3.2.1.1 Pulsed jet formation

To explain the formation of a pulsed jet produced in the range of $l_{0}^{\ast }<1$ , where bubble penetration occurs, we show figure 7, in which two series of high-speed photographs of two tandem bubbles are presented with parameter combinations of equivalent bubble size ratio $\unicode[STIX]{x1D709}=1.3$ , and almost equivalent dimensionless bubble distances $l_{0}^{\ast }\sim 0.9$ , but different values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ of 0.12 in ( $a$ ) and 0.27 in ( $b$ ). These correspond to the times from which the interaction started of $t^{\ast }=0.224$ ( $R_{A}^{\ast }=0.80$ ) in ( $a$ ) and 0.486 ( $R_{A}^{\ast }=0.90$ ) in ( $b$ ). We notice a significant difference in the radii of bubble A between ( $a$ ) and ( $b$ ) at each instant of the generation of bubble B. This leads to differences in the flow distance from the birth point of bubble B to the upper surface of bubble A, being longer in ( $a$ ) than that in ( $b$ ). For the case of the same $\unicode[STIX]{x1D709}$ values, therefore, the radial flow velocity at the upper surface of bubble A becomes lower in ( $a$ ) than that in ( $b$ ) because the radial velocity decreases with inverse proportionality to the square of flow distance. Consequently, the fluid force induced by bubble B’s motion immediately after its generation must be smaller at the upper surface of bubble A in ( $a$ ) compared with that in ( $b$ ). Furthermore, it is well recognised that a laser-induced bubble contains gas, initially at a high-pressure state from the degeneration of the plasma that drove the formation and growth of the bubble. Disregarding the surface tension and assuming that the gas obeys an adiabatic law, the gas pressure inside bubble A, $p_{g,A}^{\ast }$ , is expressed with the initial gas pressure ratio of $\unicode[STIX]{x1D6FC}=100$ at $R_{0,A}^{\ast }=0.1$ as follows (Pearson Reference Pearson2002):

where $p_{g,A}^{\ast }=p_{g,A}/\unicode[STIX]{x0394}p$ , $\unicode[STIX]{x1D6FC}=p_{0}/\unicode[STIX]{x0394}p$ , $R_{0,A}^{\ast }=R_{0}/R_{max,A}$ , $R_{A}^{\ast }=R_{A}/R_{max,A}$ , $\unicode[STIX]{x1D705}$ : ratio of specific heats of gas ( $=1.4$ ), $p_{0}$ : initial gas pressure, $R_{0}$ : initial bubble radius, $R_{A}$ : bubble A’s radius at time $t$ . From (3.1), it is readily seen that the state, $p_{g,A}^{\ast }=1$ , is satisfied at 0.026 after the birth of bubble A, where the pressure at the surface of bubble A balances with the pressure difference, $\unicode[STIX]{x0394}p$ ( $=\,p_{\infty }-p_{v}$ ), when the dimensionless bubble radius is 0.30. Of course, the state of $p_{g,A}^{\ast }=0$ implies conditions for a Rayleigh bubble, wherein the bubble interior is filled with vapour at a constant pressure. Therefore, the internal gas pressure of a Rayleigh-like single bubble differs significantly from that of a gas bubble during early bubble growth. It is, therefore, clear that the gas pressures in the first frames for both cases ( $a$ ) and ( $b$ ) are sufficiently low, namely $p_{g,A}^{\ast }<0.02$ at $t^{\ast }=0.24$ ( $R_{A}^{\ast }=0.81$ ) in ( $a$ ) and $p_{g,A}^{\ast }<0.01$ at $t^{\ast }=0.53$ ( $R_{A}^{\ast }=0.98$ ) in ( $b$ ). Consequently, the upper surface of bubble A can be easily deformed by the increasing momentum of water flow caused by the motion of bubble B. As seen in ( $b$ ), a conical pulsed jet is formed in strong interactions. The conical jet formation mechanism seems to be similar to the process of the formation of a water spike with a conical tip (Blake & Gibson Reference Blake and Gibson1981; Longuet-Higgins Reference Longuet-Higgins1983). During the pronounced penetration of the lower surface of bubble B into bubble A, the annular water flow passage between the two bubbles becomes narrower, forming a low volume and quasi-confined region of water, which is essential to the formation of a pulsed jet. A quasi-confined water volume is attained earlier for larger bubble size ratios or shorter distances between the two bubble generation sites. Under these conditions, a pulsed jet with a very small amount of water can be formed. The timing of jet impact on the lower surface of bubble A strongly depends on the time difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ between the two bubble generations. In ( $b$ ), the jet impact onto the lower surface of bubble A occurs at 1.02, while it is delayed until 1.28 in the case of ( $a$ ).

