Coupled problems on stationary flow of electrorheological fluids under the conditions of nonhomogeneous temperature distribution

TL;DR

This paper investigates the stationary flow of electrorheological fluids with nonhomogeneous temperature distribution, establishing existence results for generalized solutions under certain conditions.

Contribution

It introduces the concepts of $P$-generalized and generalized solutions for the coupled thermal and flow problem, proving their existence under weak and smooth boundary conditions.

Findings

Existence of $P$-generalized solutions under weak conditions.

Existence of generalized solutions with smooth boundary and small data.

Development of a nonlocal model based on regularized velocity fields.

Abstract

We set up and study a coupled problem on stationary non-isothermal flow of electrorheological fluids. The problem consist in finding functions of velocity, pressure and temperature which satisfy the motion equations, the condition of incompressibility, the equation of the balance of thermal energy and boundary conditions. We introduce the notions of a $P$-generalized solution and generalized solution of the coupled problem. In case of the $P$-generalized solution the dissipation of energy is defined by the regularized velocity field, which leads to a nonlocal model. Under weak conditions, we prove the existence of the $P$ -generalized solution of the coupled problem. The existence of the generalized solution is proved under the conditions on smoothness of the boundary and on smallness of the data of the problem