Darcy–Carreau Model and Nonlinear Natural Convection for Pseudoplastic and Dilatant Fluids in Porous Media

The linear and weakly nonlinear stability analyses are carried out to study instabilities in Darcy–Bénard convection for non-Newtonian inelastic fluids. The rheological model considered here is the Darcy–Carreau model, which is an extension to porous media of Carreau rheological model usually used in clear fluid media. The linear stability approach showed that the critical Rayleigh number and wave number corresponding to the onset of convection are the same as for Newtonian fluids. By employing weakly nonlinear theory, we derived a cubic Landau equation that describes the temporal evolution of the amplitude of convection rolls in the unstable regime. It is found that the bifurcation from the conduction state to convection rolls is always supercritical for dilatant fluids. For pseudoplastic fluids, however, the interplay between the macroscale properties of the porous media and the rheological characteristics of the fluid determines the supercritical or subcritical nature of the bifurcation. In the parameter range where the bifurcation is supercritical, we determined and discussed the combined effects of the fluid properties and the porous medium characteristics on the amplitude of convection rolls and the corresponding average heat transfer for both pseudoplastic and dilatant fluids. Remarkably, we found that the curves describing these effects collapse onto the universal curve for Newtonian fluids, provided the average apparent viscosity is used to define Rayleigh number.