As already described in § 3.1.2, and seen again at 0.78 in figure 7(b), a weak spray is observed at the tip of the conical jet. Spray formation in tandem-bubble systems where the bubbles are produced at different times is an interesting research subject. However, some difficulties remain in the detailed elucidation of the spray formation mechanism because few observations exist to relate the cause and effect of spray formation. In the present report, therefore, spray formation is described only briefly. Figure 8 shows two examples of a pair of frames. The first frame corresponds to the state immediately after bubble B’s generation, and the second frame indicates the phenomenon immediately before spray formation. Obviously, we can observe several slender projections, indicating two at 0.64 in ( $a$ ) as depicted by a white arrow, and four in ( $b$ ). These projections are broken, finally transforming into spray. The generation mechanism of the projections remains in question.

Figure 8.

Secondary cavitation and jet ejections: ( $a$ )  $\unicode[STIX]{x1D709}=1.1$ ( $R_{max,A}=1.9$  mm,  $R_{max,B}=2.1$  mm),  $l_{0}^{\ast }=0.74$ ( $l_{0}=2.9$  mm), and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.18$ ( $T_{A}=340~\unicode[STIX]{x03BC}\text{s}$ , $\unicode[STIX]{x0394}T=60~\unicode[STIX]{x03BC}\text{s}$ ); ( $b$ )  $\unicode[STIX]{x1D709}=1.3$ ( $R_{max,A}=1.8$  mm,  $R_{max,B}=2.3$  mm),  $l_{0}^{\ast }=0.72$ ( $l_{0}=2.9$  mm) and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.18$ ( $T_{A}=329~\unicode[STIX]{x03BC}\text{s}$ , $\unicode[STIX]{x0394}T=60~\unicode[STIX]{x03BC}\text{s}$ ). The dimensionless times noted under each frame are measured from the generation of bubble A, respectively. The white scale bars in ( $a$ ) and ( $b$ ) each denote 1 mm.

We approach the situation in ( $a$ ) in a simple manner. First, we note that, at 0.37 in the first frame, the dimensionless gas pressure inside bubble A with the radius of 1.7 mm is less than 0.01 as calculated by using (3.1), whereas bubble B is earlier in its growth period at 0.047 (i.e. 9  $\unicode[STIX]{x03BC}\text{s}$ ) after its generation. Therefore, approximately 8  $\unicode[STIX]{x03BC}\text{s}$ have elapsed since the interaction of the shock wave with the upper surface of the low-pressure bubble A, because the shock wave travels approximately 1.5 mm to reach the surface, which requires the time of 1  $\unicode[STIX]{x03BC}\text{s}$ . Consequently, a local pressure reduction is achieved in the region near the upper surface immediately after the passage of the reflected expansion wave, creating multiple black spots as seen in ( $a$ ). We now consider the central black spot, indicated by a white arrow. If this is a bubble with the maximum radius of $R_{max,a}\sim 0.12$  mm, whose period as a Rayleigh bubble is approximately 11  $\unicode[STIX]{x03BC}\text{s}$ , it should be located near the curved upper surface of bubble A with the relative distance of $\unicode[STIX]{x1D6FE}\sim 1$ . Pearson (Reference Pearson2002) elucidated that the length of a water spike developed at the surface resulting from the bubble motion linearly increased with time with a constant velocity of $V_{je}^{\ast }\sim 0.45$ over the maximum bubble state. Because the free surface deformation is about 0.8 at the maximum bubble state (Blake & Gibson Reference Blake and Gibson1987), the length of jet motion $L_{je}^{\ast }$ in the second frame of ( $a$ ) is determined as 2.7 for the dimensionless elapsed time of 4.2 when $R_{max,a}$ is used as the scale length (i.e. $0\rightarrow 4.2$ for $R_{max,a}=0.12$  mm). Since $R_{max,a}\sim 0.12$ mm, we can determine the estimated jet length of 0.32 mm, which approaches the value of 0.26 mm measured from the image in ( $a$ ). The slender projection observed at 0.64, therefore, must be a micro-jet ejection resulting from the development of the free surface caused by the motion of a minute bubble.

Figure 9 shows a special case of strong interaction between two bubbles. The parameter conditions are $\unicode[STIX]{x1D709}=1.6$ , $l_{0}^{\ast }=0.5$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ . Here, the instant at which bubble B is generated is defined as zero, and the elapsed dimensionless times are noted below each frame. Because $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ is taken as 0.5, the pressure inside bubble A is estimated to be equivalent to the vapour pressure. Immediately after 0.0092, the tip velocity of the lower surface of bubble B varies from $-440~\text{m}~\text{s}^{-1}$ (i.e. $V_{ls,B}^{\ast }=-44$ ) to $-120~\text{m}~\text{s}^{-1}$ (i.e. $V_{ls,B}^{\ast }=-12$ ), both of which are measured using two successive frames with the frame interval of 2  $\unicode[STIX]{x03BC}\text{s}$ . Thus, a large change in the momentum of the water near the lower surface of bubble B occurs much earlier in the growth of bubble B, causing a small amount of water to move forward. As clearly seen at 0.028, a black substance, located ahead of the tip of bubble B, is continuously pressed by the growing bubble B to form a conical jet. The jet moves downward with a velocity of $-194~\text{m}~\text{s}^{-1}$ (i.e. $V_{j,A}^{\ast }=-19.5$ ). During this process, an annular water flow passage between the two bubbles narrows to create a water film, due to the high-speed movement of the extending lower part of bubble B. A little water flows into the region of the pulsed jet through the annular flow passage. A spray appears at the tip of the conical jet starting at 0.028 and spreads outward over time. The water film is ruptured at 0.074, creating a circle somewhere on the lower part of bubble B, as indicated by the white arrows in figure 9. After this occurs, the circle radius is increased with increasing time, as seen at 0.11; instabilities propagate, spreading over the entire water film. The head shape of the lower surface of bubble B gradually becomes rounded, which is similar to the behaviour observed by Lauterborn & Hentschel (Reference Lauterborn and Hentschel1985). Thus, when $\unicode[STIX]{x1D709}$ is large, most of the water initially entrapped to form a jet is instead transformed into mist. Consequently, no jet impact occurs, but the tip of bubble B arrives at the lower surface of bubble A in the form of a pseudo-impact, or surface contact between the two bubbles.

