Darcy–Bénard–Bingham convection

Abstract

The present paper is the first to consider Darcy-Bénard-Bingham convection. A Bingham fluid saturates a horizontal porous layer which is subjected to heating from below. It is shown that this simple extension to the classical Darcy-Bénard problem is linearly stable to small-amplitude disturbances but nevertheless admits strongly nonlinear convection. The Pascal model for a Bingham fluid occupying a porous medium is adopted, and this law is regularized in a frame-invariant manner to yield a set of two-dimensional governing equations which are then solved numerically using finite difference approximations. A weakly nonlinear theory of the regularized Pascal model is used to show that the onset of convection is via a fold bifurcation. Some parametric studies are performed to show that this nonlinear onset of convection arises at increasing values of the Darcy-Rayleigh number as the Rees-Bingham number increases, and that for a fixed Rees-Bingham number the wavenumber at which the rate of heat transfer is maximised increases with the Darcy-Rayleigh number.

Compressibility effect on Darcy porous convection

“…The mathematical models describing the onset of convective motions in horizontal layers of fluids heated from below are well known for both clear fluids and fluid-saturated porous media (see [1,2] and references therein) and have been widely analysed under various assumptions: in [3,4,5] the authors analysed the effect of the local thermal non-equilibrium hypothesis on the onset of convection in horizontal porous layers; in [6] the Darcy-Bénard problem for Bingham fluids has been studied, while in [7,8] the authors examined the onset of convection in an inclined horizontal layer of porous medium; convective instabilities in horizontal layers of bi-disperse porous media have been analysed in [9,10,11,12,13], the onset of penetrative convection has been studied in [14,15,16]. Usually, the fluid is assumed as Newtonian and incompressible.…”

Rayleigh–Bénard Instability of an Ellis Fluid Saturated Porous Channel with an Isoflux Boundary

“…The Darcy-Bénard instability for saturating fluids with a non-Newtonian rheology has been investigated with reference to viscoelastic fluids [4][5][6], to fluids with yield-stress [7] and to purely viscous fluids [8][9][10][11][12][13][14]. In particular, Barletta and Nield [8], Alves and Barletta [9], Celli et al [10] and Petrolo et al [11] discuss the onset of the Darcy-Bénard instability in a porous medium by considering the power-law model.…”