Abstract
In this paper we consider a class of incompressible viscous fluids whose viscosity depends on the shear rate and pressure. We deal with isothermal steady flow and analyse the Galerkin discretization of the corresponding equations.
We discuss the existence and uniqueness of discrete solutions and their convergence to the solution of the original problem. In particular, we derive a priori error estimates, which provide optimal rates of convergence with respect to the expected regularity of the solution.
Finally, we demonstrate the achieved results by numerical experiments. To our knowledge, this is the first analysis of the finite element method for fluids with pressure-dependent viscosity.
The derived estimates coincide with the optimal error estimates established recently for Carreau-type models, which are covered as a special case.