Figure 9.

Jet development, spray formation and water film rupture due to strong bubble interaction: $\unicode[STIX]{x1D709}=1.6$ ( $R_{max,A}=2.2$  mm,  $R_{max,B}=3.4$  mm),  $l_{0}^{\ast }=0.5$ ( $l_{0}=2.8$  mm) and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ ( $T_{A}=397~\unicode[STIX]{x03BC}\text{s}$ , $\unicode[STIX]{x0394}T=200~\unicode[STIX]{x03BC}\text{s}$ ). The instant of the formation of bubble B is defined as zero and the first frame corresponds to the dimensionless time $t^{\ast }=0.92$ (i.e. 200  $\unicode[STIX]{x03BC}\text{s}$ ) after the generation of bubble A. White arrows indicate the circle part of film rupture. The black scale bar denotes 1 mm.

3.2.1.2 Effects of interaction parameters on jet impact velocity

Using two similar-sized bubbles with an uncertainty of $\pm 15\,\%$ (i.e. $\unicode[STIX]{x1D709}=1.0\pm 0.15$ ), the jet velocity, $V_{j,A}^{\ast }$ , in impacting the lower surface of bubble A was examined as a function of the time difference between the generation of two tandem bubbles, $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ , for the two distance parameters $l_{0}^{\ast }=0.75$ (▴) and 1.0 (●), where a spatial uncertainty of 15 % was considered in determining $l_{0}^{\ast }$ . Figure 10 shows the results and the details of these data points can be found in the figure caption. It is known that, for two bubbles generated simultaneously with equal input energies, i.e. when $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0$ (although it is difficult to generate two bubbles of exactly equal size, because the one-way focusing of the laser beam at the focal point tends to generate a non-spherical plasma volume along the optical axis), two jets are produced in opposite directions resulting from the creation of a high-pressure stagnation region between the two bubbles. Therefore, in the case where the first produced bubble, namely bubble A, is located lower, the jet formed in it is directed upward, which is a similar trend in jet direction observed for a single bubble collapsing near a rigid planar boundary above the bubble. Han et al. (Reference Han, Köhler, Jungnickel, Mettin, Lauterborn and Vogel2015) numerically obtained a jet velocity of $V_{j,A}^{\ast }=7.0$ for the parameter combination of $\unicode[STIX]{x1D709}=1.0$ , $l_{0}^{\ast }=0.87$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0$ . Although a planar stagnation region could be formed in the flow field between the two simultaneously produced bubbles with $\unicode[STIX]{x1D709}=1.0$ , a convex upward stagnation region may be formed in the case of $\unicode[STIX]{x1D709}>1.0$ ; the flow field created around the lower bubble in this case looks similar to that formed by a single bubble near a concave rigid wall (Tomita et al. Reference Tomita, Robinson, Tong and Blake2002). If bubble B is generated with a time delay $\unicode[STIX]{x0394}T$ (i.e. $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }>0$ ), bubble A grows like a Rayleigh bubble until bubble B is generated, but the symmetry of the flow field between the two bubbles is quickly lost after interaction with the delayed bubble B, causing a discrepancy in the bubble shapes; accordingly, the impact velocity of a jet in bubble A may drop drastically when $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ is slightly shifted from zero. Even under these conditions, if the gas pressure at the time $t^{\ast }$ ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ ) satisfies the condition $p_{g,A}^{\ast }\gg 1$ , the jet produced in bubble A remains directed upward because the flow field around the bubble would be similar to that created when two simultaneously produced bubbles with $\unicode[STIX]{x1D709}<1.0$ collapse; it is known that the flow field in the latter is similar to that formed by a single bubble near a convex rigid wall (Tomita et al. Reference Tomita, Robinson, Tong and Blake2002). Pearson (Reference Pearson2002) numerically found an upward-directed jet at the dimensionless time of 0.05, corresponding to $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.027$ .

Figure 10.

