Thermodynamics of viscoelastic rate-type fluids with stress diffusion

We propose thermodynamically consistent models for viscoelastic fluids with a
stress diffusion term. In particular, we derive variants of
compressible/incompressible Maxwell/Oldroyd-B models with a stress diffusion
term in the evolution equation for the extra stress tensor. It is shown that
the stress diffusion term can be interpreted either as a consequence of a
nonlocal energy storage mechanism or as a consequence of a nonlocal entropy
production mechanism, while different interpretations of the stress diffusion
mechanism lead to different evolution equations for the temperature. The
benefits of the knowledge of the thermodynamical background of the derived
models are documented in the study of nonlinear stability of equilibrium rest
states. The derived models open up the possibility to study fully coupled
thermomechanical problems involving viscoelastic rate-type fluids with stress
diffusion.