On steady flows of an incompressible fluids with implicit power-law-like rheology∗

We consider steady flows of incompressible fluids with power-law-like rheology given by an implicit constitutive equation relating the Cauchy stress and the symmetric part of the velocity gradient in such a way that it leads to a maximal monotone (possibly multivalued) graph. Such a framework includes Bingham fluids, Herschel-Bulkley fluids, and shear-rate dependent fluids with discontinuous viscosities as special cases. We assume that the fluid adheres to the boundary. Using tools such as the Young measures, properties of spatially dependent maximal monotone operators and Lipschitz approximations of Sobolev functions, we are able to extend the results concerning large data existence of weak solutions to those values of the power-law index that are of importance from the point of view of engineering and physical applications. 1 Problem formulation We consider the following problem associated to a fixed, yet arbitrary parameter q ∈ (1,∞) that is connected to its dual exponent q through the relation q = M. Buĺıček’s research is supported by the Jindřich Nečas Center for Mathematical Modeling, the project LC06052 financed by MŠMT. P. Gwiazda and A. Swierczewska-Gwiazda were supported by the Grant of Ministry of Science and Higher Education, Nr N201 033 32/2269. J. Málek’s contribution is a part of the research project MSM 0021620839 financed by MŠMT; the support of GAČR 201/06/0352 is also acknowledged.