Mechanics of gas-vapor bubbles

Most bubbles contain a mixture of vapor and incondensible gases. While the limit cases of pure vapor and pure gas bubbles are well studied, much less is known about the more realistic case of a mixture. The bubble contents continuously change due to the combined effects of evaporation and condensation and of gas diffusion in the liquid and in the bubble. This paper presents a model for this situation and illustrates by means of examples several physical processes that can occur: a bubble undergoing a temporary pressure reduction, which makes the liquid temporarily superheated; a bubble subjected to a burst of sound; and a bubble continuously growing by rectified diffusion of heat in the presence of an incondensible gas.

Figure 1

A pure vapor bubble in water with no dissolved gas at $90{}^{\circ }\mathrm{C}$ and 101.3 kPa ambient pressure quickly collapses (dotted line). The collapse is markedly slowed down if dissolved ${\text{CO}}_2$ is present in the liquid. The solid line is for water saturated with ${\text{CO}}_2$ at $100{}^{\circ }\mathrm{C}$ and the dashed line for water saturated with ${\text{CO}}_2$ at $25{}^{\circ }\mathrm{C}$. In this latter case the liquid is supersaturated at its temperature of $90{}^{\circ }\mathrm{C}$ and, after the condensation of the vapor, the bubble begins to grow by gas diffusion.

Figure 2

Radius vs time of a gas-vapor bubble in ${\text{CO}}_2$-saturated water at $100{}^{\circ }\mathrm{C}$ and 101.3 kPa. The liquid temperature is $100{}^{\circ }\mathrm{C}$ and the ambient pressure 101.3 kPa except for $5\text{ms}<t<\text{10}\text{ms}$, when it is reduced so that the liquid is superheated by $5{}^{\circ }\mathrm{C}$. The initial bubble radius $R_0$ is $100\mu \mathrm{m}$.

Figure 3

Total gas mass in the bubble vs time for the case of Fig. 2.

Figure 4

Vapor mass in the bubble vs time for the case of Fig. 2.

Figure 5

Global gas mass fraction in the bubble vs time for the case of Fig. 2.

Figure 6

Radius vs time of a gas-vapor bubble in water saturated with ${\text{CO}}_2$ at $25{}^{\circ }\mathrm{C}$. The liquid temperature is $100{}^{\circ }\mathrm{C}$. The ambient pressure is $101.3\times (1+0.3cos\omega t)$ kPa for $0\le t<30$ ms and 101.3 kPa thereafter; the sound frequency is $\omega t/2\pi =1$ kHz. The initial bubble radius $R_0$ is $100\mu \mathrm{m}$.

Figure 7

Total gas mass in the bubble vs time for the case of Fig. 6. Since the liquid is saturated with ${\text{CO}}_2$ at $25{}^{\circ }\mathrm{C}$, it is supersaturated at the temperature of $100{}^{\circ }\mathrm{C}$ to which this figure refers, and gas diffuses into the bubble.

Figure 8

Total vapor mass in the bubble vs time for the case of Fig. 6.

Figure 9

Shown on top is the growth of a bubble containing ${\text{CO}}_2$ and water vapor (upper line) compared with that of a pure vapor bubble. The bottom panel shows an enlarged picture of the interval $40\le \omega t/2\pi \le 60$, which contains the time interval in the course of which bubble executes large-amplitude oscillations. Here $T_{\infty }=100{}^{\circ }\mathrm{C},P_{\infty }=101.3$ kPa, the acoustic pressure is 30.4 kPa (0.3 atm), $R\left(0\right)=0.2$ mm, $\omega /2\pi =400$ Hz, and the bubble resonant radius is $R_{\mathrm{res}}=7.27$ mm; the liquid is saturated with dissolved gas at $25{}^{\circ }\mathrm{C}$ and is therefore supersaturated by a factor of approximately 16 at the liquid temperature used for the calculation; the initial vapor mass fraction inside the bubble is 25%.

Figure 10

Detail of Fig. 9 with circles marking the times corresponding to the mass fraction distributions shown in Fig. 11: $A,\omega t/2\pi =48.66$; $B,\omega t/2\pi =48.74$; and $C,\omega t/2\pi =49.19$.

Figure 11

Gas mass fraction distribution in the bubble at the three instants of time marked in Fig. 10. Notice the strong accumulation of gas near the interface when the vapor condenses during the compression cycle.

Figure 12

Bubble surface temperature during a portion of the oscillations shown in Fig. 9.

Figure 13

Bubble internal pressure during a portion of the oscillations shown in Fig. 9.

Figure 14

Comparison of the vapor-gas velocity averaged over the bubble volume as computed from (9) without the $E_G$ term (solid line) and corrected by including this term estimated as explained in the text.