$H$-theorem and boundary conditions for two-temperature model: Application to wave propagation and heat transfer in polyatomic gases

Polyatomic gases find numerous applications across various scientific and technological fields, necessitating a quantitative understanding of their behavior in nonequilibrium conditions. In this study, we investigate the behavior of rarefied polyatomic gases, particularly focusing on heat transfer and sound propagation phenomena. By utilizing a two-temperature model, we establish constitutive equations for internal and translational heat fluxes based on the second law of thermodynamics. A novel reduced two-temperature model is proposed, which accurately describes the system's behavior while reducing computational complexity. Additionally, we develop phenomenological boundary conditions adhering to the second law, enabling the simulation of gas-surface interactions. The phenomenological coefficients in the constitutive equations and boundary conditions are determined by comparison with relevant literature. Our computational analysis includes conductive heat transfer between parallel plates, examination of sound wave behavior, and exploration of spontaneous Rayleigh-Brillouin scattering. The results provide valuable insights into the dynamics of polyatomic gases, contributing to various technological applications involving heat transfer and sound propagation.

Figure 1

Stability analysis of the two-temperature model for ${\text{CH}}_4$ gas with Model 1 coefficients: panel (up) spatial stability and panel (down) temporal stability. In panel (up) black, red, or blue dashed lines represent different modes corresponding to the parameter $\omega$.

Figure 2

Stability analysis of the two-temperature model for ${\text{CH}}_4$ gas with Model 2 coefficients: panel (up) spatial stability and panel (down) temporal stability. In panel (up) black, red, or blue dashed lines represent different modes corresponding to the parameter $\omega$.

Figure 3

Stability analysis of the two-temperature model for ${\text{CH}}_4$ gas with Model 3 coefficients: panel (up) spatial stability and panel (down) temporal stability. In panel (up) black, red, or blue dashed lines represent different modes corresponding to the parameter $\omega$.

Figure 4

Stability analysis of the two-temperature model for ${\text{CH}}_4$ gas with Model 4 coefficients: panel (up) spatial stability and panel (down) temporal stability. In panel (up) black, red, or blue dashed lines represent different modes corresponding to the parameter $\omega$.

Figure 5

Stability analysis of the two-temperature model for ${\text{CH}}_4$ gas with reduced model coefficients: panel (up) spatial stability and panel (down) temporal stability. In panel (up) black, red, green, or blue dashed lines represent different modes corresponding to the parameter $\omega$.

Figure 6

Schematics for nitrogen (up) and oxygen (down), depicting the attenuation factor $\alpha c_0/\omega$ as a function of the rarefaction parameter $p/\mu \omega$ at a temperature of $300\text{K}$. The predictions of our theory using various coefficients and the Navier-Stokes-Fourier (NSF) theory are compared with experimental data obtained by Greenspan [28].

Figure 7

Schematics for nitrogen (up) and oxygen (down), depicting the reciprocal speed ratio $c_0/v_{\text{ph}}$ as a function of the rarefaction parameter $p/\mu \omega$ at a temperature of $300\text{K}$. The predictions of our theory using various coefficients and the Navier-Stokes-Fourier (NSF) theory are compared with experimental data obtained by Greenspan [28].

Figure 8

Schematics for nitrogen (up) and oxygen (down), depicting speed of sound $(v_{\text{ph}}-c_0)/c_0$ as a function of the rarefaction parameter $p/\mu \omega$ at a temperature of $300\text{K}$. The predictions of our theory using various coefficients and the Navier-Stokes-Fourier (NSF) theory are compared with experimental data obtained by Greenspan [28].

Figure 9

The scattered light spectrum from ${\text{CH}}_4$ for $y=18.27$ and spectrum has been normalized at the Rayleigh peak.

Figure 10

The scattered light spectrum from ${\text{CH}}_4$ for $y=4.46$ and spectrum has been normalized at the Rayleigh peak.

Figure 11

The scattered light spectrum from ${\text{CH}}_4$ for $y=2.70$ and spectrum has been normalized at the Rayleigh peak.

Figure 12

The scattered light spectrum from ${\text{N}}_2$ for $y=1.725$ and spectrum has been normalized at the Rayleigh peak.

Figure 13

The scattered light spectrum from ${\text{O}}_2$ for $y=1.633$ and spectrum has been normalized at the Rayleigh peak.

Figure 14

The schematic illustrates heat conduction, featuring a channel with two stationary parallel walls having distinct temperatures.

Figure 15

Comparison of density profiles for Kn numbers equal to 0.071 (up) and 0.71 (down). Results shown are obtained from: our theory (solid, dashed line); NSF equations (dot dashed cyan line); DSMC method (red square box) [53].

Figure 16

Comparison of translational temperature for Kn numbers equal to 0.071 (up) and 0.71 (down). Results shown are obtained from: our theory (solid, dashed line); NSF equations (dot dashed cyan line); DSMC method (red square box) [53].

Figure 17

Comparison of internal temperature for Kn numbers equal to 0.071 (up) and 0.71 (down). Results shown are obtained from: our theory (solid, dashed line); NSF equations (dot dashed cyan line); DSMC method (red square box) [53].