2.1. Experimental arrangement

The experimental arrangement for investigating the dynamics of laser-induced bubble pairs is depicted in figure 2. The bubbles were generated by Q-switched Nd:YAG lasers emitting pulses of 1064 nm wavelength with 6 ns duration and energies of up to 250 mJ. Laser 1 was a Continuum YG 671-10 system with approximately Gaussian beam profile, and laser 2 was a Spectron SL 404 system with multimode beam profile. Pulse-to-pulse energy fluctuations were less than 10 % for both devices. The laser beams were first expanded by a telescope consisting of a biconcave lens (  $f=-40~\text{mm}$ ) and a Nd:YAG laser achromat (  $f=200~\text{mm}$ ), and then focused into a water cell ( $50\times 50\times 50~\text{mm}^{3}$ ) by a second Nd:YAG laser achromat (  $f=120~\text{mm}$ ). This optical arrangement leads to a large focusing angle ( $28^{\circ }$ full angle in water), together with a large distance between laser focus and cuvette walls. The large convergence angle of the laser beam allowed the production of compact plasmas and, hence, spherical bubbles. The expanded bubbles had a radius of $R_{max}\sim 1$  mm. Owing to the different quality of the transverse beam profile, the conversion rate of laser energy into mechanical energy differs, and pulse energies of 7 mJ and 20 mJ were needed to produce such bubbles with lasers 1 and 2, respectively.

Figure 2.

Experimental arrangement for high-speed photography of laser-induced bubble pairs.

Achromats were used for beam collimation and focusing to minimize spherical aberrations, and, for the same purpose, custom-made curved glass windows were built into the cuvette walls on either side of the cell (Vogel et al. Reference Vogel, Nahen, Theisen, Birngruber, Thomas and Rockwell1999a ). For zero distance $d$ between the two laser foci, their location coincided with the centre of curvature of the windows. During the experiments, $d$ was varied between 1.5 and 4.5 mm by moving the laser beams slightly up and down (this movement in the direction perpendicular to the plane of the paper could not be depicted in figure 2, where, for the sake of simplicity, the foci are separated in the direction of the optical axis). Aiming was facilitated by helium–neon laser beams coupled into the beam paths of the Nd:YAG lasers.

Figure 3.

Model geometry and numerical settings for the axisymmetric bubble pair problem with a sketch of the mesh used (a) and two examples of initial pressure conditions for an in-phase bubble pair (b) and an anti-phase bubble pair (c). Here $N$ and $M$ are the number of steps in the $x$ direction and in the positive and negative $y$ direction, respectively. Typical values used for both $N$ and $M$ are 300, giving a total number of volume elements of 180 000. This mesh is rotated about the symmetry axis in small steps ( $5^{\circ }$ ) to give hexahedral wedges. Because of the axial symmetry, there is no flux through the front and back faces of any volume element. Along the symmetry axis, two bubbles are inserted in-phase or anti-phase with conditions that can be read from the diagrams. The static pressure (100 kPa) is always kept at the same colour (light green) for reference, whereas the extrema may change.

The dynamics of the bubble pairs was recorded with a high-speed image converter camera (Hadland Photonics, Imacon 792). In this type of camera, the optical image is converted into a spatial distribution of photoelectrons that are accelerated by a high voltage and imaged electronically on a fluorescent phosphor screen. The object is continuously illuminated, and image separation is achieved by appropriate deflection of the bunch of accelerated electrons. This way, a time-separated sequence of individual optical images is created at the fluorescent screen. The exposure time for each frame always amounts to one-fifth of the inter-framing time. A framing rate of 50 000 frames per second was chosen for taking image series of the bubble motion at twice the original magnification. For illumination, a flash lamp with $400~{\rm\mu}\text{s}$ flash duration was used that covered the most relevant part of the two-bubble interaction. The images on the fluorescent screen of the image converter camera were recorded on Kodak T Max 400 $4\times 5$  inch sheet film with a sensitivity of 400 ASA. They were digitized at 6400 pixels per inch using a photo-scanner (Epson V700), and inverted into positives by means of Adobe Photoshop software. The recorded images show a spatial resolution in object space of ${\sim}50~{\rm\mu}\text{m}$ . Spatial resolution is limited by the various image transformation and transmission processes in the Imacon camera.

