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Numerical analysis of the thermal relaxation of the dense gas between two parallel plates: the free energy monotonicity for the Enskog equation

Shigeru Takata, Soma Sakata, Aoto Takahashi, Masanari Hattori

Abstract

The thermal relaxation problem between two parallel plates with the same temperature is investigated, aiming to study the behavior of the free energy of the dense gas described by the Enskog equation. Two types of Enskog equation have been used: one is the Enskog equation with the original Enskog factor, while the other is that with a modified Enskog factor proposed recently in Takata & Takahashi, Phys. Rev. E 111, 065108 (2025). The evaluated free energy is a natural extension of the thermodynamic free energy to the non-equilibrium state. It is observed that this free energy monotonically decreases in time for the modified factor version, while it is not necessarily the case for the original version. Differences are also observed in other quantities in their time evolutions, most typically in the density profile.

Numerical analysis of the thermal relaxation of the dense gas between two parallel plates: the free energy monotonicity for the Enskog equation

Abstract

The thermal relaxation problem between two parallel plates with the same temperature is investigated, aiming to study the behavior of the free energy of the dense gas described by the Enskog equation. Two types of Enskog equation have been used: one is the Enskog equation with the original Enskog factor, while the other is that with a modified Enskog factor proposed recently in Takata & Takahashi, Phys. Rev. E 111, 065108 (2025). The evaluated free energy is a natural extension of the thermodynamic free energy to the non-equilibrium state. It is observed that this free energy monotonically decreases in time for the modified factor version, while it is not necessarily the case for the original version. Differences are also observed in other quantities in their time evolutions, most typically in the density profile.
Paper Structure (10 sections, 34 equations, 8 figures, 1 table)

This paper contains 10 sections, 34 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem and formulation
  3. Problem
  4. H-theorem
  5. Numerical method
  6. Results
  7. Free energy
  8. Density and temperature profiles
  9. Conclusion
  10. Dimensionless nomenclature

Figures (8)

  • Figure 1: Problem setting. Dashed circle indicates a molecule in contact with the plate surface.
  • Figure 2: The Enskog factor ${\mathsf{g}}(0,\Delta X_1)$ around the midpoint between the plates in the initial state in the case of $\eta_0=0.25$, $\sigma/L=0.1$ (${\mathrm{Kn}}=0.0227$), and $\lambda=0.1$ for $w=0$, $0.3$, $0.5$, and $0.7$. Solid lines: EESM. Dashed (red) lines: OEE.
  • Figure 3: Time evolution of ${\mathcal{F}}$ in the case of $\eta_0=0.25$, $\sigma/L=0.1$ (${\mathrm{Kn}}=0.0227$), and $\lambda=0.1$ for $w=0.1$, $0.3$, $0.5$, and $0.7$. Solid lines: EESM. Dashed (red) lines: OEE. $t_0=L/\sqrt{2RT_0}$. G-I in Table \ref{['tab:grid']} has been used.
  • Figure 4: Time evolution of the ideal gas part of free energy ${\mathcal{F}}-RT_0{\mathcal{H}}^{(c)}$ in the case of $\eta_0=0.25$, $\sigma/L=0.1$ (${\mathrm{Kn}}=0.0227$), and $\lambda=0.1$ for $w=0.1$, $0.3$, $0.5$, and $0.7$. See the caption of Fig. \ref{['fig:hf1']}.
  • Figure 5: Comparisons of the time evolution of ${\mathcal{F}}$ for different $\sigma/L$ in the case of $\eta_0=0.25$, $w=0.5$, and $\lambda=\sigma/L$ with $\sigma/L=0.02$, $0.05$, and $0.1$ (or ${\mathrm{Kn}}=0.0045$, $0.0114$, and $0.0227$). $t_0^\sigma=\sigma/\sqrt{2RT_0}$. See the caption of Fig. \ref{['fig:hf1']}, though G-II in Table \ref{['tab:grid']} has been used for $\sigma/L=0.02$.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2