Effect of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ on the velocity of a jet $V_{j,A}^{\ast }$ impacting the lower surface of bubble A ( $\unicode[STIX]{x1D709}=1.0$ ). ● denotes the case for $l_{0}^{\ast }=1.0$ and ▴ denotes the case for $l_{0}^{\ast }=0.75$ . ▪ indicates the case for $l_{0}^{\ast }=1.0$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ (Han et al. Reference Han, Köhler, Jungnickel, Mettin, Lauterborn and Vogel2015);  $\odot$ ( $l_{0}^{\ast }=0.95$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0$ ) (Tomita & Shima Reference Tomita and Shima1990) and $\oplus$ ( $l_{0}^{\ast }=0.87$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0$ ) (Han et al. Reference Han, Köhler, Jungnickel, Mettin, Lauterborn and Vogel2015) are experimentally and numerically obtained data for two simultaneously generated bubbles, respectively.

 ( $\unicode[STIX]{x1D6FE}=1.0$ ) (Philipp & Lauterborn Reference Philipp and Lauterborn1998) and

 ( $\unicode[STIX]{x1D6FE}=0.75$ ) (Brujan et al. Reference Brujan, Nahen, Schmidt and Vogel2001) represent the results for a single bubble near a solid boundary ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0$ ) and

 ( $\unicode[STIX]{x1D6FE}=0.94$ ) (Robinson et al. Reference Robinson, Blake, Kodama, Shima and Tomita2001) and

 ( $\unicode[STIX]{x1D6FE}=0.82$ ) (Kodama Reference Kodama1992) correspond to experiments for a single bubble near a free surface ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ ).

Considering the calculated result obtained by Pearson (Reference Pearson2002) and the data of $V_{j,A}^{\ast }$ designating a negative value for a relatively wide range of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ , exceeding the data point of 0.12, it is reasonable to estimate that the direction of a jet must change sign from positive to negative in the range of $0.027<\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{c}^{\ast }<0.12$ . Within this range, the condition $V_{j,A}^{\ast }=0$ would be satisfied for the critical birth-time difference, $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{c}^{\ast }$ , although no data for exact values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{c}^{\ast }$ have been reported. It would be interesting to know whether the existence of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{c}^{\ast }$ implies the zero conditional state of a Kelvin impulse (Blake, Leppinen & Wang Reference Blake, Leppinen and Wang2015), which allows short-duration changes in the liquid mass and momentum, induced by the motion of each laser-produced vapour bubble. The Kelvin impulse is considered an important indicator for determining the direction of bubble motion along the symmetric axis. Therefore, it is estimated that, in the range for small birth-time differences of the two bubbles, i.e. $0<\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }<\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{c}^{\ast }$ , the jet directed upward forms late in the collapse of bubble A because the water sandwiched between the two bubbles is subjected to great pressure from the inertia of the water driven by the high-pressure gases inside the two bubbles during their earlier growth periods. In fact, we observed an upward-directed jet produced during the collapse of bubble A when the birth-time difference of the two bubbles, $\unicode[STIX]{x0394}T$ , was 20  $\unicode[STIX]{x03BC}\text{s}$ , although we could not determine the parameter values, $\unicode[STIX]{x1D709}$ , $l_{0}^{\ast }$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ , because we could not calculate the exact values of the maximum bubble radii for bubble A and bubble B.

On the other hand, for the case of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }>\unicode[STIX]{x0394}\unicode[STIX]{x1D703}_{c}^{\ast }$ , when the gas pressure inside the growing bubble A is reduced to a value below the hydrostatic pressure at the instant of the generation of bubble B, the pressure gradient in the water near the upper surface of bubble A would overcome the inertia of the water driven by the growth of bubble A. This would produce a jet directed downward during the growth period of bubble A. As $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ increases up to $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ , the gas pressure within bubble A decreases to a level significantly lower than the hydrostatic pressure; for instance, the dimensionless gas pressure inside bubble A is less than 0.01 when $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.3$ . Therefore, when the upper surface of bubble A, regarded as a curved free surface, is subjected to pressure from the rapid growth of bubble B, a water spike is produced to form a conical jet directed downward into the low-pressure gas within bubble A.