The arrangement was used to investigate the dynamics of bubble pairs with equal-sized bubbles; actual slight variations in relative size around ${\it\rho}=1$ were not intended but were caused by fluctuations of the laser pulse energies. The relative initiation distance, ${\it\gamma}$ , was varied by changing the separation, $d$ , between the laser foci. Attention was focused on bubbles produced at the same time ( ${\it\tau}=0$ , in-phase bubble pair), and on bubbles produced subsequently, the second bubble being initiated at the maximum radius of the first one ( ${\it\tau}=1$ , anti-phase bubble pair). Appropriate timing for bubble generation and recording of the image series was controlled electronically by a trigger-and-delay unit.

2.2. Numerical methods

OpenFOAM (Open Field Operation And Manipulation) has been selected to perform the numerical calculations for gaining more insight into bubble pair dynamics. OpenFOAM is an open-source CFD software programmed in C $++$ based on the finite volume method. The volume-of-fluid (VOF) method is used for capturing and locating the gas–liquid interface. It is a robust method for following evolving, topologically complex interfaces (Fuster et al. Reference Fuster, Agbaglah, Josserand, Popinet and Zaleski2009) as encountered with interacting bubbles. Because the bubble interaction geometry is axisymmetric, hexahedral wedges are used. The model geometry and numerical settings are shown in figure 3 together with two examples of initial conditions. The domain of calculation is selected to be sufficiently large to avoid back-reactions from the numerical boundaries. The ideal-gas EOS is used to simulate the bubble content. Surface tension and compressibility as well as viscosity for both gas and liquid are included, whereas condensation, evaporation and heat conduction are neglected in the model. An artificial liquid with the properties of water, except for the sound velocity, is chosen. The reason is that upon laser bubble formation a strong shock wave is radiated that presents problems in being properly captured by the numerical code (see the efforts that Hawker & Ventikos (Reference Hawker and Ventikos2012) have undertaken to solve the problem of the interaction of a shock wave with a bubble). Therefore, linear acoustics and a reduced sound speed are used for the liquid, for definiteness that of the gas. This approach can be considered a better approximation than the so far mostly used incompressibility assumption for the liquid.

The fluids used are a liquid and a gas with the following parameters: density, ${\it\rho}_{liquid}=1000~\text{kg}~\text{m}^{-3}$ and ${\it\rho}_{gas}=1~\text{kg}~\text{m}^{-3}$ ; sound velocity, $c_{liquid}=316~\text{m}~\text{s}^{-1}$ (reduced as stated above) and $c_{gas}=316~\text{m}~\text{s}^{-1}$ ; surface tension between the two fluids, ${\it\sigma}=0.07~\text{N}~\text{m}^{-1}$ ; and kinematic viscosity of the liquid, ${\it\nu}_{liquid}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ , and of the gas, ${\it\nu}_{gas}=1.589\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ . The initial conditions vary for in-phase and anti-phase bubble pairs. The millimetre-sized in-phase bubbles for comparison with the experiments are initiated by starting with bubbles of radius 0.13 mm with interior pressure of 100 bar and $0~\text{m}~\text{s}^{-1}$ bubble-wall velocity expanding into water of static pressure 1 bar. The maximum bubble radius reached then is 1.04 mm (similar to the experimental radius) if the bubbles of the pair were to stay spherical. This situation is depicted in figure 3(b). In the case of millimetre-sized anti-phase bubble pairs, bubble 1 starts with a radius of 1 mm, internal pressure 0.01 bar and $0~\text{m}~\text{s}^{-1}$ bubble-wall velocity, and bubble 2 starts as do the millimetre-sized in-phase bubbles, i.e. radius 0.13 mm with interior pressure of 100 bar and $0~\text{m}~\text{s}^{-1}$ bubble-wall velocity. This situation is depicted in figure 3(c).

