Some Remarks on Non-Newtonian Fluids Including Nonconvex Perturbations of the Bingham and Powell–Eyring Model for Viscoplastic Fluids

Abstract

We consider quasi-static flows of certain viscoplastic materials for which the velocity field v can be found as a minimizer of the functional [Formula: see text] in classes of functions u : ℝn ⊃ Ω → ℝn satisfying div u = 0 and also the appropriate boundary conditions. The density ω is characteristic for the material under consideration and ℰv denotes the symmetric gradient of v. In case of a Bingham fluid we have for example ω(ℰv) = η|ℰv|2 + g|ℰv| with positive constants η and g. We also consider various perturbations of ω which are not assumed to be convex so that we 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.