As is well known, when an isolated vapour bubble grows near a free surface, the surface becomes elevated and evolves to form a water spike with a conical tip (Blake & Gibson Reference Blake and Gibson1981; Longuet-Higgins Reference Longuet-Higgins1983). The motion of this water spike strongly depends on the Kelvin impulse, whose absolute value is known to increase as the stand-off parameter $\unicode[STIX]{x1D6FE}$ is decreased for the case of a single bubble (Blake et al. Reference Blake, Leppinen and Wang2015). Pearson et al. (Reference Pearson, Cox, Blake and Otto2004) investigated bubble interactions near a free surface and elucidated that the maximum Kelvin impulse was attained not during final collapse, but as the bubble became entrained in the raised free surface. They also demonstrated that the Kelvin impulse was almost constant with less variation in time for the dimensionless times $t^{\ast }$ from 0.458 to 0.915 in the range of $0.8<\unicode[STIX]{x1D6FE}<1.3$ (i.e. $0.8<l_{0}^{\ast }<1.3$ for the present study employing two tandem bubbles with $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ ). A similar physical situation may occur when two tandem bubbles are produced at different times, $0.25<\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }<0.5$ (corresponding to $0.458<t^{\ast }<0.915$ ), under the bubble distances of $0.8<l_{0}^{\ast }<1$ . At $t^{\ast }=0.458$ , the radius of bubble A reaches $R_{A}^{\ast }=0.885$ ; using (3.1), we obtain the corresponding gas pressure inside bubble A as 0.011 ( ${<}1$ ). Therefore, if bubble B is generated very close to bubble A, the upper surface of bubble A could deform notably because of the large pressure difference between the inside and outside of the bubble, as well as the increasing stagnation pressure. Consequently, pulsed jets with almost the same values of $V_{j,A}^{\ast }$ were generated (see figure 10). On the other hand, in the range of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }>0.5$ , flow superposition occurs because of the radial flow induced by the explosive growth of bubble B coupled with the inward flow from the collapse motion of bubble A, forming a so-called liquid jet instead of a pulsed jet. Therefore, the absolute values of the jet velocities in these regions are somewhat larger than those generated in the region $0.25<\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }<0.5$ . Meanwhile, for smaller bubble distances (i.e. $l_{0}^{\ast }=0.75$ ), the absolute value of water velocity becomes larger than the case of $l_{0}^{\ast }=1$ ; the resultant stagnation pressure would act on the surface of the large, low-pressure bubble A, thereby producing jet impact velocities higher than those for $l_{0}^{\ast }=1$ , as clearly seen in figure 10.

Figure 11.

Effect of the dimensionless bubble distance, $l_{0}^{\ast }$ , on the velocity of a jet impacting the lower surface of bubble A, $-V_{j,A}^{\ast }$ , denoted by ● for the parameter combination $\unicode[STIX]{x1D709}=1.0$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.3$ . Three data points, denoted by ▵, are numerically obtained for two bubbles at $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ (Han et al. Reference Han, Köhler, Jungnickel, Mettin, Lauterborn and Vogel2015), and one data point (○) indicates the case $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.27$ (Pearson Reference Pearson2002). Furthermore, two data points for a single bubble near a free surface ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ ) are added for comparison;

corresponds to the experimental value (Robinson et al. Reference Robinson, Blake, Kodama, Shima and Tomita2001) and

corresponds to the calculated value (Pearson Reference Pearson2002).

Figure 11 shows the relationship between the value of the dimensionless jet impact velocity, $-V_{j,A}^{\ast }$ , and the dimensionless bubble distance, $l_{0}^{\ast }$ , for a specified parameter, $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.3$ , where the direction of $V_{j,A}^{\ast }$ , is negative as shown in figure 10. As defined in (2.1), $l_{0}^{\ast }$ depends on $\unicode[STIX]{x1D709}$ . Therefore, we investigated the effect of $l_{0}^{\ast }$ on $V_{j,A}^{\ast }$ for a fixed bubble size ratio $\unicode[STIX]{x1D709}=1.0\pm 0.2$ ; six data points denoted by (●) are available to plot in the figure. To compare our experiments with calculations, one available result (○) obtained for $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.27$ (Pearson Reference Pearson2002) is included. Three data points numerically obtained for $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ , denoted by (▵) (Han et al. Reference Han, Köhler, Jungnickel, Mettin, Lauterborn and Vogel2015) are included for comparison. Two data points for the jet impact velocities obtained for single bubbles collapsing near free surfaces, which can create flow fields similar to those induced by the motion of two bubbles generated with $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ , are also added for reference. One of the two is an experimental value (Robinson et al. Reference Robinson, Blake, Kodama, Shima and Tomita2001), denoted by (

