Two-temperature Navier-Stokes equations for a polyatomic gas derived from kinetic theory

A polyatomic gas with slow relaxation of the internal modes is considered, and the Navier-Stokes equations with two temperatures, the translational and internal temperatures, are derived for such a gas on the basis of the ellipsoidal-statistical (ES) model of the Boltzmann equation for a polyatomic gas, proposed by Andries et al. [Eur. J. Mech. B, Fluids 19, 813 (2000)], by the Chapman-Enskog procedure. Then, the derived equations are applied to numerically investigate the structure of a plane shock wave in ${\mathrm{CO}}_2$ gas, which is known to have slowly relaxing internal modes. The results show good agreement with those obtained by the direct numerical analysis of the ES model for moderately strong shock waves. In particular, the results perfectly reproduce the double-layer structure of the shock profiles consisting of a thin front layer with rapid change and a thick rear layer with slow relaxation of the internal modes.

Figure 1

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=1.05$ for ${\mu }_b/\mu =1000$. In (a), the red solid line indicates $\check{\rho }$, the green dashed line $\check{v}$, and the blue dotted-dashed line $\check{T}$, and in (b), the red solid line indicates ${\check{T}}_{\text{tr}}$, the green dashed line ${\check{T}}_{\text{int}}$, and the blue dotted-dashed line $\check{T}$.

Figure 2

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=1.138$ for ${\mu }_b/\mu =1000$. See the caption of Fig. 1.

Figure 3

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=1.2$ for ${\mu }_b/\mu =1000$. Panels (a) and (c) show profiles for $-2000\le X_1/l_{-}\le 15000$; (b) and (d) those for $-30\le X_1/l_{-}\le 300$. In (a) and (b), the red solid line indicates $\check{\rho }$, the green dashed line $\check{v}$, and the blue dotted-dashed line $\check{T}$, and in (c) and (d), the red solid line indicates ${\check{T}}_{\text{tr}}$, the green dashed line ${\check{T}}_{\text{int}}$, and the blue dotted-dashed line $\check{T}$. The dotted lines of the same color as the respective macroscopic quantities in (b) and (d) indicate the profiles obtained from Eq. (77) with $\theta =0$ (i.e., ${\mu }_b/\mu =\infty$).

Figure 4

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=2$ for ${\mu }_b/\mu =10$. See the caption of Fig. 1.

Figure 5

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=2$ for ${\mu }_b/\mu =100$. Panels (a) and (c) show profiles for $-40\le X_1/l_{-}\le 340$; (b) and (d) those for $-8\le X_1/l_{-}\le 30$. See the caption of Fig. 3 for the types of the lines. The symbol $\times$ indicates the profiles based on the ES model.

Figure 6

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=2$ for ${\mu }_b/\mu =1000$. Panels (a) and (c) show profiles for $-500\le X_1/l_{-}\le 4000$; (b) and (d) those for $-20\le X_1/l_{-}\le 60$. See the caption of Fig. 3 for the types of the lines. The symbol $\times$ indicates the profiles based on the ES model.

Figure 7

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=5$ for ${\mu }_b/\mu =1000$. Panels (a) and (c) show profiles for $-200\le X_1/l_{-}\le 2400$; (b) and (d) those for $-15\le X_1/l_{-}\le 50$. See the caption of Fig. 3 for the types of the lines. The symbol $\times$ indicates the profiles based on the ES model.

Figure 8

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=2$ for ${\mu }_b/\mu =1000$ in the case of $\delta =3$. Panels (a) and (c) show profiles for $-500\le X_1/l_{-}\le 4000$; (b) and (d) those for $-20\le X_1/l_{-}\le 60$. See the caption of Fig. 3 for the types of the lines.

Figure 9

Profiles of $\check{\rho }, \check{v}, \check{T}, {\check{T}}_{\text{tr}}$, and ${\check{T}}_{\text{int}}$ at $M_{-}=2$ for ${\mu }_b/\mu =1000$ in the case of $\delta =5$. Panels (a) and (c) show profiles for $-500\le X_1/l_{-}\le 4000$; (b) and (d) those for $-20\le X_1/l_{-}\le 60$. See the caption of Fig. 3 for the types of the lines.