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.

In this paper, a weakly nonlinear stability of viscoplastic fluid flow is performed. The system consists of a plane Rayleigh–Bénard–Poiseuille (RBP) flow of a Bingham fluid. The basic flow is characterized by a central plug zone, of 2y_b width, in which the stresses are smaller than the Bingham number B, the dimensionless yield stress. The Bingham model assumes that inside this zone the material moves as a rigid solid, and that outside this zone it behaves as a viscous fluid. The aim of this study is to investigate the influence of the yield stress on the instability conditions. The linear stability analysis is performed using a modal method and provides critical values of Rayleigh and wavenumbers, from which the system becomes unstable. The critical mode, i.e. the least stable mode, is also determined. This mode, also called the fundamental mode, creates perturbation harmonics which cannot be neglected above criticality. The weakly nonlinear analysis is performed for small-amplitude perturbations. In this study, the quadratic modes of the perturbation are determined. Results indicate that the nonlinear modes perturbation can attain high maximal values, which is the consequence of the high variations of viscosity in the flow. The characterization of the complex Landau equation sheds light on a transition in terms of the bifurcation nature above a critical Péclet number Pe_c = O(1). Below Pe_c, it is found that a supercritical equilibrium state could exist, such as in the Newtonian case, while above Pe_c, the bifurcation becomes subcritical. One observes a sharp transition from supercritical to subcritical bifurcation as the Péclet value is increased. A dependence of Pe_c on the yield stress is highlighted since the subcritical bifurcation is first observed for weak values of y_b (y_b < O(10⁻¹)). For this range of values, the transition is mainly due to the presence of the unyielded region via non-homogeneous boundary conditions at the yield surfaces. Then for y_b > O(10⁻¹), the change of the bifurcation nature is due to the variations of the effective viscosity in the unyielded regions.

This paper deals with the Taylor-Couette flow of shear-thinning fluids. It focuses on the first principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity μ with the shear rate γ̇ on the flow structure in the Taylor-vortex flow regime. A wide gap configuration (η=0.4) is mainly considered. A weakly nonlinear analysis, using the amplitude expansion method at high order, is adopted as a first approach to study nonlinear effects. For the numerical computation, the shear-thinning behavior is described by the Carreau model. The rheological parameters are varied in a wide range. The results indicate that the flow field undergoes a significant change as shear-thinning effects increase. First, vortices are squeezed against the inner wall and the center of the patterns is shifted axially toward the radial outflow boundaries (z=0,z/λz=1). This axial shift leads to increasing concentration of vorticity at these positions. The outflow becomes stronger than the inflow and the extent of the inflow zone where the vorticity is low increases acoordingly. Nevertheless, the strength of the vortices relative to the velocity of the inner cylinder is weaker. Second, the pseudo-Nusselt number, ratio of the torque to that obtained in the laminar flow, decreases. Third, higher harmonics become more relevant and grow faster with Reynolds number. Finally, the modification of the viscosity field is described.

We present an experimental and theoretical study of Rayleigh–Bénard convection in shear-thinning fluids with temperature-dependent properties. Experiments were performed using a cylindrical cell with a radius $\widehat{R}=60\hspace{0.17em}\hspace{0.17em}\text{mm}$ and height adjustable at $\widehat{d}=15$ and $20\hspace{0.17em}\hspace{0.17em}\text{mm}$ giving a radius-to-height ratio L = 4 and 3, respectively. The fluids used are glycerol (Newtonian fluid) and aqueous xanthan gum solutions (shear-thinning fluids) at 1000 and 1200 ppm. Convection patterns are visualized by the shadowgraph method. In the theoretical part of this study, the weakly nonlinear analysis performed by Varé et al. [J. Fluid Mech. 905, A33 (2020)] is extended to take into account the variation of the thermal expansion coefficient with temperature. For the xanthan gum solutions used, the temperature dependence of the fluid parameters is sufficiently strong to obtain hexagonal cells at the onset of convection. It has been observed that their size decreases with the increase in the temperature difference across the fluid layer above the critical value. This result provides an experimental support to our theoretical study where it is shown that for hexagons, the band of stable wavenumbers is bent toward higher wavenumbers. For the glycerol, Newtonian fluid with a large Prandtl number, a slight increase in the wavelength of rolls is observed in agreement with the literature.