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On three dimensional flows of viscoelastic fluids of Giesekus type

Miroslav Bulíček, Tomáš Los, Josef Málek

Abstract

Viscoelastic rate-type fluids are popular models of choice in many applications involving flows of fluid-like materials with complex micro-structure. A well-developed mathematical theory for the most of these classical fluid models is however missing. The main purpose of this study is to provide a complete proof of long-time and large-data existence of weak solutions to unsteady internal three-dimensional flows of Giesekus fluids subject to a no-slip boundary condition. As a new auxiliary tool, we provide the identification of certain biting limits in the parabolic setting, presented here within the framework of evolutionary Stokes problems. We also generalize the long-time and large-data existence result to higher dimensions, to viscoelastic models with multiple relaxation mechanisms and to viscoelastic models with different type of dissipation.

On three dimensional flows of viscoelastic fluids of Giesekus type

Abstract

Viscoelastic rate-type fluids are popular models of choice in many applications involving flows of fluid-like materials with complex micro-structure. A well-developed mathematical theory for the most of these classical fluid models is however missing. The main purpose of this study is to provide a complete proof of long-time and large-data existence of weak solutions to unsteady internal three-dimensional flows of Giesekus fluids subject to a no-slip boundary condition. As a new auxiliary tool, we provide the identification of certain biting limits in the parabolic setting, presented here within the framework of evolutionary Stokes problems. We also generalize the long-time and large-data existence result to higher dimensions, to viscoelastic models with multiple relaxation mechanisms and to viscoelastic models with different type of dissipation.
Paper Structure (16 sections, 12 theorems, 193 equations)

This paper contains 16 sections, 12 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Basic notation
  3. Mathematical and convergence tools
  4. Formulation of the Main Result
  5. Proof of Theorem \ref{['bitingT']}
  6. Proof of Theorem \ref{['Burgmain']}
  7. System with evolutionary equation for the tensor F
  8. Approximation and the first a priori estimates
  9. Weak convergence results as e0
  10. Compactness of Fs in LQ
  11. Positivity of detF
  12. Extensions
  13. Mathematical tools
  14. Proof of Theorem \ref{['maini']} (main ideas)
  15. ejk-approximations and the limits epsk and ep
  16. ...and 1 more sections

Key Result

Theorem 1

For an arbitrary $\Omega$ and $T$ satisfying p000 and for arbitrary data $\boldsymbol{v}_0$, $\mathbb{F}_0$ and $\boldsymbol{f}$ that enables the right-hand side of pepa98 to be finite, there exists a global-in-time weak solution to the problem 1Burg--ic.

Theorems & Definitions (19)

  • Theorem
  • Proposition 1.1: Parabolic Lipschitz truncation, BuBuSch19
  • Proposition 1.2: Chacon's biting lemma, see BaMu89
  • Proposition 1.3: Stokes problem I, see Wolf or BuMaLo22
  • Proposition 1.4: Stokes problem II
  • Proposition 1.5
  • proof
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 3.1
  • ...and 9 more