Coupling the Navier-Stokes-Fourier equations with the Johnson-Segalman stress-diffusive viscoelastic model: Global-in-time and large-data analysis

TL;DR

This paper establishes the existence of global-in-time weak solutions for a complex thermomechanical fluid model coupling Navier-Stokes-Fourier equations with a viscoelastic stress-diffusive model, under large data and strengthened dissipation assumptions.

Contribution

It introduces a new analytical framework for proving global solutions to a coupled thermoviscoelastic system with temperature-dependent coefficients and energy considerations.

Findings

Proved existence of weak solutions for large initial data.

Developed a thermodynamically consistent derivation of the model.

Handled temperature-dependent coefficients and energy inequalities.

Abstract

We prove that there exists a~large-data and global-in-time weak solution to a~system of partial differential equations describing an unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up a~mechanically and thermally isolated container of any dimension. To overcome the~principle difficulties connected with ill-posedness of the~diffusive Oldroyd-B model in three dimensions, we assume that the~fluid admits a~strengthened dissipation mechanism, at least for excessive elastic deformations. All the~relevant material coefficients are allowed to depend continuously on the~temperature, whose evolution is captured by a~thermodynamically consistent equation. In fact, the~studied model is derived from scratch using only the~balance equations for linear momentum and energy, the~formulation of the~second law of thermodynamics and the~constitutive equation for the~internal energy. The~latter is assumed to be a~linear function of temperature, which simplifies the~model. The~concept of our weak solution incorporates both the~temperature and entropy inequalities, and also the~local balance of total energy provided that the~pressure function exists.