Motion of an array of plates in a rarefied gas caused by radiometric force

In a rarefied gas in an infinitely long channel between two parallel plates, an array of infinitely many plates, arranged longitudinally with uniform interval, is placed along the channel. The array is assumed to be freely movable along the channel. If one side of each plate is heated, the radiometric force acts on it, and the array starts moving toward the cold sides of the plates. The final steady motion of the array, as well as the corresponding behavior of the gas, is investigated numerically on the basis of kinetic theory using the ellipsoidal statistical model of the Boltzmann equation. As the solution method, a finite-difference method, with a method of characteristics incorporated, that is able to capture the discontinuity in the velocity distribution function is employed. As the result, the local flow field near the edges of the plates and the terminal velocity of the array are obtained accurately for relatively small Knudsen numbers.

Figure 1

Problems I and II. (a) Problem I: An array of plates moving with velocity $(-v^{*},0,0)$ in a gas between two walls at rest because of the radiometric force, (b) Problem II: An array of plates at rest in a gas between two walls moving with velocity $(v_w,0,0)$.

Figure 2

Net force $F_1/p_{0}D$ vs $v_w/{\left(2R{T}_0\right)}^{1/2}$ in Problem II for several $\mathrm{Kn}$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$). The numerical results are shown by the symbols. The dashed line indicates $F_1=0$.

Figure 3

Terminal speed of the array $v^{*}$ vs $\mathrm{Kn}$ in Problem I ($L/D=4$). Here, $•$ indicates the result for $T_1/T_0=2,H/D=4$, $\circ$ that for $T_1/T_0=2,H/D=2$, and $\diamond$ that for $T_1/T_0=1.5,H/D=4$.

Figure 4

Flow velocity field $(v_1,v_2)$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$). (a) $\mathrm{Kn}=0.5$ (Problem I, or $v_w=v^{*}$ in Problem II); (b) $\mathrm{Kn}=0.05$ (Problem I, or $v_w=v^{*}$ in Problem II); (c) $\mathrm{Kn}=0.5$ ($v_w=0$ in Problem II); (d) $\mathrm{Kn}=0.05$ ($v_w=0$ in Problem II). The arrow indicates the flow velocity vector $(v_1,v_2)/{\left(2R{T}_0\right)}^{1/2}$ at its starting point, and the scale is shown at the top right corner of each figure.

Figure 5

Magnified figure of the flow velocity field near the edge in the case of $T_1/T_0=2$ ($L/D=H/D=4$). (a) $\mathrm{Kn}=0.5$ (Problem I, or $v_w=v^{*}$ in Problem II); (b) $\mathrm{Kn}=0.05$ (Problem I, or $v_w=v^{*}$ in Problem II); (c) $\mathrm{Kn}=0.5$ ($v_w=0$ in Problem II); (d) $\mathrm{Kn}=0.05$ ($v_w=0$ in Problem II). See the caption of Fig. 4.

Figure 6

Isolines of the flow speed $|v_i|/{\left(2R{T}_0\right)}^{1/2}=\mathrm{const}$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$). (a) $\mathrm{Kn}=0.5$ (Problem I, or $v_w=v^{*}$ in Problem II), $|v_i|/{\left(2R{T}_0\right)}^{1/2}=0.01+0.01m$ ($m=0,...,5$); (b) $\mathrm{Kn}=0.05$ (Problem I, or $v_w=v^{*}$ in Problem II), $|v_i|/{\left(2R{T}_0\right)}^{1/2}=0.005+0.005m$ ($m=0,...,12$); (c) $\mathrm{Kn}=0.5$ ($v_w=0$ in Problem II), $|v_i|/{\left(2R{T}_0\right)}^{1/2}=0.005+0.005m$ ($m=0,...,8$); (d) $\mathrm{Kn}=0.05$ ($v_w=0$ in Problem II), $|v_i|/{\left(2R{T}_0\right)}^{1/2}=0.005+0.005m$ ($m=0,...,12$).

Figure 7

Isolines of the density $\rho /{\rho }_{av}=\mathrm{const}$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$). (a) $\mathrm{Kn}=0.5$ (Problem I, or $v_w=v^{*}$ in Problem II), $\rho /{\rho }_{av}=0.75+0.05m$ ($m=0,...,6$); (b) $\mathrm{Kn}=0.05$ (Problem I, or $v_w=v^{*}$ in Problem II), $\rho /{\rho }_{av}=0.65+0.05m$ ($m=0,...,9$); (c) $\mathrm{Kn}=0.5$ ($v_w=0$ in Problem II), $\rho /{\rho }_{av}=0.75+0.05m$ ($m=0,...,6$); (d) $\mathrm{Kn}=0.05$ ($v_w=0$ in Problem II), $\rho /{\rho }_{av}=0.65+0.05m$ ($m=0,...,9$).