The subsequent evolution of bubble shape and of the distribution of velocity and pressure in the two-phase fluid is then calculated for in-phase and anti-phase equal-sized bubble pairs with different relative distances in bulk water, i.e. $\{({\it\rho}=1,{\it\gamma}~\text{varied},~{\it\tau}=0)\}$ and $\{({\it\rho}=1,~{\it\gamma}~\text{varied},~{\it\tau}=1)\}$ , respectively, for comparison with the experiments. Afterwards, the calculations are extended to micrometre-sized bubble pairs.

Figure 4.

Growth and collapse of in-phase bubble pairs ( ${\it\tau}=0$ ) with bubbles of approximately equal size ( ${\it\rho}=1$ ) in each pair for two different relative bubble distances ${\it\gamma}$ . The bubble distance $d$ at initiation is (a)  $d=4.5~\text{mm}$ ( $R_{max}=0.9~\text{mm}$ , ${\it\gamma}=2.5$ ) and (b)  $d=2~\text{mm}$ ( $R_{max}=1.15~\text{mm}$ , ${\it\gamma}=0.87$ ). The frame interval is $40~{\rm\mu}\text{s}$ between the first seven frames depicting the expansion phase and the start of the collapse phase, and $20~{\rm\mu}\text{s}$ afterwards for the final collapse and rebound.

3.1. In-phase bubble pairs

For in-phase bubble pairs (figure 4), jetting occurs towards the plane of symmetry between the two bubbles. The dynamics of each individual bubble resembles that of a single bubble adjacent to a solid boundary, and the variation of jetting with ${\it\gamma}$ is also very similar to the behaviour near a solid boundary. There should be an equivalence except for the boundary layer at a real solid surface that is absent in the case of two bubbles in the bulk of the liquid. It is also seen that the two bubbles approach each other, again as in the case of a bubble near a solid boundary. With decreasing ${\it\gamma}$ , jetting becomes more pronounced and involves a larger liquid mass. Interestingly, for ${\it\gamma}<1$ , the bubbles do not merge but a thin liquid film remains between them (figure 4 b).

3.2. Anti-phase bubble pairs

The behaviour of anti-phase bubble pairs is very different (see figures 5–9) from the in-phase bubble pairs. Jetting occurs again in opposite directions but the two jets are now directed away from each other. The expansion of bubble 2 accelerates the collapse of bubble 1 and induces a jet penetrating bubble 1 in outside direction. The collapse of bubble 1, in turn, pulls on the wall of bubble 2, creating a conical protrusion. After the collapse of bubble 1, the protrusion of bubble 2 flips back and forms a jet penetrating bubble 2 in outside direction. The jet becomes visible on the bright-field images because it is surrounded by a gaseous protrusion of the cavity, which is formed by jet impact. The general features of the bubble dynamics remain the same regardless of ${\it\gamma}$ (in the range considered and when the bubbles do not merge), but the details vary strongly with the actual bubble distance.

Figure 5.

Dynamics of an anti-phase bubble pair ( ${\it\tau}=1$ ) with bubbles of approximately equal size ( ${\it\rho}=1$ ). The bubble distance $d$ at initiation is $d=3.0~\text{mm}$ ; $R_{max}=1.0~\text{mm}$ , ${\it\gamma}=1.5$ . The frame interval is $20~{\rm\mu}\text{s}$ .

Figure 6.

Dynamics of an anti-phase bubble pair ( ${\it\tau}=1$ ) with bubbles of approximately equal size ( ${\it\rho}=1$ ). The bubble distance $d$ at initiation is $d=2.5~\text{mm}$ ; $R_{max}=1.0~\text{mm}$ , ${\it\gamma}=1.25$ . The frame interval is $20~{\rm\mu}\text{s}$ .