), and the other corresponds to a calculated value (Pearson Reference Pearson2002), denoted by (). Although our data are restricted to the range $l_{0}^{\ast }\leqslant 1.1$ , it is apparent that $V_{j,A}^{\ast }$ strongly depends on $l_{0}^{\ast }$ , and its absolute value increases as $l_{0}^{\ast }$ decreases in the range $l_{0}^{\ast }<1.0$ . When $l_{0}^{\ast }=0.5$ , the absolute jet velocity can reach approximately twice its value at $l_{0}^{\ast }=1.0$ , and the minimum value of $-V_{j,A}^{\ast }$ is achieved at $l_{0}^{\ast }\sim 1.0$ . Good agreement obtained for $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }\sim 0.3$ is found between the calculated result obtained by (○) for $l_{0}^{\ast }=0.8$ (Pearson Reference Pearson2002) and our experimental result at the same $l_{0}^{\ast }$ value. As seen in figure 4(a,c), in this region of $l_{0}^{\ast }<1.0$ , a pulsed jet is created by the accelerated increase in momentum of a quasi-confined water mass located ahead of the lower tip of bubble B, predominantly resulting from the increase in the lower tip velocity of bubble B. As $l_{0}^{\ast }$ decreases, a confined water mass is accelerated significantly to reach a much higher downward velocity. Meanwhile, when $l_{0}^{\ast }=1.0$ , less difference occurs in the values of $V_{j,A}^{\ast }$ between the two cases of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.3$ , as denoted by (●), for the present experimental value, and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ , denoted by (▵), for the numerically calculated result obtained by Han et al. (Reference Han, Köhler, Jungnickel, Mettin, Lauterborn and Vogel2015). For these parameter ranges, the bubble pressures drop sufficiently below the hydrodynamic pressure, namely less than 0.01, and the distances from the birth point of bubble B to the upper surface of bubble A are almost the same for the two cases; subsequently a certain rate of change of water momentum is imposed on the upper surface of bubble A, thereby producing jets with similar impact velocity magnitudes for a specified $l_{0}^{\ast }$ , such as $l_{0}^{\ast }=1.0$ . In contrast, for a region $l_{0}^{\ast }>1.0$ , the so-called jet, instead of a pulsed jet, now forms inside bubble A; its velocity, $V_{j,A}^{\ast }$ , becomes higher in the negative direction with increasing $l_{0}^{\ast }$ . The flow field around bubble A can form because of the superimposition of the flow field induced by the motion of bubble B on that induced by the motion of bubble A. The existence of bubble B may block the flow induced by the motion of bubble A. For shorter $l_{0}^{\ast }$ in the region $l_{0}^{\ast }>1.0$ , the blockage effect increases and more water particles are blocked, especially during the collapse process of bubble A. Subsequently, a small mass of water creates a jet with a relatively low velocity. However, when $l_{0}^{\ast }$ becomes larger, the blockage effect decreases; the water mass accumulated near the upper portion of bubble A may be adequate to produce a jet of higher velocity during the collapse process, although a weak flow effect from the growth motion of bubble B exists because the radial flow velocity decreases with proportionality inverse to the square of the distance.

Figure 12.

Effect of the bubble size ratio, $\unicode[STIX]{x1D709}$ , on the velocity of a jet impacting the lower surface of bubble A, $-V_{j,A}^{\ast }$ , for two parameter combinations; ●: $l_{0}^{\ast }=0.75$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.3$ ; ▲: $l_{0}^{\ast }=0.6$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ . ○ denotes the calculated result for $l_{0}^{\ast }=0.8$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.27$  (Pearson Reference Pearson2002).

Regarding applications of tandem-bubble sonoporation, an example concerning two tandem bubbles on the micrometre scale is worth considering. Sankin et al. (Reference Sankin, Yuan and Zhong2010) performed an experiment using these two-dimensional bubbles. For the parameter combination of bubble size ratio $\unicode[STIX]{x1D709}\approx 1$ ( $R_{max}=25~\unicode[STIX]{x03BC}\text{m}$ ), dimensionless bubble distance $l_{0}^{\ast }=0.87$ ( $l_{0}=40~\unicode[STIX]{x03BC}\text{m}$ ) and birth-time difference between the two bubbles $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.31$ ( $\unicode[STIX]{x0394}T=4~\unicode[STIX]{x03BC}\text{s}$ ), they obtained a maximum flow velocity $V_{B1}$ of 10  $\text{m}~\text{s}^{-1}$ near the jet tip. According to Hsiao et al. (Reference Hsiao, Choi, Singh, Chahine, Hay, Ilinskii, Zabolotskaya, Hamilton, Sankin, Yuan and Zhong2013), the three-dimensional jet velocity can be estimated by multiplying the two-dimensional jet velocity by a two-dimensional correction factor, such as $\{\log (R_{D}/R_{max})\}^{1/2}$ , in which $R_{D}$ is the disk size ( $=\,0.75$  mm). From this estimation, we obtain a three-dimensional jet velocity of $-18.4~\text{m}~\text{s}^{-1}$ , corresponding to $V_{j,A}^{\ast }=-1.8$ . As seen from figure 11, this value is slightly lower than the lower edge of the $-V_{j,A}^{\ast }$ versus $l_{0}^{\ast }$ relationship for $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.3$ .

