  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 suit-able power law
depending on $p \in (1,2)$ (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.
