Rayleigh–Bénard convection in non-Newtonian Carreau fluids with arbitrary conducting boundaries *

The objective of the present work is to investigate the Rayleigh–Bénard convection in non-Newtonian fluids with arbitrary conducting boundaries. A linear and weakly nonlinear analysis is performed. The rheological behavior of the fluid is described by the Carreau model. As a first step, the critical Rayleigh number and wavenumber for the onset of convection are computed as a function of the ratios ξb and ξt of the thermal conductivities of the bottom and top slabs to that of the fluid. In the second step, the preferred convection pattern is determined using an amplitude equation approach. The stability of rolls and squares is investigated as a function of (ξb,ξt) and the rheological parameters. The bounded region of (ξb,ξt) space where 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 [1]. For a significant deviation from the critical conditions, the nonlinear convection terms and nonlinear viscous terms become stronger, reducing overall the stability domain of squares. The largest Nusselt number, Nu, is obtained for perfectly conducting boundaries. For a given (ξb,ξt), the stable solution yields the largest Nusselt number. The enhancement of heat transfer due to shear-thinning effects is significantly reduced for poorly heat conducting plates.