The authors consider the isothermal steady motion of an incompressible fluid whose viscosity depends on the pressure and the shear rate. The system is completed by suitable boundary conditions involving non-homogeneous Dirichlet, Navier's slip and inflow/outflow parts. The authors prove the existence of weak solutions and show that the resulting level of the pressure is fixed by the boundary conditions. The paper is motivated by the journal bearing lubrication problem and extends the earlier results for homogeneous boundary conditions.