Slow approach to steady motion of a concave body in a free-molecular gas

A body in a free-molecular gas accelerated by a constant external force is considered on the basis of kinetic theory. The body is an infinitely long rectangular hollow column with one face removed, and thus it has a squarish $\mathsf{U}$-shaped cross section. The concave part of the body points toward the direction of motion, and thus the gas molecules may be trapped in the concavity. Gas molecules undergo diffuse reflection on a base part, whereas specular reflection on two lateral parts. It is numerically shown that the velocity of the body approaches a terminal velocity, for which a drag force exerted by the gas counterbalances the external force, in such a way that their difference decreases in proportion to the inverse square of time for a large time. This rate of approach is slower than the known rate proportional to the inverse cube of time in the case of a body without concavity [Aoki et al., Phys. Rev. E 80, 016309 (2009)]. Based on the detailed investigation on the velocity distribution function of gas molecules impinging on the body, it is clarified that the concavity prevents some molecules from escaping to infinity. This trapping enhances the effect of recollision between the body and the gas molecules, which is the cause of the inverse power laws, and thus leads to the slower approach.

Figure 1

A concave body considered in Ref. [15] and in the present problem. The case of $H\to 0$ corresponds to the previous studies [10, 11, 13, 16, 17, 18, 20].

Figure 2

Schematic description of $\mathrm{\Gamma }={\mathrm{\Gamma }}_1\times {\mathrm{\Gamma }}_2$. Solid zigzag curve [type (i)] is the molecular path of the molecule that collides with the base plate at time $t^b$ and time $t$ (recollision). Dashed zigzag curve [type (ii)] is the molecular path of the molecule coming from the initial condition, i.e., $t^b=0$.

Figure 3

Schematic diagrams for the definitions of (a) $t^b$ and ${\mathrm{\Gamma }}_1$; (b) ${\tilde{x}}_2$; and (c) ${\overline{\mathrm{\Gamma }}}_1,\tau$, and ${\tau }_{\pm }$.

Figure 4

The velocity distribution function $\mathcal{Q}$ [cf. Eq. (30d)] versus ${\zeta }_2(>0)$ for several ${\zeta }_1^{\mathrm{nor}}$ at $x_2=0$ and time (a) $t=0.8$, (b) $t=1$, (c) $t=2$, (d) $t=3$, (e) $t=4$, (f) $t=5$, (g) $t=10$, (h) $t=20$, and (i) $t=50$ in the case of $v_{B\infty }=2$ and $h=0.1$. The red (dark gray) curve represents $\mathcal{Q}$ with ${\zeta }_1^{\mathrm{nor}}={\check{v}}_{B,\tau }^{\mathrm{nor}}$ [cf. Eq. (21) and Fig. 3]. The magnified figure around ${\zeta }_2\approx 0$ for (g)–(i) is also shown at the bottom row. For the better visualization, we set $n_p=2$ and $N_{1min}=40$.

Figure 5

The velocity distribution function $\mathcal{Q}$ [cf. Eq. (30d)] versus ${\zeta }_2(>0)$ for several ${\zeta }_1^{\mathrm{nor}}$ at $x_2=0$ and time $t=5$. (a) $h={10}^3$ and (b) $h={10}^{-5}$ in the case of $v_{B\infty }=2$ and $h=0.1$. For better visualization, we set $N_{1min}=40$.

Figure 6

The profile of ${\overline{Q}}_2$ versus ${\zeta }_1^{\mathrm{nor}}$ at $x_2=0$ and for various $t$ in the case of $v_{B\infty }=2$ and $h=0.1$. (a) $t=0.5,0.6,...,1$; (b) $t=1$ (dashed) and $t=2$, 3, 4, 5 (solid); (c) $t=5$ (dashed) and $t=6,8,...,20$ (solid); (d) $t=20$ (dashed) and $t=30$, 40, 50 (solid). In panels (c) and (d), the magnified figures around ${\zeta }_1^{\mathrm{nor}}\approx {\check{v}}_{B,\tau }^{\mathrm{nor}}$ are shown in the insets with circles at the grid points. For better visualization, we set $N_{1min}=40$.

Figure 7

(a) Profiles of the history term ${\mathcal{H}}_2$ (solid), ${\mathcal{H}}_2^{\infty }$ (dashed), and ${\overline{\mathcal{H}}}_2$ (dash-dot) [cf. Eq. (30a)] versus $t$ in the double logarithmic scale for cases $h\to \infty ,h=0.1$, and $h\to 0$. (b) The gradients of curves in panel (a) [cf. Eq. (36)]. For $\alpha \left({\overline{\mathcal{H}}}_2\right)$, two curves are presented: the one obtained with $N_{1min}=10$ (dash-dot) and the one obtained with $N_{1min}=40$ (solid).

Figure 8

The long-time behavior of $v_d$ [cf. (35)] and $\alpha \left(v_d\right)$ in the case of $v_{B\infty }=2$. The results with $h\to \infty$ and $h\to 0$ are shown with a bold solid line; those with $h=1\times {10}^{-1},5\times {10}^{-2}$, and $2.5\times {10}^{-2}$ are shown by a dashed line; those with $h=1\times {10}^{-2},5\times {10}^{-3},2.5\times {10}^{-3}$ are shown by a long-dashed line; those with $h=1\times {10}^{-3},5\times {10}^{-4},2.5\times {10}^{-4}$ are shown by a dash-dot line. The curves for $h={10}^{-1},{10}^{-2}$, and ${10}^{-3}$ are slightly thickened. (a) $v_d$, (b) $\alpha \left(v_d\right)$, (c) $-\alpha \left(v_d\right)-2$, and (d) $-\alpha \left(v_d\right)-3$.

Figure 9

The long-time behavior of $v_d$ [cf. (35)] and $\alpha \left(v_d\right)$ in the case of $v_{B\infty }=0.5$. See the caption to Fig. 8.

Figure 10

Schematic diagram of ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2$, and ${\mathrm{\Omega }}_3$, which are represented by bold segments. Depending on the relation between $a_{k+1}$ and $b_k$, the number and the range of segments, i.e., the support of $g_{B+}-E$ [cf. Eq. (15)], vary.