Figure 7.

Dynamics of an anti-phase bubble pair ( ${\it\tau}=1$ ) with bubbles of approximately equal size ( ${\it\rho}=1$ ). The bubble distance $d$ at initiation is $d=2~\text{mm}$ ; $R_{max}=1.0~\text{mm}$ , ${\it\gamma}=1.0$ . The frame interval is $20~{\rm\mu}\text{s}$ .

Figure 8.

Dynamics of an anti-phase bubble pair ( ${\it\tau}=1$ ) with bubbles of approximately equal size ( ${\it\rho}=1$ ). The bubble distance $d$ at initiation is $d=1.5~\text{mm}$ ; $R_{max}=1.0~\text{mm}$ , ${\it\gamma}=0.75$ . The frame interval is $20~{\rm\mu}\text{s}$ .

Figure 9.

Dynamics of an anti-phase bubble pair ( ${\it\tau}=1$ ) with bubbles of approximately equal size ( ${\it\rho}=1$ ). The bubble distance $d$ at initiation is $d=3~\text{mm}$ ; $R_{max}=2.1~\text{mm}$ , ${\it\gamma}=0.715$ . The frame interval is $50~{\rm\mu}\text{s}$ . Taken from Lauterborn & Hentschel (Reference Lauterborn and Hentschel1985).

With decreasing ${\it\gamma}$ , the involution of the upper wall of bubble 1 occurs earlier during the collapse phase, and the jet becomes broader and contains more mass. While in figure 5 ( ${\it\gamma}=1.5$ ) the jet flow through bubble 1 remains relatively concentrated after the jet has penetrated the bubble, it rapidly evolves into a ring vortex in figure 8 ( ${\it\gamma}=0.75$ ). With regard to bubble 2, decreasing ${\it\gamma}$ results in an ever stronger elongation of the conical protrusion and thus a stronger jet in the upward direction. While in figure 5, the jet becomes visible only after the collapse of bubble 2, it shows up in the middle of the collapse phase in figure 6, and already at the beginning of the collapse phase in figures 7 and 8. In the latter case, the jet flow rushes through the expanded, empty bubble and enters far into the liquid beyond the opposite bubble wall. It can be deduced from figure 8 that, under optimum conditions, the penetration depth amounts to at least three times the maximum radius of bubble 2; the exact value is not known because the jet tip is outside the frame boundaries.

When the bubbles are too close, part of bubble 2 is pulled out too strongly by the collapsing bubble 1, and the protrusion enters the ring vortex formed upon collapse of bubble 1, such as in figure 9. This way, both the formation of the conical tip of bubble 2 and of the jet tip belonging to bubble 1 are disturbed. The protrusion is intruded into the jet that now occupies a ring shape and splashes through the bottom of bubble 1. The irregular shape of the protrusion of bubble 2 results in irregularities also during jet formation. The jet penetrating bubble 2 is now slower and broader than in figure 8.

