On three-dimensional flows of thermo-viscoelastic fluids of Giesekus type

TL;DR

This paper introduces a thermodynamically consistent model for heat-conducting thermo-viscoelastic fluids of Giesekus type and proves the existence of global weak solutions in three dimensions without restrictive assumptions.

Contribution

It develops a comprehensive 3D model for heat-dependent viscoelastic fluids and establishes existence results without smallness or regularity restrictions on initial data.

Findings

Proved existence of global weak solutions in 3D for the model.

No need for artificial stress diffusion or small initial data.

Successfully handled full nonlinear coupling of velocity, temperature, and stress.

Abstract

Viscoelastic rate-type fluid models constitute a fundamental framework for the mathematical description of complex materials exhibiting coupled elastic and viscous effects, with a wide range of applications in engineering, biomaterials, and medicine. In realistic regimes, thermal effects are essential and lead to strongly coupled systems in which heat conduction and temperature-dependent constitutive laws play a decisive role. In this paper, we develop a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids. We establish the existence of a global weak solution in the full three-dimensional setting. In contrast to the existing literature, no smallness, regularity, or structural restrictions on the initial data are imposed beyond natural energy and entropy bounds, and no additional regularising mechanisms such as artificial stress diffusion are required. The analysis is based on a weak-strong framework combining energy and entropy balances with compactness tools, allowing us to treat the full nonlinear coupling between the fluid velocity, the temperature, and the elastic stress.