A linear and weakly nonlinear analysis of convection in a layer of shear-thinning fluids between two horizontal plates heated from below is performed. The objective is to examine the effects of the nonlinear variation of the viscosity with the shear rate on the nature of the bifurcation, the planform selection problem between rolls, squares and hexagons, and the consequences on the heat transfer coefficient. Navier’s slip boundary conditions are used at the top and bottom walls. The shear-thinning behaviour of the fluid is described by the Carreau model. By considering an infinitesimal perturbation, the critical conditions, corresponding to the onset of convection, are determined. At this stage, non-Newtonian effects do not come into play. The critical Rayleigh number decreases and the critical wavenumber increases when the slip increases. For a finite-amplitude perturbation, nonlinear effects enter in the dynamic. Analysis of the saturation coefficients at cubic order in the amplitude equations shows that the nature of the bifurcation depends on the rheological properties, i.e. the fluid characteristic time and shear-thinning index. For weakly shear-thinning fluids, the bifurcation is supercritical and the heat transfer coefficient increases, as compared with the Newtonian case. When the shear-thinning character is large enough, the bifurcation is subcritical, pointing out the destabilizing effect of the nonlinearities arising from the rheological law. Departing from the onset, the weakly nonlinear analysis is carried out up to fifth order in the amplitude expansion. The flow structure, the modification of the viscosity field and the Nusselt number are characterized. The competition between rolls, squares and hexagons is investigated. Unlike Albaalbaki & Khayat (J. Fluid. Mech., vol. 668, 2011, pp. 500–550), it is shown that in the supercritical regime, only rolls are stable near onset.

Abstract

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 ξ 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 ξ decreases from infinity to zero. In the second step, the critical value α(c) of the shear-thinning degree above which the bifurcation becomes subcritical is determined. It is shown that α(c) increases with decreasing ξ. The stability of rolls and squares is then investigated as a function of ξ and the rheological parameters. The limit value ξ(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. Fluid Mech. 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 ξ(c). The dependency of the heat transfer on ξ 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.

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

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