4.1. In-phase bubble pairs

The dynamics of an in-phase bubble pair is depicted in figure 10 for ${\it\gamma}=2.5$ and in figure 11 for ${\it\gamma}=0.87$ , both for ${\it\rho}=1$ . The evolution of bubble shape (figures 10 a and 11 a) is given together with selected frames of the pressure and velocity distribution both in the liquid and inside the bubble (figures 10 b–e and 11 b–e). The bubble shape sequences reproduce the experimental finding (figure 4) that during collapse and rebound each bubble develops a liquid jet towards the other bubble. However, in the numerical simulation of figure 10 the jet just reaches the opposite bubble wall, whereas in the experiment the jet produces a noticeable protrusion. As jet formation is very sensitive to initial and boundary conditions, and these are not precisely known for the experiment (the conditions in the bubble after laser-induced breakdown are not known exactly and the bubbles are of slightly different size), the agreement between experimental and numerical results seems reasonable. Similarly, when the bubble shapes of figure 11 are compared with the respective experimental ones of figure 4, the different bubble sizes will lead to jet formation at slightly different times and with different strength, so that a complete match cannot be expected. The main features, however, are well modelled in figure 11, for instance the squeezing of the two bubbles when the expanding walls approach each other. The hitting of the two approaching jets with a flat disk-like radial jet of outflowing liquid is visible only in the simulations, but can be guessed also from the experiments. For comparison with the experiment it has to be noticed that the bubble shapes in figures 10 and 11 are cuts through the bubbles along the vertical axis of rotational symmetry. The shapes have to be rotated around this axis, yielding torus-like bubbles after collapse that in the experiments are looked at in side view.

Besides the bubble shapes, the simulations also supply the evolution of the pressure and the velocity fields in the bubble and in the liquid. Just a few frames are shown in figures 10 and 11 for selected times. Figure 10(b,c) shows the pressure and velocity fields shortly before and shortly after collapse. The pressure field is shown in the left half of each panel, the velocity field in the right half of each panel, with the velocity magnitude given by the (varying) background colour and the direction given by the arrows (all of equal length). As expected, shortly before collapse the bubbles of figure 10 (bubbles far away from each other) are surrounded by a high-pressure region, the pressure inside the bubble being lower. Whereas the pressure inside the bubbles remains homogeneous, the velocity of the gas already shows signs of jet formation. The flow field in the liquid is symmetric with respect to the plane of symmetry between the two bubbles and finally directed towards the bubbles with a stagnation point on the vertical axis of rotational symmetry in the middle between the two bubbles. Shortly after collapse (figure 10 c) the pressure field is similar to that before collapse. The velocity field, however, must change from inflow to outflow. In the frame chosen the switching has just started, so that only the part of the liquid near the bubble is flowing away from the bubble, while the main body of the liquid is still inflowing. In the later stages of expansion (figure 10 d,e) the outflow region has expanded throughout the frame region. The jet develops further during rebound of the bubble with the highest velocity at the tip, where it is smaller in diameter. In figure 10(d) the jet tip has a velocity of $16.7~\text{m}~\text{s}^{-1}$ , the peak velocity reached for the given parameter set being about $25~\text{m}~\text{s}^{-1}$ . In figure 10(e) the pressure has fallen to a low value inside the bubble and the jet has come to about a stop as the high velocity of the jet tip has ceased.

For the case of ${\it\gamma}=0.87$ (figure 11) the pressure and velocity fields before and after bubble collapse are shown. They are easily discernible by the inflow and outflow velocity fields of the liquid (figure 11 b–e). In figure 11(b) the two bubbles are still expanded, but the pressure field in the liquid is quite inhomogeneous, showing two regions of higher pressure on the vertical symmetry axis immediately above the upper bubble and below the lower bubble. This is the reason for the liquid jets being initiated. The jets are already noticeable in the gas flow field inside the bubble. In figure 11(c) before collapse, the two liquid jets are flowing inwards with high velocity (about $45~\text{m}~\text{s}^{-1}$ ) and have created a region of high pressure (about 20 bar) in the liquid around the meeting point (stagnation point). Thereby, two torus bubbles are formed. Shortly after collapse (as seen by the outflow of the liquid from the bubble region) each bubble develops a (radial) splash and the liquid between the two torus bubbles is squeezed out at high speed in the form of a flat disk, the liquid being supplied by the inflowing jets (figure 11 d). This situation prevails during much of the rebound time, as can be seen from figure 11( e), and the bubble shapes at later times.

