Non-isothermal non-Newtonian fluids: The stationary case

The stationary Navier-Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on p ? (1,2) (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier-Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.