Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids

• We study 2D Rayleigh–Bénard thermoconvection of strongly shear-thinning fluids.
• The transition from static state to thermoconvection flow is typically subcritical.
• A simple expression to calculate the subcritical Rayleigh number is proposed.
• A correlation law for the Nusselt number is proposed.
• A relevant viscosity is the bulk average of the viscosity.

The Rayleigh–Bénard thermoconvection of Newtonian fluids has been extensively studied. The transition from the conductive, static state to thermoconvection flows corresponds in this case to a supercritical bifurcation. In shear-thinning fluids, on the contrary, recent weakly nonlinear studies have shown that the transition may become subcritical. Using a custom numerical code developed with Freefem++ to compute bidimensional, fully nonlinear roll solutions in Carreau fluids, for a large range of rheological parameters, and more particularly for strongly shear-thinning fluids, approaching power-law fluids, we confirm this result. A simple expression of the value of the Rayleigh number at which subcritical convection rolls appear is proposed. This law suggests to reconsider the choice of the reference viscosity for shear-thinning fluids. Indeed, when the shear-thinning effects increase, the critical Rayleigh number increases or decreases depending on the choice of the reference viscosity. The ‘neutral’ or ‘effective’ viscosity, which gives a constant value of the Rayleigh number at the onset of subcritical convection rolls, is close to the bulk average viscosity. In addition, a correlation is proposed to estimate the Nusselt number of subcritical rolls.