Darcy–Carreau–Yasuda rheological model and onset of inelastic non-Newtonian mixed convection in porous media

An extension of Carreau and Carreau–Yasuda rheological models to porous media is proposed to study the onset of mixed convection of both pseudoplastic fluids (PF) and dilatant fluids (DF) in a porous layer heated from below in the presence of a horizontal throughflow. In comparison with Newtonian fluids, three more dimensionless parameters are introduced, namely, the Darcy–Weissenberg number Wi, the power–law index n, and the Yasuda parameter a. Temporal stability analysis of the basic state showed that in the absence of a throughflow (Wi = 0), the critical Rayleigh number and the critical wavenumber at the onset of convection are the same as for Newtonian fluids, namely, R a c = 4 π 2 and k c = π, respectively. When the throughflow is added (Wi > 0), it is found that moving transverse rolls (stationary longitudinal rolls) are the dominant mode of the instability for PF (for DF). Furthermore, depending on Wi, two regimes of instability were identified. In the weakly non-Newtonian regime (i.e., W i < W i t ≈ 1), a destabilizing effect is observed for PF, while the reverse occurs for DF. These effects are more intense by reducing (increasing) the index n for PF (for DF). In this regime, a significant qualitative difference is found between the Darcy–Carreau model and the power–law model. However, in the strongly non-Newtonian regime, the two models lead to similar results. A mechanical energy budget analysis is performed to understand the physical effects of the interaction between the basic throughflow and the disturbances. It is also shown that the intrinsic macroscale properties of the porous medium may play a key role in the stabilizing/destabilizing effect. Finally, a comparison is made between the present theoretical predictions and recent mixed convection experiments in a Hele–Shaw cell.