Weakly nonlinear analysis of Rayleigh-Benard convection in a non-Newtonian fluid between plates of finite conductivity: Influence of shear-thinning effects

Finite-amplitude thermal convection in a shear-thinning fluid layer between two horizontal plates of finite thermal conductivity is considered. Weakly nonlinear analysis is adopted as a first approach to investigate nonlinear effects. The rheological behavior of the fluid is described by the Carreau model. As a first step, the critical conditions for the onset of convection are computed as a function of the ratio xi of the thermal conductivity of the plates to the thermal conductivity of the fluid. In agreement with the literature, the critical Rayleigh number Ra-c and the critical wave number k(c) decrease from 1708 to 720 and from 3.11 to 0, when xi decreases from infinity to zero. In the second step, the critical value alpha(c) of the shear-thinning degree above which the bifurcation becomes subcritical is determined. It is shown that alpha(c) increases with decreasing xi. The stability of rolls and squares is then investigated as a function of xi and the rheological parameters. The limit value xi(c), below which squares are stable, decreases with increasing shear-thinning effects. This is related to the fact that shear-thinning effects increase the nonlinear interactions between sets of rolls that constitute the square patterns [M. Bouteraa et al., J. FluidMech. 767, 696 (2015)]. For a significant deviation from the critical conditions, nonlinear convection terms and nonlinear viscous terms become stronger, leading to a further diminution of xi(c). The dependency of the heat transfer on xi and the rheological parameters is reported. It is consistent with the maximum heat transfer principle. Finally, the flow structure and the viscosity field are represented for weakly and highly conducting plates.