Figure 8

Isolines of the temperature $T/T_0=\mathrm{const}$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$). (a) $\mathrm{Kn}=0.5$ (Problem I, or $v_w=v^{*}$ in Problem II), $T/T_0=1.1+0.1m$ ($m=0,...,5$); (b) $\mathrm{Kn}=0.05$ (Problem I, or $v_w=v^{*}$ in Problem II), $T/T_0=1.1+0.1m$ ($m=0,...,7$); (c) $\mathrm{Kn}=0.5$ ($v_w=0$ in Problem II), $T/T_0=1.1+0.1m$ ($m=0,...,5$); (d) $\mathrm{Kn}=0.05$ ($v_w=0$ in Problem II), $T/T_0=1.1+0.1m$ ($m=0,...,7$).

Figure 9

Isolines of the pressure $p/p_0=\mathrm{const}$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$). (a) $\mathrm{Kn}=0.5$ (Problem I, or $v_w=v^{*}$ in Problem II), $p/p_0=1.09+0.01m$ ($m=0,...,6$); (b) $\mathrm{Kn}=0.05$ (Problem I, or $v_w=v^{*}$ in Problem II), $p/p_0=1.16+0.005m$ ($m=0,...,7$); (c) $\mathrm{Kn}=0.5$ ($v_w=0$ in Problem II), $p/p_0=1.09+0.01m$ ($m=0,...,9$); (d) $\mathrm{Kn}=0.05$ ($v_w=0$ in Problem II), $p/p_0=1.16+0.005m$ ($m=0,...,7$).

Figure 10

Marginal velocity distribution functions $g$ and $h$ for $\mathrm{Kn}=0.5$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$) (Problem I). (a) $(X_1/D,X_2/D)=(-0.05,0.5)$; (b) $(0.05,0.5)$; (c) $(0.5,0.5)$; (d) $(1,0.5)$.

Figure 11

Marginal velocity distribution functions $(g,h)$ for $\mathrm{Kn}=0.05$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$) (Problem I). (a) $(X_1/D,X_2/D)=(-0.05,0.5)$; (b) $(0.05,0.5)$; (c) $(0.5,0.5)$; (d) $(1,0.5)$.

Figure 12

Distributions of the normal stress $p_{11}/p_0$ along $X_1/D=\mathrm{const}$ for $\mathrm{Kn}=0.5$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$) (Problem I). (a) $X_1/D=0_{-}$, $-0.01$, $-0.05$, and $-0.1$; (b) $X_1/D=0_{+}$, $0.01$, $0.05$, and $0.1$.

Figure 13

Distributions of the normal stress $p_{11}/p_0$ along $X_1/D=\mathrm{const}$ for $\mathrm{Kn}=0.05$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$) (Problem I). (a) $X_1/D=0_{-}$, $-0.01$, $-0.05$, and $-0.1$; (b) $X_1/D=0_{+}$, $0.01$, $0.05$, and $0.1$.

Figure 14

Distributions of the normal stress along each side of the plate for different $\mathrm{Kn}$ in the case of $T_1/T_0=2$ ($L/D=H/D=4$) (Problem I). (a) $p_{11}^{-}=p_{11}(X_1=0_{-},X_2)$ and $p_{11}^{+}=p_{11}(X_1=0_{+},X_2)$; (b) ${\left[p_{11}\right]}_{+}^{-}=p_{11}^{-}-p_{11}^{+}$. In (a), $p_{11}^{+}$ is shown by the solid line and $p_{11}^{-}$ by the dashed line.

Figure 15

Dimensionless mass flow rate $M_f/{\rho }_{av}{\left(2R{T}_0\right)}^{1/2}D$ through the channel vs $\mathrm{Kn}$ for a stationary array ($v_w=0$) in the case of $T_1/T_0=2$ (Problem II). Here, $•$ indicates the result based on the ES model, and the open symbols ($\square$, $▵$, and $\diamond$) connected by the solid lines indicate the results for hard-sphere molecules obtained by the DSMC method.

Figure 16

Illustrative figures explaining the cause of the radiometric force. (a) High-speed molecules reflected on the heated surface, (b) high-speed molecules hitting the incident molecules away, (c) near the edge.

Figure 17

Distribution of the normal component of the local force ${\left[p_{11}\right]}_{+}^{-}$ along the plate for several $\mathrm{Kn}$ ($L/D=H/D=4$) (Problem II). (a) $T_1/T_0=2$ and $v_w/{\left(2R{T}_0\right)}^{1/2}=0$, (b) $T_1/T_0=1$ and $v_w/{\left(2R{T}_0\right)}^{1/2}=0.05$.