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.
The aim of this survey is to give a concise but technical and, as much as possible, comprehensive introduction to the resolution of certain eigenvalue problems occurring in the research field of hydrodynamics via the Chebyshev-$$\tau$$
τ
method. While many details on the construction of mathematical models (for which we will refer to notable and well-known references as reported by Chandrasekhar (Hydrodynamic and hydromagnetic stability, Dover, London, 1981); Straughan (The energy method, stability, and nonlinear convection, Springer, New York, 2004); Nield and Bejan (Convection in porous media, Springer, New York, 2017)) will not be given, much attention will be paid to the practical and theoretical aspects of the discretization of the continuum problem. Chebyshev polynomials will be employed to expand solutions of the differential eigenvalue problem and end up with a discrete eigenvalue problem. Finally, MATLAB codes for the considered problems are shown in detail and available on GitHub.
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 $$\beta$$
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 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 $$\widehat{\beta }$$
β
^
proportional to the compressibility factor $$\beta$$
β
, proving that $$\widehat{\beta }$$
β
^
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.
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