Steady flow for shear thickening fluids with arbitrary fluxes

Abstract

We solve the stationary Navier–Stokes equations for non-Newtonian incompressible fluids with shear dependent viscosity in domains with unbounded outlets, in the case of shear thickening viscosity, i.e. the viscosity is given by the shear rate raised to the power p − 2 where p > 2 . The flux assumes arbitrary given values and the Dirichlet integral of the velocity field grows at most linearly in the outlets of the domain. Under some smallness conditions on the “energy dispersion” we also show that the solution of this problem is unique. Our results are an extension of those obtained by O.A. Ladyzhenskaya and V.A. Solonnikov [O.A. Ladyzhenskaya, V.A. Solonnikov, Determination of the solutions of boundary value problems for steady-state Stokes and Navier–Stokes equations in domains having an unbounded Dirichlet integral, J. Soviet Math. 21 (1983) 728–761] for Newtonian fluids ( p = 2 ).