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On various models describing behaviour of thermoviscoelastic rate-type fluids

Miroslav Bulìček, Jakub Woźnicki

TL;DR

The article establishes global weak existence for a thermodynamically consistent model of heat-conducting viscoelastic rate-type fluids in a planar setting, without stress diffusion. By formulating the system through a Helmholtz free energy and an entropy inequality, the authors derive robust a priori estimates and deploy renormalization, div-curl, and compactness tools (Aubin–Lions, biting limits) to pass to the limit in a four-step Galerkin approximation. They first address the general coupling case $\text{general\_g\_case}$, proving weak sequential stability, then prove existence for the constant-g case, with detailed convergence analysis for $\boldsymbol{v}$, $\mathbb{F}$, and $\theta$, including strong convergence of $\theta$ and $\mathbb{F}$. The results demonstrate global-in-time existence for large data in 2D and provide a framework adaptable to 3D, relying on energy-entropy balance and thermodynamic consistency rather than stress diffusion. Overall, the work advances rigorous understanding of thermo-viscoelastic fluids with temperature-dependent coefficients, offering solid mathematical foundations for these complex materials.

Abstract

Viscoelastic rate-type fluid models are essential for describing the behavior of a wide range of complex materials, with applications in fields such as engineering, biomaterials, and medicine. These models are particularly useful for understanding the rheological properties of materials that exhibit both elastic and viscous behavior under deformation. However, many real-world applications involve significant thermal effects, where heat conduction and the temperature dependence of material properties must also be considered. In this paper, we introduce a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids and establish the existence of a global weak solution in a two-dimensional setting. The result holds under the condition that the initial energy and entropy are controlled in appropriate natural norms.

On various models describing behaviour of thermoviscoelastic rate-type fluids

TL;DR

The article establishes global weak existence for a thermodynamically consistent model of heat-conducting viscoelastic rate-type fluids in a planar setting, without stress diffusion. By formulating the system through a Helmholtz free energy and an entropy inequality, the authors derive robust a priori estimates and deploy renormalization, div-curl, and compactness tools (Aubin–Lions, biting limits) to pass to the limit in a four-step Galerkin approximation. They first address the general coupling case , proving weak sequential stability, then prove existence for the constant-g case, with detailed convergence analysis for , , and , including strong convergence of and . The results demonstrate global-in-time existence for large data in 2D and provide a framework adaptable to 3D, relying on energy-entropy balance and thermodynamic consistency rather than stress diffusion. Overall, the work advances rigorous understanding of thermo-viscoelastic fluids with temperature-dependent coefficients, offering solid mathematical foundations for these complex materials.

Abstract

Viscoelastic rate-type fluid models are essential for describing the behavior of a wide range of complex materials, with applications in fields such as engineering, biomaterials, and medicine. These models are particularly useful for understanding the rheological properties of materials that exhibit both elastic and viscous behavior under deformation. However, many real-world applications involve significant thermal effects, where heat conduction and the temperature dependence of material properties must also be considered. In this paper, we introduce a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids and establish the existence of a global weak solution in a two-dimensional setting. The result holds under the condition that the initial energy and entropy are controlled in appropriate natural norms.

Paper Structure

This paper contains 15 sections, 7 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak sequential stability of strong solutions for the case \ref{['general_g_case']}
  4. Auxiliary identities for B and
  5. Uniform estimates
  6. Convergence results based on the uniform estimates
  7. Compactness of the temperature t
  8. Compactness of F
  9. Existence of the weak solutions for the case \ref{['contant_g_case']}
  10. Approximating scheme
  11. Galerkin approximation
  12. Convergence with m
  13. Convergence with n
  14. Convergence with e and r.
  15. Auxiliary propositions

Key Result

Theorem 1

Let $g$ in general_g_case be properly chosen. Then for the relevant initial data, there is a weak global-in-time solution to main_sys_for_g_theta--eq:boundaryconditions_for_g_theta.

Theorems & Definitions (8)

  • Theorem
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • Lemma A.1
  • Lemma A.2
  • Lemma A.3