Non-Newtonian fluids with discontinuous-in-time stress tensor

TL;DR

This paper proves the existence of weak solutions for incompressible non-Newtonian fluids with a discontinuous-in-time stress tensor, modeling materials with instantaneously changing properties such as electric field effects.

Contribution

It introduces a framework for analyzing non-Newtonian fluids with time-discontinuous stress tensors, extending existence results under minimal regularity assumptions.

Findings

Existence of weak solutions for large data

Long-time behavior established

Applicable to materials with instant property changes

Abstract

We consider the system of equations describing the flow of incompressible fluids in bounded domain. In the considered setting, the Cauchy stress tensor is a monotone mapping and has asymptotically $(s-1)$-growth with the parameter $s$ depending on the spatial and time variable. We do not assume any smoothness of $s$ with respect to time variable and assume the log-H\"{o}lder continuity with respect to spatial variable. Such a setting is a natural choice if the material properties are instantaneously, e.g. by the switched electric field. We establish the long time and the large data existence of weak solution provided that $s\ge(3d+2)(d+2)$.