Some remarks on non-Newtonian fluids including nonconvex perturbations of the Bingham and Powell-Eyring model for viscoplastic fluids

We consider quasi-static hows of certain viscoplastic materials for which the velocity held u can be found as a minimizer of the functional J(v) = integral(Omega)omega(epsilon v)dx in classes of functions u : R-n superset of Omega --> R-n satisfying div u = 0 and also the appropriate boundary conditions. The density omega is characteristic for the material under consideration and epsilon v denotes the symmetric gradient of v. In case of a Bingham fluid Ne have for example omega(Ev) = eta\epsilon v\(2) + g\epsilon v\ with positive constants eta and g. We also consider various perturbations of omega which are not assumed to be convex so that Ne have to study the relaxed variational problem. Our main result states that in all cases the symmetric derivative of the velocity field is a locally bounded function.