Frictional Contact Problems for Steady Flow of Incompressible Fluids in Orlicz Spaces

The chapter is devoted to the study of steady-state flow problems of isotropic, isothermal, inhomogeneous, viscous, and incompressible fluids in a bounded domain with subdifferential boundary conditions in Orlicz spaces. Two general cases are investigated. First, we study the non-Newtonian fluid flow with a non-polynomial growth of the extra (viscous) part of the Cauchy stress tensor together with multivalued nonmonotone slip boundary conditions of frictional type described by the Clarke generalized gradient. Second, we analyze the Newtonian fluid flow with a multivalued nonmonotone leak boundary condition of frictional type which is governed by the Clarke generalized gradient with a non-polynomial growth between the normal velocity and normal stress. In both cases, we provide abstract results on existence and uniqueness of solution to subdifferential operator inclusions with the Clarke generalized gradient and the Navier–Stokes type operator which are associated with hemivariational inequalities in the reflexive Orlicz–Sobolev spaces. Moreover, our study, in both aforementioned cases, is supplemented by similar results for the Stokes flows where the convective term is negligible. Finally, the results are applied to examine hemivariational inequalities arising in the study of the flow phenomenon with frictional boundary conditions. The chapter is concluded with a continuous dependence result and its application to an optimal control problem for flows of Newtonian fluids under leak boundary condition of frictional type.