Figure 12 indicates the value of the dimensionless jet impact velocity $-V_{j,A}^{\ast }$ as a function of the bubble size ratio $\unicode[STIX]{x1D709}$ for two parameter combinations ( $l_{0}^{\ast }$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ ), i.e. (0.75, 0.3) denoted by (●) and (0.6, 0.5) denoted by (▴), although only two data points are used for the latter. It is obvious that the magnitude of $V_{j,A}^{\ast }$ increases with increasing $\unicode[STIX]{x1D709}$ , especially in the range of $\unicode[STIX]{x1D709}>1.0$ . For instance, when we examine the case of $\unicode[STIX]{x1D709}=1.4$ under the same parameter combination, $l_{0}^{\ast }=0.75$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.3$ , corresponding to the dimensionless time at $t^{\ast }=0.549$ , the lower surface of bubble B was exceedingly elongated along the symmetric axis due to the shortness of the distance between the two bubbles, as well as the presence of the nearby bubble A, whose internal pressure is low; subsequently, the bubble B’s lower tip velocity of $-5.7$ was reached. Under this situation, the pressure produced inside bubble A has a little effect on the velocity of a conical jet. Consequently, the water located between the two bubbles was accelerated more strongly to move downward, producing a pulsed jet with the impact velocity of $-7.2$ , which is approximately twice the velocity for the case of $\unicode[STIX]{x1D709}=1.0$ (i.e. $V_{j,A}^{\ast }=-3.4\rightarrow -7.2$ ), as previously explained for figure 4. This trend becomes more remarkable for the shorter bubble distance cases, e.g. $l_{0}^{\ast }=0.6$ , because $V_{j,A}^{\ast }$ varies with approximately the same value for the two cases, $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.3$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.5$ , as clearly seen in figure 10. Pearson (Reference Pearson2002) calculated the evolutions of bubble shapes for two bubbles with the parameter combination, $l_{0}^{\ast }=0.8$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.27$ , and $\unicode[STIX]{x1D709}=1.0$ , from which the value of $V_{j,A}^{\ast }$ can be determined as $-2.7$ . This value denoted by (○) falls inside the range of our experiments, ranging from $-2.6$ to $-3.4$ for the parameter range $0.7<\unicode[STIX]{x1D709}\leqslant 1$ . It is noted that for larger values of $\unicode[STIX]{x1D709}$ , such as $\unicode[STIX]{x1D709}>1.5$ , the impact jet velocity becomes high; however, the conical water mass consisting of the jet will be completely transformed into mist before impacting the lower surface of bubble A.

3.2.1.3 Jet impact time and jet impact pressure

The jet impact time, $t_{imp}^{\ast }$ , is an important indicator for evaluating the effectiveness of perforation in the cell membrane during tandem-bubble sonoporation. As described in §§ 3.1.1 and 3.1.2, the lower surface of bubble A behaved similarly to a Rayleigh bubble until jet impact occurred. Naturally, a desirable jet impact must be achieved at $t_{imp}^{\ast }=0.915$ : if a cell is present, the lower surface might be in contact with the membrane surface when bubble A reaches its maximum volume.

With this limit, we explore the dimensionless time difference between the jet impact time and the time at which bubble A reaches its maximum volume, $\unicode[STIX]{x0394}t_{imp}^{\ast }$ , defined as

Therefore, an optimum jet impact occurs when $\unicode[STIX]{x0394}t_{imp}^{\ast }=0$ .

Figure 13 shows the relationship between $\unicode[STIX]{x0394}t_{imp}^{\ast }$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ for three selected parameter combinations with uncertainties of 15 % in all cases. The parameters for the first data group, denoted by a solid circle (●), are $\unicode[STIX]{x1D709}=1.0$ and $l_{0}^{\ast }=1.0$ ; those for the second data group, indicated by a solid triangle (▴), are $\unicode[STIX]{x1D709}=1.0$ and $l_{0}^{\ast }=0.75$ ; those for the third data group, denoted by a solid square (▪), are $\unicode[STIX]{x1D709}=1.5$ and $l_{0}^{\ast }=0.5$ . We note that the jets produced in bubble A for $\unicode[STIX]{x1D709}=1.0$ and $l_{0}^{\ast }=1.0$ are not pulsed jets but instead jets induced by the weak interaction of two bubbles, as shown in figure 3. Under these conditions, $\unicode[STIX]{x0394}t_{imp}^{\ast }$ is always positive and varies linearly with respect to $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ in a range of $0.15<\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }<0.7$ , as denoted by the broken line. Thus, the jet impact never occurs before bubble A reaches its maximum volume. As $l_{0}^{\ast }$ decreases for a constant bubble size ratio $\unicode[STIX]{x1D709}=1.0$ , bubble interaction increases in strength, as was clearly shown in figures 4 and 11. Accordingly, pulsed jets are created by the pronounced growth of the lower surface of bubble B. In the case where $l_{0}^{\ast }=0.75$ , $\unicode[STIX]{x0394}t_{imp}^{\ast }$ increases linearly with respect to $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ according to the function $\unicode[STIX]{x0394}t_{imp}^{\ast }=1.3\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }-0.26$ , as determined by applying the least squares method. This expression suggests that, if a cell is present, an optimum jet impact time satisfying the condition $\unicode[STIX]{x0394}t_{imp}^{\ast }=0$ occurs at the dimensionless time difference of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.2$ . For larger bubble size ratios $\unicode[STIX]{x1D709}>1$ , the optimum condition of $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ is shifted to values exceeding 0.2. In fact, we obtain the optimum $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ of 0.34 when $\unicode[STIX]{x1D709}=1.5$ and $l_{0}^{\ast }=0.5$ , although this value is estimated using only two data points.

Figure 13.

Dimensionless time difference between the jet impact time and the time at which bubble A reaches its maximum volume, $\unicode[STIX]{x0394}t_{imp}^{\ast }$ , versus dimensionless time difference between the two bubble generations, $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ . ●: $\unicode[STIX]{x1D709}=1.0$ and $l_{0}^{\ast }=1.0$ ; ▴: $\unicode[STIX]{x1D709}=1.0$ and $l_{0}^{\ast }=0.75$ ; ▪: $\unicode[STIX]{x1D709}=1.5$ and $l_{0}^{\ast }=0.5$ . The two solid lines, linearly approximated by the least squares method, correspond to cases of pulsed jets produced by strong interactions of the two bubbles. The dashed line indicates the cases of jets caused by weak interactions.

