Unsteady Flows of Fluids with Pressure Dependent Viscosity in Unbounded Domains

TL;DR

This paper investigates the mathematical properties of unsteady, pressure-dependent viscosity fluid flows in unbounded domains, establishing the existence of weak solutions and advancing understanding of such complex fluid models.

Contribution

It provides the first analysis of flows of pressure-dependent viscosity fluids in unbounded domains, improving upon previous results even for shear rate dependent fluids.

Findings

Established large data existence of weak solutions

Analyzed three-dimensional flows with pressure and shear rate dependent viscosity

Improved upon earlier results for power-law type shear rate dependent fluids

Abstract

In order to describe behavior of various liquid-like materials at high pressures, incompressible fluid models with pressure dependent viscosity seem to be a suitable choice. In the context of implicit constitutive relations involving the Cauchy stress and the velocity gradient these models are consistent with standard procedures of continuum mechanics. Understanding mathematical properties of governing equations is connected with various types of idealization, some of them lead to studies in unbounded domains. In this paper, we first bring up several characteristic features concerning fluids with pressure dependent viscosity. Then we study three-dimensional flows of a class of fluids with the viscosity depending on the pressure and the shear rate. By means of higher differentiability methods we establish large data existence of a weak solution for the Cauchy problem. This seems to be a first result that analyzes flows of considered fluids in unbounded domains. Even in the context of purely shear rate dependent fluids of a power-law type the result presented here improves some of earlier works.