Influence of a small amount of noncondensable gas on shock wave generation inside a collapsing vapor bubble

In this paper, the influence of a small amount of noncondensable gas on shock wave generation inside a collapsing vapor bubble is examined by combining numerical analysis of a bubble dynamics equation and of the Boltzmann equation for gas flow inside a bubble. We show the importance of the high Knudsen number in the flow, i.e., the nonequilibrium gas flow, to the temperature field inside the collapsing bubble. We also show that the maximum mean Knudsen number inside the collapsing bubble can be evaluated from the initial Knudsen number and the initial number density ratio of noncondensable gas and vapor molecules. As a result, we conclude that a small amount of noncondensable gas strongly affects the temperature field inside the collapsing bubble, preventing shock wave generation inside the bubble in the final stage of collapse.

Figure 1

Temporal evolution of (a) bubble radius, $R\left(t\right)$, and (b) bubble wall velocity, $\dot{R}\left(t\right)$, for each value of ${\overline{n}}_0^r$ when the bubble wall velocity $|\dot{R}\left(t\right)|/c_0\le 2.0$. The enlarged view for each figure shows the bubble radius and bubble wall velocity in the final stage of collapse. The bubble radius decreases with time, and exhibits the same tendency in spite of the difference in ${\overline{n}}_0^r$. However, in the final stage, there is a slight difference shown in enlarged view.

Figure 2

Temperature fields inside a collapsing bubble for each ${\overline{n}}_0^r$ when the bubble wall velocity is $1.0\le |\dot{R}\left(t\right)|/c_0\le 2.0$. The filled circles for each line denote the position of the bubble wall at each time. $r/R_0=0$ denotes the bubble center. In the case of small ${\overline{n}}_0^r$, the temperature near the bubble wall becomes higher. On the other hand, in the case of large ${\overline{n}}_0^r$, the temperature at the center of the bubble takes the maximum value.

Figure 3

Vapor and noncondensable gas temperature fields inside a collapsing bubble for each ${\overline{n}}_0^r$ when the bubble wall velocity is $1.0\le |\dot{R}\left(t\right)|/c_0\le 2.0$. The filled circles for each line denote the position of the bubble wall at each time. Additionally, $r/R_0=0$ denotes the bubble center. In the case of small ${\overline{n}}_0^r$, the vapor and noncondensable gas temperatures take different profile. On the other hand, in the case of large ${\overline{n}}_0^r$, these temperature profiles agree with each other except for the slight difference at the bubble wall.

Figure 4

Number density fields of vapor and noncondensable gas inside a collapsing bubble for each ${\overline{n}}_0^r$ when the bubble wall velocity is $1.0\le |\dot{R}\left(t\right)|/c_0\le 2.0$. The filled circles for each line denote the position of the bubble wall at each time. Additionally, $r/R_0=0$ denotes the bubble center. The noncondensable gas molecules drift on the bubble wall surface in spite of the difference of ${\overline{n}}_0^r$ as the bubble collapses.

Figure 5

Temporal evolution of the mean Knudsen number inside the collapsing bubble for each ${\overline{n}}_0^r$: (a) the Knudsen number obtained from the numerical simulation and (b) that obtained from Eq. (9). The enlarged view is shown in each figure in the final stage of bubble collapse. As shown in the figure, Eq. (9) can evaluate the value of the mean Knudsen number inside the bubble in the case of large ${\overline{n}}_0^r$.

Figure 6

Temporal evolution of the ratio of the number of noncondensable gas molecules, $N_g$, and that of vapor molecules, $N_v$, inside the collapsing bubble for each ${\overline{n}}_0^r$ obtained from the numerical simulation. In the final stage of the collapse, the number of noncondensable gas becomes larger than that of vapor molecules in the cases of ${\overline{n}}_0^r={10}^{-3}$ and ${10}^{-4}$.