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The Asymptotic Behaviour of Oldroyd-B Fluids is Almost Newtonian

Matthias Hieber, Thieu Huy Nguyen, César J. Niche, Cilon F. Perusato

Abstract

Consider a viscoelastic fluid of Oldroyd-B type. It is shown that its stress tensor $τ$ and its Newtonian deformation tensor $D(u)$ decay at the same rate, while the elastic part $\varepsilon=τ-2ωD(u)$ decays faster. As a consequence, the stress tensor of a viscoelastic fluid exhibits an almost Newtonian behaviour for large times.

The Asymptotic Behaviour of Oldroyd-B Fluids is Almost Newtonian

Abstract

Consider a viscoelastic fluid of Oldroyd-B type. It is shown that its stress tensor and its Newtonian deformation tensor decay at the same rate, while the elastic part decays faster. As a consequence, the stress tensor of a viscoelastic fluid exhibits an almost Newtonian behaviour for large times.
Paper Structure (10 sections, 8 theorems, 119 equations)

This paper contains 10 sections, 8 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Results
  3. Preliminaries
  4. Main results
  5. Analysis of the Linear Part
  6. Proof of the Main Theorems
  7. Proof of Theorem \ref{['thm-a-equal-zero']}
  8. Proof of Theorem \ref{['general-a']}
  9. Proof of Theorem \ref{['thm:error']}
  10. Proof of Theorem \ref{['thm:lower']}

Key Result

Theorem 2.1

Let $a = 0$ and $(u_0, \tau_0) \in L^2_\sigma (\mathbb{R}^3) \times L^2 (\mathbb{R}^3)$, with $-\frac{3}{2} < r^{\ast} (u_0), r^{\ast}(\tau_0) < \infty$. Then, for any weak solution $z = (u, \tau)$ to eqn:oldroyd-b, there exists a constant $C>0$ such that

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • proof
  • ...and 3 more