Steady Flow for Shear Thickening Fluids with Arbitrary Fluxes

TL;DR

This paper establishes existence and uniqueness results for steady shear thickening non-Newtonian fluid flows with arbitrary fluxes in unbounded domains, extending classical results from Newtonian to shear thickening fluids.

Contribution

It extends the analysis of steady Navier-Stokes solutions to shear thickening fluids with arbitrary fluxes, including new existence and uniqueness results.

Findings

Existence of solutions under small energy dispersion conditions

Uniqueness of solutions in unbounded domains

Extension of classical Newtonian results to shear thickening fluids

Abstract

We solve the stationary Navier-Stokes equations for non-Newtonian incompressible fluids with shear dependent viscosty in domains with unbounded outlets, in the case of shear thickening viscosity, i.e. the viscosity is given by the shear rate 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 (J. Soviet Math., 21 (1983) 728-761) for Newtonian fluids (p=2).