The effect of temperature dependence of the viscosity on stationary convective flows in Hele–Shaw cell

The results of direct numerical simulation of stationary convective flows in a vertical Hele–Shaw cell under the uniform heating from below are presented in this paper. The calculations have been fulfilled for realistic values of the heat-transfer coefficient on vertical wide boundaries and model thermal conditions on narrow vertical walls. The approximation of plane trajectories has been applied to calculate the flows in the Hele–Shaw cell. The linear stability analysis is executed for the situation when the viscosity depends on the temperature. An analytical formula for critical values of Rayleigh number has been deduced which determine the threshold of convection in dependence on parameters of the problem. It has been shown that the numerical simulation imitating the full-scale experiment gives adequate description of the transition from one-vortex stationary flow to the two-vortex steady regime when the dependence of viscosity on the temperature is taken into account in mathematical model. The equations system of thermal convection in Boussinesq approximation was solved by the method of finite differences at the “PGU-Tesla” supercomputer of the Research Academic Center “Parallel and Distributed Calculations” in Perm State University. The fields of stream function in vertical section have been calculated which confirm the effect of the vortices centers displacement to the bottom of the cavity.