Onset of Darcy–Bénard convection using a thermal non-equilibrium model

In this paper we use a two-field model for the separate modelling of the solid and fluid phase temperature fields in a fluid-saturated porous medium, and, in particular, we consider how the onset criterion for convection in a horizontal layer is affected by the adoption of such a model. In general we find that both the critical Rayleigh number and wavenumber are modified by the presence of thermal non-equilibrium effects. It is shown that the well-known result of Lapwood [Proc. Cambridge Philos. Soc. 44 (1948) 508] which corresponds to local thermal equilibrium (LTE), is recovered when taking the thermal equilibrium limit of the non-equilibrium analysis. We also present asymptotic solutions for both small and large values of H the inter-phase heat transfer coefficient, H, and compare this with the numerical solutions. For intermediate values of H we find that the critical wavenumber is always larger than π, the critical value for the LTE case. In some cases this critical wavenumber may be very large compared with π.