In general, when a water jet impacts the boundary surface, the impact occurs with the water hammer pressure of $p_{wh}$ . This pressure forms because of the action of compression waves from the compressibility of water. The water hammer pressure can be defined as follows (Heymann Reference Heymann1968):

where $c$ : sound velocity in water, $c_{m}$ : sound velocity in medium, $V_{j}$ : impact velocity of a jet and $\unicode[STIX]{x1D70C}_{m}$ : density of the medium. In the case where a jet with the velocity of $V_{j,A}$ impacts the lower surface of bubble A, which moves with the velocity of $V_{ls,A}$ , the water hammer pressure becomes (Johnsen & Colonius Reference Johnsen and Colonius2009; Kobayashi, Kodama & Takahira Reference Kobayashi, Kodama and Takahira2011):

Immediately after the jet impact, the jet flow stops instantaneously at the surface, creating the stagnation pressure, $p_{s}$ . Thus, the full waveform of the pressure resulting from jet impact exhibits a sharp impulse from $p_{wh}$ followed by a lower level stagnation pressure of $p_{s}$ with a relatively broad pulse width. We can calculate the water hammer pressures for two parameter combinations, corresponding to figure 4(a,b), such as ( $a$ ) $\unicode[STIX]{x1D709}=1.0$ , $l_{0}^{\ast }=0.74$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.26$ and ( $b$ ) $\unicode[STIX]{x1D709}=1.4$ , $l_{0}^{\ast }=0.6$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }=0.28$ , both of which lead to results close to the optimum jet impact conditions, as clearly seen in figure 13. We obtain $p_{wh}=27$  MPa for ( $a$ ), where $V_{j,A}=-34~\text{m}~\text{s}^{-1}$ and $V_{ls,A}=2~\text{m}~\text{s}^{-1}$ and $p_{wh}=55$  MPa for ( $b$ ), where $V_{j,A}=-72~\text{m}~\text{s}^{-1}$ and $V_{ls,A}=2.4~\text{m}~\text{s}^{-1}$ . The duration of the water hammer pressure, $t_{d,wh}$ , can be roughly evaluated by the expression $d_{j}/2c$ , with $d_{j}$ being the jet diameter of approximately one-tenth of the bubble diameter, namely $2R_{max,A}$ (Tomita & Shima Reference Tomita and Shima1986). Thus, we obtain the duration $t_{d,wh}$ of approximately 0.1  $\unicode[STIX]{x03BC}\text{s}$ for both cases, in which $R_{max,A}$ is determined as 1.7 mm for ( $a$ ) and 1.6 mm for ( $b$ ). Of course, the water hammer pressure $p_{wh}$ approaches $\unicode[STIX]{x1D70C}c|V_{j,A}|/2$ when jet impact occurs at the condition that exactly satisfies $\unicode[STIX]{x0394}t_{imp}^{\ast }=0$ , in which $V_{ls,A}$ is zero. The stagnation pressure, $p_{s}=\unicode[STIX]{x1D70C}V_{j,A}^{2}/2$ , is inevitably followed by the water hammer pressure and can be calculated as ( $a$ ) 0.6 MPa and ( $b$ ) 2.6 MPa for the same parameter conditions. If a cell or tissue is present near the first produced bubble, the water hammer pressure due to jet impact is slightly greater than that generated when a jet impacts the lower surface of the bubble without any boundary, because the acoustic impedances of the bio-surfaces should be slightly larger than that of a water surface. Cell membrane failure occurs at approximately 3 kPa for red blood cells (Rotsch, Jacobson & Radmacher Reference Rotsch, Jacobson and Radmacher1999; Prentice et al. Reference Prentice, Cuschieri, Dholakia, Prausnitz and Campbell2005); if the membrane surface is subject to a micro-streaming induced by bubble oscillation, the surface could be destroyed at the critical shear stress. The water hammer pressures from jet impact can easily penetrate the cell membrane, despite the very short duration of the water hammer pressure for small bubbles. For instance, when the bubble diameter is 10  $\unicode[STIX]{x03BC}\text{m}$ , the pressure duration is of the order of 0.1 ns.

In general, it is true that the jet velocity formed inside the bubble first produced in a system of tandem bubbles tends to be smaller than that produced from a single bubble for the same bubble distance condition. Nevertheless, the impact pressure of a jet with the velocity of several tens of metres per second seems sufficiently large to make a small hole in a cell membrane. One merit of using a jet produced in the bubble first produced, like that in the present system of two tandem bubbles with different birth times, is the ability to create a jet at a definite position and time by controlling the three parameters $\unicode[STIX]{x1D709}$ , $l_{0}^{\ast }$ , and $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}^{\ast }$ . This permits the determination of the optimum jet impact condition where the jet impact occurs. Ideally, this would occur as bubble A reaches its maximum volume with its lower surface in contact with the membrane surface of a cell.