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A kinetic interpretation of thermomechanical restrictions of continua

Patrick E. Farrell, Josef Málek, Ondřej Souček, Umberto Zerbinati

Abstract

Rajagopal and Srinivasa's thermodynamic framework derives constitutive relations in continuum mechanics from two scalar functions describing energy storage and entropy production via a constrained optimization principle. In parallel, kinetic theory obtains constitutive laws through moment closure, most notably via the Chapman-Enskog expansion. In this work we establish a connection between these approaches by providing a kinetic interpretation of the Rajagopal-Srinivasa principle of maximal entropy production. For Bhatnagar-Gross-Krook-type kinetics, we show that this principle is equivalent to a minimal relaxation-time principle, selecting among admissible constitutive responses the one with the fastest compatible relaxation toward equilibrium. We show how the caloric equation of state, entropy balance, and entropy production can be computed from kinetic theory, and we propose a hybrid methodology in which the Chapman--Enskog expansion is used only to compute entropy production and equilibrium thermodynamic relations, while full constitutive closures are obtained through constrained optimization. The proposed hybrid Chapman-Enskog-Rajagopal-Srinivasa approach recovers the standard Euler and Navier-Stokes-Fourier constitutive laws for monatomic gases. We demonstrate how different choices of selection procedure can be more informative than the classical Chapman-Enskog closure in the context of an inviscid compressible Leslie-Ericksen model arising in liquid crystals.

A kinetic interpretation of thermomechanical restrictions of continua

Abstract

Rajagopal and Srinivasa's thermodynamic framework derives constitutive relations in continuum mechanics from two scalar functions describing energy storage and entropy production via a constrained optimization principle. In parallel, kinetic theory obtains constitutive laws through moment closure, most notably via the Chapman-Enskog expansion. In this work we establish a connection between these approaches by providing a kinetic interpretation of the Rajagopal-Srinivasa principle of maximal entropy production. For Bhatnagar-Gross-Krook-type kinetics, we show that this principle is equivalent to a minimal relaxation-time principle, selecting among admissible constitutive responses the one with the fastest compatible relaxation toward equilibrium. We show how the caloric equation of state, entropy balance, and entropy production can be computed from kinetic theory, and we propose a hybrid methodology in which the Chapman--Enskog expansion is used only to compute entropy production and equilibrium thermodynamic relations, while full constitutive closures are obtained through constrained optimization. The proposed hybrid Chapman-Enskog-Rajagopal-Srinivasa approach recovers the standard Euler and Navier-Stokes-Fourier constitutive laws for monatomic gases. We demonstrate how different choices of selection procedure can be more informative than the classical Chapman-Enskog closure in the context of an inviscid compressible Leslie-Ericksen model arising in liquid crystals.
Paper Structure (17 sections, 5 theorems, 120 equations)

This paper contains 17 sections, 5 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. The Boltzmann equation
  3. Deriving balance laws from kinetic theory
  4. H-Theorem and entropy production
  5. Chapman--Enskog approach to constitutive equations for entropy and entropy production
  6. Chapman--Enskog expansions and its consequences for macroscopic moments
  7. Constitutive equation for entropy
  8. Thermodynamic temperature and caloric equation of state
  9. Thermodynamic pressure and the thermal equation of state (ideal gas law)
  10. Zeroth and first order entropy flux and entropy production: Clausius--Duhem inequality
  11. Euler and Fourier--Navier--Stokes equations
  12. Zeroth order approximation: Euler equations
  13. First order approximation: Navier--Stokes equations
  14. Relaxation time and maximization of the rate of entropy production principles
  15. Beyond Euler and Navier--Stokes: Leslie--Ericksen equations
  16. ...and 2 more sections

Key Result

Proposition 1

Each of $\rho(\bm{x},t)$, $\bm{u}(\bm{x},t)$, $e(\bm{x},t)$ are expansion-independent under the solvability condition eq:solvability.

Theorems & Definitions (26)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5: Various formulations of the second law of thermodynamics
  • Remark 6
  • Remark 7
  • Definition 1: Expansion independence
  • Proposition 1
  • proof
  • ...and 16 more