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A BGK-type model for multi-component gas mixtures undergoing a bimolecular chemical reaction

Giorgio Martalò, Ana Jacinta Soares, Romina Travaglini

TL;DR

The paper develops a BGK-type kinetic model for a reactive four-component gas mixture undergoing a bimolecular reversible reaction, introducing a clear separation between mechanical and chemical relaxation terms and ensuring that Maxwellian attractors reflect the macroscopic fields. By matching momentum and energy exchange rates to the Boltzmann operators, it constructs auxiliary parameters that preserve conservation laws and equilibria, and proves an $\mathcal{H}$-theorem under specific simplifying hypotheses. The work also provides a quasi-equilibrium approach for the chemical contributions and offers numerical simulations that illustrate relaxation to equilibrium and the relative timing of mechanical versus chemical equilibration. This framework enables multi-scale analysis of reactive gas dynamics and paves the way for studying transport properties across fast and slow chemical regimes with computational tractability. The results have potential impact on modeling re-entry and industrial processes where reactive gas mixtures exhibit distinct mechanical and chemical time scales.

Abstract

We propose a new kinetic BGK-type model for a mixture of four monatomic gases, undergoing a bimolecular and reversible chemical reaction. The elastic and reactive interactions are described separately by distinct relaxation terms and the mechanical operator is the sum of binary BGK contributions, one for each pair of interacting species. In this way, our model separately incorporates the effects of mechanical processes and chemical reactions. Additionally, it retains the effects of inter-species interactions which are proper of the mixture. The dependence of Maxwellian attractors on the main macroscopic fields is explicitly expressed by assuming that the exchange rates for momentum and energy of mechanical and chemical operators coincide with the ones of the corresponding Boltzmann terms. Under suitable hypotheses, the relaxation of the distribution functions to equilibrium is shown through entropy dissipation. Some numerical simulations are included to investigate the trend to equilibrium.

A BGK-type model for multi-component gas mixtures undergoing a bimolecular chemical reaction

TL;DR

The paper develops a BGK-type kinetic model for a reactive four-component gas mixture undergoing a bimolecular reversible reaction, introducing a clear separation between mechanical and chemical relaxation terms and ensuring that Maxwellian attractors reflect the macroscopic fields. By matching momentum and energy exchange rates to the Boltzmann operators, it constructs auxiliary parameters that preserve conservation laws and equilibria, and proves an -theorem under specific simplifying hypotheses. The work also provides a quasi-equilibrium approach for the chemical contributions and offers numerical simulations that illustrate relaxation to equilibrium and the relative timing of mechanical versus chemical equilibration. This framework enables multi-scale analysis of reactive gas dynamics and paves the way for studying transport properties across fast and slow chemical regimes with computational tractability. The results have potential impact on modeling re-entry and industrial processes where reactive gas mixtures exhibit distinct mechanical and chemical time scales.

Abstract

We propose a new kinetic BGK-type model for a mixture of four monatomic gases, undergoing a bimolecular and reversible chemical reaction. The elastic and reactive interactions are described separately by distinct relaxation terms and the mechanical operator is the sum of binary BGK contributions, one for each pair of interacting species. In this way, our model separately incorporates the effects of mechanical processes and chemical reactions. Additionally, it retains the effects of inter-species interactions which are proper of the mixture. The dependence of Maxwellian attractors on the main macroscopic fields is explicitly expressed by assuming that the exchange rates for momentum and energy of mechanical and chemical operators coincide with the ones of the corresponding Boltzmann terms. Under suitable hypotheses, the relaxation of the distribution functions to equilibrium is shown through entropy dissipation. Some numerical simulations are included to investigate the trend to equilibrium.
Paper Structure (16 sections, 2 theorems, 110 equations, 2 figures)

This paper contains 16 sections, 2 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Kinetic equations
  3. Boltzmann formulation
  4. A BGK model
  5. Computation of auxiliary parameters
  6. Parameters in mechanical terms
  7. Parameters in chemical terms
  8. Entropy properties and trend to equilibrium
  9. Entropy growth
  10. Numerical insight
  11. Conclusions
  12. Appendix -- computation of integrals
  13. Some preliminaries
  14. Production of mass
  15. Production of momentum
  16. ...and 1 more sections

Key Result

Theorem 4.1

( $\cal H$-theorem) Let us assume conditions equal_cond and MAL_chem on the auxiliary parameters of chemical terms. Under space homogeneous conditions, for all measurable distribution functions $f_i\ge 0$, $i=1,2,3,4$, we have that

Figures (2)

  • Figure 1: Auxiliary temperatures in mechanical and chemical BGK terms, rescaled with respect to species temperature.
  • Figure 2: H-functional given in \ref{['eq:HchemK']}.

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 4.2