Shock wave emission during the collapse of cavitation bubbles

Abstract

Shock wave emission induced by intense laser pulses is investigated experimentally. The present work focuses on the conditions of shock wave emission in glycerine and distilled water during the first bubble collapse. Experimental investigations are carried out in liquids as a function of temperature and viscosity. Comparison is made with the theoretical work of Poritsky (Proc 1st US Natl Congress Appl Mech 813–821, 1952) and Brennen (Cavitation and bubble dynamics, Oxford University Press 1995). To the best knowledge of the authors, this is the first experimental verification of those theories.

Appendices

Appendix 1: Estimation of the sensitivity of the LSM

From Fig. 11 the refraction angle $\epsilon$ can be calculated as:

$$\begin{aligned} \mathrm{sin}(\epsilon )=\frac{1}{1+\frac{{\varDelta }n}{n_\mathrm{w}}}\frac{y}{R}\sqrt{\left( 1+\frac{{\varDelta }n}{n_\mathrm{w}}\right) ^2-\left( \frac{y}{R}\right) ^2} \end{aligned}$$

(7)

Fig. 11

Refraction of the incident beam by a spherical shock wave in water. The index of refraction for water is $n_\mathrm{w}= 1.33$; the index of refraction behind the shock wave is $n_\mathrm{w} + {\varDelta }n(t) = n_\mathrm{s}(t,M_\mathrm{s})$

When $\frac{y}{R}=1$ there is the maximum refraction angle $\epsilon _\mathrm{max}$.

$$\begin{aligned} \mathrm{sin}(\epsilon )=\frac{1}{1+\frac{{\varDelta }n}{n_\mathrm{w}}}\sqrt{\left( 1+\frac{{\varDelta }n}{n_\mathrm{w}}\right) ^2-1} \approx \epsilon _\mathrm{max} \end{aligned}$$

(8)

If $\epsilon _\mathrm{max} \ge r_\mathrm{a}/A$ (acceptance angle of the diode, refer to Fig. 11), the illumination of the diode will be affected by the shock wave refraction.

For water, the sensitivity (in terms of the weakest detectable pressure wave) of the experimental setup in Fig. 11 can be determined by solving the Lorentz–Lorenz equation (9), a lithotripter equation (10) and the Tait equation (11).

The Lorentz–Lorenz equation is

$$\begin{aligned} \frac{n^2-1}{n^2-2}=K(\lambda )\rho , \end{aligned}$$

(9)

where $K(\lambda )=\frac{n_{0}^2-1}{n_{0}^2-2}\frac{1}{\rho _{0}}$.

An equation obtained by the shock wave physics of lithotripters [12] is

$$\begin{aligned} M_\mathrm{s}^2=\frac{r}{n}\frac{r^n-1}{r-1}, \end{aligned}$$

(10)

where $r=\frac{\rho }{\rho _0}=1+\frac{{\varDelta }\rho }{\rho _0}$.

Fig. 12

Temporal development of the bubble radius R (left) and the pressure $P(t)-P_{1}$ (right), for glycerine

Fig. 13

Temporal development of the bubble radius R (left) and the pressure $P(t)-P_{1}$ (right), for water

The Tait equation is

$$\begin{aligned} P=B\left[ \left( \frac{\rho }{\rho _0}\right) ^n-1\right] , \end{aligned}$$

(11)

where P is the pressure. B and n are constants. $B_\mathrm{water}=3000\hbox { bar}$ and $n_\mathrm{water}=7.15$ (it is not the index of refraction!).

Thus, the sensitivity of the measuring arrangement can be estimated:

$$\begin{aligned} \frac{{\varDelta }\rho _\mathrm{w}}{\rho _\mathrm{w}}_\mathrm{min}\approx 3.7\left[ \frac{1}{\sqrt{1-\left( \frac{r_\mathrm{a}}{A}\right) ^2}}-1\right] \end{aligned}$$

(12)

and

$$\begin{aligned} M_\mathrm{s}^2=\frac{1+\frac{{\varDelta }\rho _\mathrm{w}}{\rho _\mathrm{w}}}{7.15}\frac{(1+\frac{{\varDelta }\rho _\mathrm{w}}{\rho _\mathrm{w}})^{7.15}-1}{\frac{{\varDelta }\rho _\mathrm{w}}{\rho _\mathrm{w}}} \end{aligned}$$

(13)

With $r_\mathrm{a} = 0.5\hbox { mm}$ and $A = 500\hbox { mm}$, one obtains $\frac{{\varDelta }\rho _\mathrm{w}}{\rho _\mathrm{w}}_\mathrm{min} \approx 5.56 \times 10^{-6}$ and $M_\mathrm{s} \approx 1.0002$

Appendix 2: Role of the non-condensable gases

In this appendix, we discuss the role of the non-condensable gases for collapsing bubbles. It should be mentioned that, our investigation is focused on extreme initial conditions.

