Thermodynamics of viscoelastic rate-type fluids with stress diffusion

Abstract

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.

On diffusive variants of some classical viscoelastic rate-type models

“…In this work the authors have analysed the stability of the equilibrium steady state of a fluid described by a diffusive variant of Oldroyd-B model. The successful identification of the energy storage and entropy producing mechanisms for other diffusive viscoelastic-type models, see above, opens up the possibility to redo the stability analysis by Málek et al [31] for these models as well.…”

“…(Note that in the case of more complicated ansatz for the Helmholtz free energy one can get non-constant specific heat both with respect to the temperature and B 2,GS , see for example Hron et al [33] or Málek et al [31].) Consequently, using the chain rule we see that Eq.…”

“…In their analysis Málek et al [31] have explored the possibility to interpret the stress diffusion effect in both ways. Namely, the stress diffusion term has been identified either as a symptom of a nonstandard energy storage mechanism or a nonstandard entropy production mechanism.…”

“…Namely, the stress diffusion term has been identified either as a symptom of a nonstandard energy storage mechanism or a nonstandard entropy production mechanism. In both cases one obtains a stress diffusion term in the evolution equation for the extra stress tensor, but the final models differ in the structure of the full Cauchy stress tensor and the specific heat capacity, see Málek et al [31] for details.…”

“…Consequently it is reasonable to interpret the stress diffusion as a dissipative (entropy producing) process. Concerning the characterisation of the entropy production, we in principle use the same ansatz as that used by Málek et al [31]. In particular the corresponding term in the entropy production ansatz is quadratic in the gradient of the "extra stress" field.…”