  Motivated by extrusion problems, we consider a non-stationary incompress-ible
3D fluid flow with a non-constant (temperature dependent) viscosity, subjected
to mixed boundary conditions with a given time dependent velocity on a part of
the boundary and Tresca's friction law on the other part. We construct a
sequence of approximate solutions by using a regularization of the free
boundary condition due to friction combined with a particular penalty method,
reminiscent of the " incompressibility limit " of compressible fluids, allowing
to get better insights into the links between the fluid velocity and pressure
fields. Then we pass to the limit with compactness arguments to obtain a
solution to our original problem.
