Comment on “Effect of liquid temperature on sonoluminescence”

It has been suggested that bubble-wall velocities cannot exceed the sound speed in the liquid at the bubble wall [K. Yasui, Phys. Rev. E 64, 016310 (2001).]. Here we show that this upper bound was derived omitting the partial derivatives with respect to time, i.e., assuming that the flow was in the steady state. For collapsing bubbles, however, the steady-flow assumption requires justification, as the partial time derivatives appear to have the same orders of magnitude as the other terms in Euler's and the continuity equations. Furthermore, numerical solutions of the hydrodynamic equations with a hyades hydrocode yielded supersonic velocities in the liquid at and near the collapsing bubble. We also show the spatial distributions of pressure, density, sound speed, and mass flux density around the supersonically collapsing bubble.

Figure 1

Bubble radius $R\left(t\right)$, bubble-wall velocity $U\left(t\right)$, sound speed in the liquid at the bubble wall $C\left(t\right)$, and the bubble-wall Mach number $U/C$. The bubble-wall velocity $U\left(t\right)$ was negative during the implosion phase ($t<0$) until the moment of the collapse (dashed line). A dotted-line rectangle highlights the time interval ($-36.7$ to $-4.9$ ns) when the bubble-wall velocity was supersonic ($\left|U\right|>C$).

Figure 2

Velocity magnitude $\left|u\right|$ and sound speed $c$ in the liquid at three moments in time: $t_1=-14.7\text{ns},t_2=-12.7\text{ns}$, and $t_3=-10.7\text{ns}$. Bubble-wall positions are shown by the dashed lines: $r_1=68.5\mu \mathrm{m}$ at $t_1,r_2=63.3\mu \mathrm{m}$ at $t_2$, and $r_3=57.8\mu \mathrm{m}$ at $t_3$. Bubble's interior (to the left of the dashed lines) is not shown. Dotted lines mark the locations of $\left|u\right|=c:73.7\mu \mathrm{m}$ at $t_1,67\mu \mathrm{m}$ at $t_2$, and $59.2\mu \mathrm{m}$ at $t_3$. The highlights (between the dashed and the dotted lines) indicate regions with supersonic velocities.

Figure 3

Mass flux density $j$, velocity magnitude $\left|u\right|$, sound speed $c$, pressure $P$, and density $\rho$ in the liquid near the bubble wall (leftmost position) at $t=-14.7\text{ns}$. The supersonic region (highlighted) extends from the bubble wall to $r=73.7\mu \mathrm{m}$ (vertical dotted line).

Figure 4

Sound speed vs pressure calculated using Eq. (17) with three sets of parameters: (i) $B=300$ MPa and $n=7$ (red dashed line, Ref. [34]), (ii) $B=314$ MPa and $n=7$ (dotted blue, Ref. [35]), and (iii) $B=295.5$ MPa and $n=7.44$ (solid black, Ref. [36]).