Existence of steady solutions for a model for micropolar electrorheological fluid flows with not globally $\log$--H\"older continuous shear exponent

TL;DR

This paper proves the existence of steady solutions for a micropolar electrorheological fluid model with a shear exponent that is not globally log-Hölder continuous, using weighted variable exponent Sobolev spaces.

Contribution

It establishes the existence of solutions under less restrictive conditions on the shear exponent and electric field, expanding the mathematical understanding of such complex fluids.

Findings

Existence of weak solutions for the fluid model.

Solutions exist even when the shear exponent is not globally log-Hölder continuous.

Application of weighted variable exponent Sobolev spaces.

Abstract

In this paper, we study the existence of weak solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted variable exponent Sobolev spaces. We establish the existence of solutions for a material function $\hat p$ that is $\log$--H\"older continuous and an electric field $\mathbf{E}$ for that $\vert\mathbf{E}\vert^2$ is bounded and smooth. Note that these conditions do not imply that the variable shear exponent $p=\hat p\circ\vert \mathbf{E}\vert^2$ is globally $\log$--H\"older continuous.