Slip boundary conditions for the compressible Navier-Stokes equations for a polyatomic gas

The slip boundary conditions for the compressible Navier-Stokes equations for a polyatomic gas are derived from kinetic theory using the ellipsoidal statistical model of the Boltzmann equation for a polyatomic gas. The analysis, which follows the previous work by the present authors and others for a monatomic gas [Aoki et al., J. Stat. Phys. 169, 744 (2017)], is based on the Chapman-Enskog expansion and the analysis of the Knudsen layer adjacent to the boundary. The resulting slip boundary conditions are presented with explicit slip coefficients for some typical polyatomic gases.

Figure 1

Coordinate systems. (a) Coordinate system on the boundary; (b) coordinate system for the Knudsen layer.

Figure 2

Knudsen-layer functions for the BGK model. (a) $\alpha =1.0$, (b) $\alpha =0.5$, (c) $\alpha =0.2$. The solid line indicates $Y_{v\mathrm{BGK}}\left(y\right)$, the dashed line $H_{v\mathrm{BGK}}\left(y\right)$, the dot-dashed line $Y_{T\mathrm{BGK}}\left(y\right)$, and the dotted line $H_{T\mathrm{BGK}}\left(y\right)$.

Figure 3

Knudsen-layer function ${\mathrm{\Omega }}_v\left(y\right)$. (a) $\alpha =1.0$, (b) $\alpha =0.5$, (c) $\alpha =0.2$. The solid line indicates the result for the BGK model for a monatomic gas, the dashed line for nitrogen, the dot-dashed line for methanol, and the dotted line for water vapor.

Figure 4

Knudsen-layer function ${\mathrm{\Omega }}_T\left(y\right)$. (a) $\alpha =1.0$, (b) $\alpha =0.5$, (c) $\alpha =0.2$. See the caption of Fig. 3.

Figure 5

Knudsen-layer function ${\mathrm{\Theta }}_v^{\mathrm{tr}}\left(y\right)$. (a) $\alpha =1.0$, (b) $\alpha =0.5$, (c) $\alpha =0.2$. See the caption of Fig. 3.

Figure 6

Knudsen-layer function ${\mathrm{\Theta }}_v^{\mathrm{int}}\left(y\right)$. (a) $\alpha =1.0$, (b) $\alpha =0.5$, (c) $\alpha =0.2$. The solid line indicates the result for nitrogen, the dashed line for methanol, and the dot-dashed line for water vapor.

Figure 7

Knudsen-layer function ${\mathrm{\Theta }}_T^{\mathrm{tr}}\left(y\right)$. (a) $\alpha =1.0$, (b) $\alpha =0.5$, (c) $\alpha =0.2$. See the caption of Fig. 3.

Figure 8

Knudsen-layer function ${\mathrm{\Theta }}_T^{\mathrm{int}}\left(y\right)$. (a) $\alpha =1.0$, (b) $\alpha =0.5$, (c) $\alpha =0.2$. See the caption of Fig. 6.

Figure 9

Knudsen-layer functions ${\mathrm{\Omega }}_v\left(y\right), {\mathrm{\Theta }}_v^{\mathrm{tr}}\left(y\right)$, and ${\mathrm{\Theta }}_v^{\mathrm{int}}\left(y\right)$ in the case of $\alpha =1$ for ${\mathrm{CO}}_2\left(n\right)$. (a) ${\mathrm{\Omega }}_v\left(y\right)$, (b) ${\mathrm{\Theta }}_v^{\mathrm{tr}}\left(y\right)$, (c) ${\mathrm{\Theta }}_v^{\mathrm{int}}\left(y\right)$. The solid line indicates the result for ${\mathrm{CO}}_2\left(5\right)$, the dashed line for ${\mathrm{CO}}_2\left(10\right)$, the dot-dashed line for ${\mathrm{CO}}_2\left(20\right)$, and the dotted line for ${\mathrm{CO}}_2\left(50\right)$.

Figure 10

Knudsen-layer functions ${\mathrm{\Omega }}_T\left(y\right), {\mathrm{\Theta }}_T^{\mathrm{tr}}\left(y\right)$, and ${\mathrm{\Theta }}_T^{\mathrm{int}}\left(y\right)$ in the case of $\alpha =1$ for ${\mathrm{CO}}_2\left(n\right)$. (a) ${\mathrm{\Omega }}_T\left(y\right)$, (b) ${\mathrm{\Theta }}_T^{\mathrm{tr}}\left(y\right)$, (c) ${\mathrm{\Theta }}_T^{\mathrm{int}}\left(y\right)$. See the caption of Fig. 9.