| The thermo-gravitational instability in a fluid layer of a non-Newtonian medium heated from below is investigated. The non-Newtonian fluids under study are the generalized Newtonian (shear-thinning and –thickening) fluids with and without elasticity. The fluid is assumed to obey the Carreau-Bird model of viscosity for the case of inelastic shear-thinning/thickening fluids, and for the case of elastic shear-thinning fluids the Oldroyd-B constitutive equation is applied. For the dependence of the viscosity on the rate of strain, the Carreau-Bird model of viscosity is applied on the polymeric solute contribution to the solution viscosity. This model is traditionally called the Phan-Thien-Tanner-B constitutive equation. Linear and weakly nonlinear analyses are successively presented.;Next, the weakly nonlinear method of amplitude equations is implemented to investigate the convective flow patterns of inelastic shear-thinning and –thickening fluids in a slightly supercritical range. Although the critical threshold is the same as for a Newtonian fluid, it is found that non-Newtonian fluids can convect in the form of rolls, squares or hexagons depending on the shear-thinning level. Similarly to Newtonian fluids, shear-thickening fluids convect only in the form of rolls. The stability of the convective steady branches is carried out to determine under which specific conditions a pattern is preferred. The influence of the rheological and physical parameters is examined and discussed in detail. Next, the weakly nonlinear method of amplitude equations is implemented to investigate the convective flow patterns of shear-thinning and – thickening fluids in a slightly supercritical range.;Finally, the low order dynamical system approach (the modified Lorenz model) is implemented to investigate the Rayleigh-Bénard convection of the elastic shear-thinning fluids. It is found that interaction between elasticity and shear-thinning limits the domain of stability of rolls further compared to the inelastic shear-thinning fluids or viscoelatsic fluids with constant viscosity.;Keywords: Rayleigh-Bénard convection, generalized Newtonian fluids, elastic shear-thinning fluids, linear stability analysis, amplitude equations approach, the Lorenz model.;First, the conventional nonlinear approaches for the Rayleigh-Bénard convection are reviewed. Two of the famous nonlinear approaches for the slightly post-critical range, namely, the Lorenz model and the method of amplitude equations are emphasized here. Similarities and differences of the results of these approaches on steady two-dimensional flow and its stability, and the transient behavior are investigated. Moreover, the role of the non-dimensional groups, the Rayleigh number and the Prandtl number are scrutinized. |
