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A Lagrangian formulation for the Oldroyd B fluid and the second principle of thermodynamics

Hervé Le Dret, Annie Raoult

Abstract

We show that the Oldroyd B fluid model is the Eulerian form of a Lagrangian model with an internal variable that satisfies the second law of thermodynamics under some conditions on the initial value of the internal variable. We similarly derive several new nonlinear versions of the Oldroyd B model as well as Lagrangian formulations of the Zaremba-Jaumann and Oldroyd A fluid models. We discuss whether or not these other models satisfy the second law.

A Lagrangian formulation for the Oldroyd B fluid and the second principle of thermodynamics

Abstract

We show that the Oldroyd B fluid model is the Eulerian form of a Lagrangian model with an internal variable that satisfies the second law of thermodynamics under some conditions on the initial value of the internal variable. We similarly derive several new nonlinear versions of the Oldroyd B model as well as Lagrangian formulations of the Zaremba-Jaumann and Oldroyd A fluid models. We discuss whether or not these other models satisfy the second law.
Paper Structure (10 sections, 23 theorems, 104 equations, 2 figures)

This paper contains 10 sections, 23 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General notation
  3. Objective derivatives
  4. The Oldroyd B complex fluid model
  5. The conditional Coleman-Noll procedure for incompressible viscoelastic materials with internal variables
  6. A Lagrangian formulation for the Oldroyd B model
  7. The Oldroyd B fluid and the second law of thermodynamics
  8. A few variants of the Oldroyd B model
  9. Nonlinear Oldroyd B models
  10. The Zaremba-Jaumann and Oldroyd A fluids

Key Result

Proposition 3.2

An operator of the form forme a priori is objective if and only if with $w=\mathop{\mathrm{Skew}}(h)$, $d=\mathop{\mathrm{Sym}}(h)$, and $\mathrm{Ob}_s\colon \mathop{\mathrm{Sym}}_3\times\mathop{\mathrm{Sym}}_3\to\mathop{\mathrm{Sym}}_3$ is an objective function.

Figures (2)

  • Figure 1: Left: Oldroyd B, center: Zaremba-Jaumann, right: Oldroyd A.
  • Figure 2: Smallest eigenvalue of $\sigma_p$, Oldroyd B case.

Theorems & Definitions (52)

  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Definition 5.1
  • Proposition 5.2
  • proof
  • Proposition 5.3
  • proof
  • Proposition 5.4
  • proof
  • ...and 42 more