On incompressible heat-conducting viscoelastic rate-type fluids with stress-diffusion and purely spherical elastic response

TL;DR

This paper proves the existence of large-data global weak solutions for a complex PDE system modeling heat-conducting, stress-diffusing viscoelastic fluids with spherical elastic response, integrating thermodynamics and novel mathematical techniques.

Contribution

It provides the first rigorous proof of global weak solutions for coupled thermo-mechanical equations in heat-conducting viscoelastic fluids with stress diffusion and spherical elastic response.

Findings

Existence of large-data global-in-time weak solutions established.

Thermodynamic foundations guide the mathematical analysis.

Novel mathematical methods developed for complex internal energy and entropy structures.

Abstract

We prove the existence of large-data global-in-time weak solutions to an evolutionary PDE system describing flows of incompressible \emph{heat-conducting} viscoelastic rate-type fluids with stress-diffusion, subject to a stick-slip boundary condition for the velocity and a homogeneous Neumann boundary condition for the extra stress tensor. In the introductory section we develop the thermodynamic foundations of the proposed model, and we document the role of thermodynamics in obtaining critical structural relations between the quantities of interest. These structural relations are then exploited in the mathematical analysis of the governing equations. In particular, the definition of weak solution is motivated by the thermodynamic basis of the model. The extra stress tensor describing the elastic response of the fluid is in our case purely spherical, which is a simplification from the physical point of view. The model nevertheless exhibits features that require novel mathematical ideas in order to deal with the technically complex structure of the associated internal energy and the more complicated forms of the corresponding entropy and energy fluxes. The paper provides the first rigorous proof of the existence of large-data global-in-time weak solutions to the governing equations for \emph{coupled thermo-mechanical processes} in viscoelastic rate-type fluids.