algebraic form via linear instability analysis. The critical Rayleigh-Darcy number is shown to be a decreasing function of the dimensionless compressibility factor.