Rayleigh-Benard flow for a Carreau fluid in a parallelepiped cavity

A continuation method is used to study Rayleigh-Benard convection in a non-Newtonian fluid inside a parallelepiped cavity. The cavity has its length equal to twice the side of the square cross-section. Shear-thinning and shear-thickening Carreau fluids are considered. The focus is put on the two stable branches which exist in the Newtonian case, a stable primary branch of transverse rolls B-1 and a primary branch of longitudinal roll B-2, stabilized beyond a secondary bifurcation point S-2. Although the primary bifurcation points are unchanged, the non-Newtonian properties strongly modify the bifurcation diagram. Indeed, for a shear-thinning fluid, the stable solutions can exist at much smaller Rayleigh numbers Ra, on subcritical branches beyond saddle-node points SN1 and SN2, and small perturbations can be sufficient to reach them. In agreement with Bouteraa et al. (J. Fluid Mech., vol. 767, 2015, pp. 696-734), the change of the primary bifurcations from supercritical to subcritical occurs at given values of what they define as the degree of shear-thinning parameter a. Moreover, the value of the Rayleigh number at the saddle-node points can be approximated by a simple expression, as proposed by Jenny et al. (J. Non-Newtonian Fluid Mech., vol. 219, 2015, pp. 19-34). In the case of a shear-thickening fluid, the branches remain supercritical, but the secondary point S-2 is strongly moved towards larger Ra, making it more difficult to reach the longitudinal roll solution. Energy analyses at the bifurcations SN1, SN2 and S-2 show that the changes of the corresponding critical thresholds Ra-c are connected with the changes of the viscous properties, but also with changes of the buoyancy effect.