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Two-temperature fluid models for a polyatomic gas based on kinetic theory for nearly resonant collisions

Kazuo Aoki, Niclas Bernhoff

TL;DR

This work derives two-temperature fluid models for a polyatomic gas by starting from a Boltzmann equation in which the collision kernel blends resonant (elastic) and standard (inelastic) collisions through a parameter θ. Using Chapman–Enskog expansion in near-fluid regimes, the authors obtain Euler and Navier–Stokes-type equations that feature translational and internal temperatures with explicit relaxation terms proportional to (T_tr − T_int). Two asymptotic scalings, θ = O(Kn^2) and θ = O(Kn), yield two-temperature Navier–Stokes systems with different orderings of transport and relaxation terms, and in some cases enable closed-form expressions for transport coefficients in terms of the collision-kernel parameters. The results provide a rigorous kinetic-theory basis for two-temperature fluids and connect to ES-model-based analyses, offering a framework applicable to shock-wave problems and high-temperature polyatomic flows while highlighting avenues for alternative collision operators. Overall, the paper clarifies how weak translational–internal coupling manifests in macroscopic two-temperature dynamics and supplies explicit constitutive formulas for practical use.

Abstract

A polyatomic ideal gas with weak interaction between the translational and internal modes is considered. For the purpose of describing the behavior of such a gas, a Boltzmann equation is proposed in the form that the collision integral is a linear combination of inelastic and elastic (or resonant) collisions, and its basic properties are discussed. Then, in the case where the elastic collisions are dominant, fluid dynamic equations of Euler and Navier--Stokes type including two temperatures, i.e., translational and internal temperatures, as well as relaxation terms are systematically obtained by means of the Chapman--Enskog expansion. The obtained equations are different depending on the degree of weakness of the interaction between the translational and internal modes.

Two-temperature fluid models for a polyatomic gas based on kinetic theory for nearly resonant collisions

TL;DR

This work derives two-temperature fluid models for a polyatomic gas by starting from a Boltzmann equation in which the collision kernel blends resonant (elastic) and standard (inelastic) collisions through a parameter θ. Using Chapman–Enskog expansion in near-fluid regimes, the authors obtain Euler and Navier–Stokes-type equations that feature translational and internal temperatures with explicit relaxation terms proportional to (T_tr − T_int). Two asymptotic scalings, θ = O(Kn^2) and θ = O(Kn), yield two-temperature Navier–Stokes systems with different orderings of transport and relaxation terms, and in some cases enable closed-form expressions for transport coefficients in terms of the collision-kernel parameters. The results provide a rigorous kinetic-theory basis for two-temperature fluids and connect to ES-model-based analyses, offering a framework applicable to shock-wave problems and high-temperature polyatomic flows while highlighting avenues for alternative collision operators. Overall, the paper clarifies how weak translational–internal coupling manifests in macroscopic two-temperature dynamics and supplies explicit constitutive formulas for practical use.

Abstract

A polyatomic ideal gas with weak interaction between the translational and internal modes is considered. For the purpose of describing the behavior of such a gas, a Boltzmann equation is proposed in the form that the collision integral is a linear combination of inelastic and elastic (or resonant) collisions, and its basic properties are discussed. Then, in the case where the elastic collisions are dominant, fluid dynamic equations of Euler and Navier--Stokes type including two temperatures, i.e., translational and internal temperatures, as well as relaxation terms are systematically obtained by means of the Chapman--Enskog expansion. The obtained equations are different depending on the degree of weakness of the interaction between the translational and internal modes.
Paper Structure (37 sections, 12 theorems, 234 equations)

This paper contains 37 sections, 12 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Kinetic model
  3. Velocity-energy distribution function and macroscopic quantities
  4. Boltzmann equation and collision operator
  5. Boltzmann equation
  6. Binary collisions
  7. Collision operator
  8. Borgnakke-Larsen-type model
  9. Collision invariants and equilibrium distributions
  10. Linearized collision operator
  11. Fredholmness of the linearized collision operator
  12. Nearly resonant collisions and two-temperature fluid models
  13. Preliminaries
  14. Collision frequency and mean free path
  15. Nondimensionalization
  16. ...and 22 more sections

Key Result

Lemma 1

The measure $\mathrm{d}A_{\theta }$ is invariant under the interchanges of variables respectively.

Theorems & Definitions (22)

  • Lemma 1
  • Proposition 1
  • Definition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Proposition 5
  • ...and 12 more