Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Investigating dynamics and asymptotic trend to equilibrium in a reactive BGK model

Giorgio Martalò, Ana Jacinta Soares, Romina Travaglini

TL;DR

This work analyzes the dynamics toward thermal and chemical equilibrium in a reactive BGK framework for a four-species mixture undergoing $G_1+G_2 \leftrightarrows G_3+G_4$ with endothermic heat $\Delta E>0$. The authors formulate a BGK-type kinetic model with separate mechanical and chemical relaxation operators, using Maxwellian attractors parameterized by auxiliary fields to preserve conservation laws and exchange rates. They prove an $\mathcal{H}$-theorem under quasi-equilibrium conditions and demonstrate, via space-homogeneous isotropic simulations, that the $\mathcal{H}$-functional is a Lyapunov functional in near-equilibrium regimes while remaining informative in far-from-equilibrium cases where the chemical transients are pronounced. Numerically, the study reveals that mechanical temperature equilibration occurs prior to chemical temperature equilibration, and that chemical exchanges govern the initial transient behavior, with relaxation to a common Maxwellian temperature observed even when initial data are far from equilibrium. The results provide a tractable framework for analyzing reactive gas mixtures and guide future extensions to space-dependent problems and hybrid Boltzmann-BGK models.

Abstract

We investigate numerically a recent BGK-type model for a multi-component mixture of monatomic gases, undergoing a reversible bimolecular chemical reaction. The model replaces each collisional term of the Boltzmann equation with a relaxation term, thereby describing separately the effects of the mechanical processes and the chemical reaction. Additionally, the model exhibits consistency properties. The correct entropy production is ensured when auxiliary temperatures in the chemical contributions share a common value. We assume isotropic distributions and perform numerical simulations for the macroscopic fields to appraise how the dynamics push the mixture toward thermalization and chemical equilibrium. We show that the hypothesis on the equalization of fictitious species temperatures is justifiable to ensure the monotonicity of the classical $H$-Boltzmann functional. Simulations show that, when initial temperatures are far from equilibrium, the relaxation towards equilibrium occurs at a later stage and the classical $H$-Boltzmann functional is not monotone during the initial transient.

Investigating dynamics and asymptotic trend to equilibrium in a reactive BGK model

TL;DR

This work analyzes the dynamics toward thermal and chemical equilibrium in a reactive BGK framework for a four-species mixture undergoing with endothermic heat . The authors formulate a BGK-type kinetic model with separate mechanical and chemical relaxation operators, using Maxwellian attractors parameterized by auxiliary fields to preserve conservation laws and exchange rates. They prove an -theorem under quasi-equilibrium conditions and demonstrate, via space-homogeneous isotropic simulations, that the -functional is a Lyapunov functional in near-equilibrium regimes while remaining informative in far-from-equilibrium cases where the chemical transients are pronounced. Numerically, the study reveals that mechanical temperature equilibration occurs prior to chemical temperature equilibration, and that chemical exchanges govern the initial transient behavior, with relaxation to a common Maxwellian temperature observed even when initial data are far from equilibrium. The results provide a tractable framework for analyzing reactive gas mixtures and guide future extensions to space-dependent problems and hybrid Boltzmann-BGK models.

Abstract

We investigate numerically a recent BGK-type model for a multi-component mixture of monatomic gases, undergoing a reversible bimolecular chemical reaction. The model replaces each collisional term of the Boltzmann equation with a relaxation term, thereby describing separately the effects of the mechanical processes and the chemical reaction. Additionally, the model exhibits consistency properties. The correct entropy production is ensured when auxiliary temperatures in the chemical contributions share a common value. We assume isotropic distributions and perform numerical simulations for the macroscopic fields to appraise how the dynamics push the mixture toward thermalization and chemical equilibrium. We show that the hypothesis on the equalization of fictitious species temperatures is justifiable to ensure the monotonicity of the classical -Boltzmann functional. Simulations show that, when initial temperatures are far from equilibrium, the relaxation towards equilibrium occurs at a later stage and the classical -Boltzmann functional is not monotone during the initial transient.
Paper Structure (9 sections, 1 theorem, 21 equations, 7 figures)

This paper contains 9 sections, 1 theorem, 21 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A BGK reactive model
  3. Kinetic equations
  4. Relaxation to equilibrium and entropy estimate
  5. Numerical experiments
  6. Asymptotic trend to equilibrium
  7. Far from equilibrium case
  8. Conclusions
  9. Acknowledgments.

Key Result

theorem thmcountertheorem

Let us assume that auxiliary parameters of chemical attractors given in (eq:atract) satisfy the conditions and Under space homogeneous conditions, for all measurable distribution functions $f_i\ge 0$, $i=1,2,3,4$, we have that (a) $\dfrac{d\mathcal{H}}{d t}\le 0\,, \quad \text{for all} \;\; t\ge 0$; (b) $\dfrac{d{\cal H}}{dt}(t) = 0 \qquad \hbox{if and only if} \qquad f_i = f_i^M, \ \ \hbox{for}

Figures (7)

  • Figure 1: Scenario 1 -- Trend to equilibrium. Chemical production terms for mass (top) and deviation from chemical equilibrium measured by the left-hand side of the mass action law (\ref{['MAL']}) (bottom).
  • Figure 2: Scenario 1 -- Trend to equilibrium. Auxiliary temperatures in mechanical and chemical BGK terms, rescaled with respect to species temperature, for the first component (top) and the third component (bottom).
  • Figure 3: Scenario 1 -- Trend to equilibrium. Species temperatures scaled with respect to the global temperature.
  • Figure 4: Scenario 2 -- Far from equilibrium. Distribution function $f_3$ in $(t,v)-$plane (top). Initial distribution $f_i^{0}$ and asymptotic distribution functions $f_i^{\infty}$ for species 1,2 and 4 (bottom).
  • Figure 5: Scenario 2 -- Far from equilibrium. H-functional given in (\ref{['eq:Hchem']}).
  • ...and 2 more figures

Theorems & Definitions (1)

  • theorem thmcountertheorem