In the case of water, a vapor-filled bubble with an initial radius of $R_{1} = 1.12\hbox { mm}$ and an initial vapor pressure of $P_{\mathrm{v}1} = 385\hbox { mbar}$ ($T_{1} = 75\,^{\circ }\hbox {C}$) shrinks to a bubble with a minimal radius of 0.2 mm and “rebounds” with a closed loop into a new expanding bubble. Although the present measurement method is very sensitive to density gradients, no shock is detected (the bubble does not collapse , even though there might be a weak pressure wave which emits into the liquid). On the contrary, the experiments in water with an initial temperature $T_{1} < 75\,^{\circ }\hbox {C}$ always show a weak collapse shock wave.

From the experimental results [refer to Fig. 13 (left) and Fig. 12 (left)], d R / d t and $d^2{R}/{d{t}}^2$ were calculated and the bubble pressures P(t) were obtained by applying the Rayleigh–Plesset equation (14):

$$\begin{aligned} P(t)-P_{1} = \rho _{1}\left[ R\ddot{R}+\frac{3}{2}\dot{R}^{2}\right] + \frac{2\sigma }{R}+4\eta \frac{\dot{R}}{R} \end{aligned}$$

(14)

with surface tension $\sigma$, viscosity $\eta$, liquid density $\rho _{1}$ and ambient pressure $P_{1}$. This yields a maximum bubble pressure $P_\mathrm{max} \approx 125\hbox { bars}$ for glycerine (Fig. 12, right) and $P_\mathrm{max} \approx 113\hbox { bars}$ for water (Fig. 13, right). Assuming that a non-condensable gas (mainly air) results in $P_\mathrm{max}$, the corresponding initial pressure P(0) can be calculated for glycerine and water, respectively.

1. For glycerine: The non-condensated water vapor and air are responsible for the deceleration of the bubble wall during the (shock wave free) rebound without the bubble-collapsing. Since the water vapor pressure at $30\,^{\circ }\hbox {C}$ is negligibly small ($P(0)_\mathrm{vap} < 1\hbox { mbar}$), non-condensated air must be present. The initial gas pressure $P(0)_\mathrm{gas}$ inside the bubble (assuming an adiabatic process), as the bubble shrinks from $R_{0}$ to $R_\mathrm{min}$, can be calculated by the following equation:

$$\begin{aligned} P(0)_\mathrm{gas} = P_\mathrm{max}\left( \frac{R_\mathrm{min}}{R_{0}}\right) ^{3\gamma } \end{aligned}$$

(15)

with the ratio of specific heat $\gamma = 1.4$, $R_\mathrm{min} =0.08\hbox { mm}$ and $R_{0} = 1.03\hbox { mm}$. Thus $P(0)_\mathrm{gas} \approx 2.7\hbox { mbar}$. This is an acceptable value for the partial pressure of a non-condensable gas (mainly air) in glycerine at $T_{1} \approx 30\,^{\circ }\hbox {C}$, whereupon this initial pressure might be the sum of partial pressures of air and water vapor because glycerine is hygroscopic.

2. For water: the situation is different from that of glycerine. At the initial temperature $T_{1} \ge 75\,^{\circ }\hbox {C}$, there is a relatively high initial vapor pressure $P(0)_\mathrm{vap} = 385\hbox { mbar}$. However, the initial pressure of a non-condensable gas is low. If we assume a non-condensable gas (air and some small fraction of the non-condensated water vapor) stops the bubble wall at the minimum bubble radius, the required initial pressure can be calculated by the relation (15) with $R_\mathrm{min} = 0.2\hbox { mm}$, $R_{0} = 1.12\hbox { mm}$. Thus $P(0)_\mathrm{gas} \approx 81\hbox { mbar}$.

The maximum partial initial pressure of air in water for $T_{1} = 50\,^{\circ }\hbox {C}$ is $P(0)_\mathrm{air} \approx 5.7\hbox { mbar}$ [13] and for $T_{1} = 75\,^{\circ }\hbox {C}$ is even lower. It is comprehensible that the bubble shrinking is stopped by the remaining compressed water vapor inside the bubble which cannot condense fast enough before the bubble reaches the minimum volume [14, 15]. The main reasons for this may be the increasing wall velocity and the decreasing bubble surface area which prevent the transport of the latent heat from the condensed vapor to the water surface.