On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate

We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. The existence of weak solutions for steady flows of such fluids subject to homogeneous Dirichlet boundary conditions is established in Franta, Málek, Rajagopal M. Franta, J. Málek, K.R. Rajagopal, On steady flows of fluids with pressure- and shear- dependent viscosities, Proc. Roy. Soc. A Math. Phys. Eng. Sci. 461 (2055) (2005) 651–670]. In this paper we treat non-homogeneous Dirichlet boundary conditions, assuming either that the normal part of velocity on the boundary is equal to zero or that the boundary data are small. We also relax the requirement concerning how to fix the pressure. Such a model has relevance to some important engineering applications.