Sound waves propagating in a slightly rarefied gas over a smooth solid boundary

A time-evolution of a slightly rarefied gas from a uniform equilibrium state at rest is investigated on the basis of the linearized Boltzmann equation under the acoustic time scaling. By a systematic asymptotic analysis, linearized Euler sets of equations and acoustic-boundary-layer equations are derived, together with their slip and jump boundary conditions, as well as the correction formula in the Knudsen layer. Analysis is done up to the first order of the Knudsen number ($\mathrm{Kn}$), with ${\mathrm{Kn}}^{1/2}$ being the small parameter. Several rarefaction effects, which are known as the effects of the second order in $\mathrm{Kn}$ in the diffusion scaling, are enhanced to be of the first order in $\mathrm{Kn}$. This is because the variation of the macroscopic quantities along the normal direction is steep in the boundary layer and the compressibility of the gas is comparatively strong. The occurrence of secular terms associated with the Hilbert expansion is pointed out and a remedy for it is also given. Finally, as an application example, a sound propagation in a half space caused by a sinusoidal oscillation of flat boundary is examined on the basis of the Bhatnagar–Gross–Krook equation. The asymptotic solution agrees well with the direct numerical solution.

Figure 1

The profile of macroscopic quantities. (a) $\omega /a$ for $k=0.02$, (b) $\omega /a$ for $k=0.01$, (c) $\tau /a$ for $k=0.02$, (d) $\tau /a$ for $k=0.01$, (e) $u/a$ for $k=0.02$, and (f) $u/a$ for $k=0.01$. The solid line indicates the N solution, the dashed line the HBK2 solution, the dash-dotted line the HBK1 solution, and the dash-double-dotted line the HBK0 solution.

Figure 2

The profile of $\omega$ in a far field at $t=2n\pi$ for $k=0.01$. (a) N, HBK2, CI, and CII solutions and (b) N, HBK1, and HBK0 solutions. The N, HBK2, CI, CII, HBK1, and HBK0 solutions are indicated by the solid, dashed, long-dashed, dotted, dash-dotted, and dash-double-dotted lines, respectively. In panel (a), the long-dashed and dotted lines are invisible because they are close to the solid line. In panel (b), the dash-dotted and dash-double-dotted lines overlap each other.

Figure 3

The profile of macroscopic quantities near the plate at $t=2n\pi$. (a) $\omega /a$ for $k=0.02$, (b) $\omega /a$ for $k=0.01$, (c) $\tau /a$ for $k=0.02$, (d) $\tau /a$ for $k=0.01$, (e) $u/a$ for $k=0.02$, and (f) $u/a$ for $k=0.01$. The solid line indicates the N solution, the dashed line the HBK2 solution, the dash-dotted line the HBK1 solution, the dash-double-dotted line the CI solution, and the long-dashed line the CII solution.

Figure 4

The profile of macroscopic quantities near the plate at $t=[2n+(3/2\left)\right]\pi$. (a) $\omega /a$ for $k=0.02$, (b) $\omega /a$ for $k=0.01$, (c) $\tau /a$ for $k=0.02$, (d) $\tau /a$ for $k=0.01$, (e) $u/a$ for $k=0.02$, and (f) $u/a$ for $k=0.01$. See the caption of Fig. 3.

Figure 5

The profile of macroscopic quantities near the plate for $k=0.02$. (a) $\omega /a$ at $t=2n\pi$, (b) $\omega /a$ at $t=[2n+(3/2\left)\right]\pi$, (c) $\tau /a$ at $t=2n\pi$, (d) $\tau /a$ at $t=[2n+(3/2\left)\right]\pi$, (e) $u/a$ at $t=2n\pi$, and (f) $u/a$ at $t=[2n+(3/2\left)\right]\pi$. The solid line indicates the CI solution, the dashed line the CII solution, and the dash-dotted line the V solution.