Rayleigh–Bénard flow for a Carreau fluid in a parallelepiped cavity

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A continuation method is used to study Rayleigh–Bénard 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 B₁ and a primary branch of longitudinal roll B₂ , stabilized beyond a secondary bifurcation point S₂ . 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 α . 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 S₂ is strongly moved towards larger Ra , making it more difficult to reach the longitudinal roll solution. Energy analyses at the bifurcations SN₁ , SN₂ and S₂ show that the changes of the corresponding critical thresholds Rac are connected with the changes of the viscous properties, but also with changes of the buoyancy effect.