PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion

Abstract

We establish the long-time existence of large-data weak solutions to a system of nonlinear partial differential equations. The system of interest governs the motion of non-Newtonian fluids described by a simplified viscoelastic rate-type model with a stress-diffusion term. The simplified model shares many qualitative features with more complex viscoelastic rate-type models that are frequently used in the modeling of fluids with complicated microstructure. As such, the simplified model provides important preliminary insight into the mathematical properties of these more complex and practically relevant models of non-Newtonian fluids. The simplified model that is analyzed from the mathematical perspective is shown to be thermodynamically consistent, and we extensively comment on the interplay between the thermodynamical background of the model and the mathematical analysis of the corresponding initial-boundary-value problem.2000 Mathematics Subject Classification. Primary 35Q35, 76A05, 76A10. The authors acknowledge the support of the ERC-CZ project LL1202, financed by MŠMT.

Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid

“…Fifth, thermodynamical type considerations such as the identification of the energy storage mechanisms and entropy producing mechanisms are known to play an important role in the rigorous mathematical theory of nonlinear governing equations for viscoelastic fluids, see for example Hu and Lelièvre (2007), Boyaval et al (2009), Boyaval (2011, 2017), Barrett and Süli (2018) or Bulíček et al (2018). On the other hand, rigorous mathematical analysis of long-time behaviour of viscoelastic fluids is usually done without a direct appeal to thermodynamics, and the available results are quite limited especially if one considers thermodynamically open systems, see for example Guillopé and Saut (1990), Nohel and Pego (1993), Jourdain et al (2006) or Renardy (2009).…”

Global Existence of Large Data Weak Solutions for a Simplified Compressible Oldroyd-B Model Without Stress Diffusion

“…Note in this context that in [25] a special non-linear diffusion was used and global-in-time large-data solution for the corresponding Oldroyd-B model was possible to prove for a large variety of power-law models for the solvent stress tensor (from shear thinning to shear thickening, including, indeed, the linear dependence). Another interesting recent result is paper [11], where a slightly modified model was considered. The model was developed as a simplification of the general model based on the approach from [33] and global-in-time existence of weak solutions was shown for large data.…”

On a class of compressible viscoelastic rate-type fluids with stress-diffusion

“…We use a methodology developed within the context of incompresssible viscoelastic fluids in [13]. Here, we follow more recent studies concerning compressible viscoelastic rate-type fluids without or with stress diffusion presented in [11] and in [10,2], where the reader can found more detailed expositions. The methodology is based on the tenet that as the body dissipates the energy (or more generally produce the entropy) there is an underlying evolving natural configuration associated with the current configuration and the response between these configurations is purely elastic.…”

“…where ̺ is the density and T is the Cauchy stress. Restricting ourselves to isothermal processes, the formulation of the first and the second law thermodynamics (the balance of energy and the balance of entropy) reduces to (see [2] for details)…”