Non-isothermal non-Newtonian fluids: the stationary case

TL;DR

This paper investigates stationary non-Newtonian fluid flow coupled with heat transfer, establishing existence and regularity of solutions, and validating finite element approximations through numerical experiments.

Contribution

It provides new existence, regularity results, and error estimates for stationary non-isothermal non-Newtonian fluids with temperature-dependent viscosity.

Findings

Existence of weak solutions for the coupled system.

Regularity results for Navier-Stokes and Stokes cases.

Finite element error estimates validated by numerical experiments.

Abstract

The stationary Navier-Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suit-able power law depending on $p \in (1,2)$ (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier-Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.