Micropolar meets Newtonian. The Rayleigh–Bénard problem

We consider the Rayleigh–Bénard problem for the two-dimensional Boussinesq system for the micropolar fluid. Our main goal is to compare the value of the critical Rayleigh number, and estimates of the Nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical Navier–Stokes system. Our estimates reveal the stabilizing effects of micropolarity in comparison with the homogeneous Navier–Stokes fluid. In particular, the critical Rayleigh number for the micropolar model is larger then that for the Navier–Stokes one, and large micropolar viscosity may even prevent the averaged heat transport in the upward vertical direction. The estimate of the fractal dimension of the global attractor for the considered problem is better then that for the same problem for the Navier–Stokes system. Besides, we obtain also other results which relate the heat convection in the micropolar fluids to the Newtonian ones, among them, relations between the Nusselt number and the energy dissipation rate and upper semicontinuous convergence of global attractors to the Rayleigh–Bénard attractor for the considered model.