4.2. Anti-phase bubble pairs

The larger the relative distance between two bubbles, the less intense is the interaction between them. This is also valid for anti-phase bubble pairs, where the two jets are directed away from the other bubble. In the case of ${\it\gamma}=1.5$ , the largest ${\it\gamma}$ value considered here (figure 12), each liquid jet just reaches the opposite bubble wall. The apparent penetration of the opposite bubble wall is considered an artifact, as the thin gas bridge between jet tip and bubble wall cannot be resolved by the VOF method (used for interface capturing) unless the mesh is strongly refined. The jets are more pronounced in the corresponding experiment (figure 5) than in the simulation but the main jetting features are reproduced, with the thin jet of bubble 1 reaching a higher velocity than the broad jet of bubble 2. Upon second collapse of bubble 1, the upper bubble part around the jet base involutes and liquid rushes into the bubble as a ring-like jet inclined towards the axis of rotational symmetry. Thereby, the bubble disrupts into two torus bubbles. Upon rebound, a complicated gas structure is formed with a thin dimple sticking upwards out of the main bubble part after ${\approx}300~{\rm\mu}\text{s}$ . Even this peculiar gas structure after the second collapse of bubble 1 showing up in the experiment is present in the calculation. The pressure and velocity fields show the expected behaviour. A collapsed bubble is immersed in a high-pressure field with an inflow velocity field that reverses upon rebound. However, as one bubble is collapsing while the other bubble is expanding, a near-field flow from the expanding bubble to the collapsing one is generated.

Figure 12.

Dynamics of an anti-phase bubble pair. Simulation parameters are $R_{max1}=1~\text{mm}$ , $R_{max2}=1.04~\text{mm}$ and ${\it\gamma}=1.5$ . (a) Bubble shapes; and (b–e) selected frames from the simulation with pressure and velocity fields. To be compared with experimental results from figure 5 with $R_{max1}\approx 1~\text{mm}$ and ${\it\gamma}\approx 1.52$ . The numerical peak jet velocities of bubble 1 and bubble 2 are about $65~\text{m}~\text{s}^{-1}$ and $15~\text{m}~\text{s}^{-1}$ , respectively. For further explanations concerning the pressure and velocity fields, see figure 10.

At ${\it\gamma}=1.25$ (figure 13) the jet of bubble 1 gets broader and slower, but otherwise the dynamics resembles that of figure 12. More pronounced changes occur when the relative distance is reduced to ${\it\gamma}=1.0$ (figure 14) and the interaction gets stronger. Now, the expanding bubble 2 elongates towards bubble 1, as also seen in the experiments (figure 7). The jet of bubble 1 is very broad, but hardly reaches the opposite bubble wall. Its ‘tip’ looks like a nose and hits the opposite bubble wall along a ring with the inclusion of gas in between. The same type of contact has also been observed by Lindau & Lauterborn (Reference Lindau and Lauterborn2003) for a laser-induced bubble near a solid boundary in the case when the bubble upon expansion just touched the boundary. The pressure field at $t=60~{\rm\mu}\text{s}$ reveals a zone of high pressure between the two bubbles from the inflow of liquid, which is considered the main driving force for the two jets in opposite directions. The flow field is mainly directed to bubble 1. At $t=80~{\rm\mu}\text{s}$ , shortly after collapse, bubble 1 is still surrounded by a high-pressure field with a compression wave emitted. A jet is forming downwards into bubble 1 and the jet of bubble 2 is just starting. The velocity field exhibits a near-field flow directed from bubble 1 to bubble 2. At $160~{\rm\mu}\text{s}$ , both jets are well developed, with a high pressure at the tip of the jet of bubble 2 as the jet has already hit the opposite bubble wall. The velocity field is complicated but the peak velocities are about the same for both jets, $30~\text{m}~\text{s}^{-1}$ . At $280~{\rm\mu}\text{s}$ , shortly after the second collapse of bubble 1, the jet through bubble 2 has deeply entered the liquid. The velocity field shows an extended near-field flow from bubble 1 to bubble 2 that feeds the jet of bubble 2. This jet becomes visible in the bubble shape plots when the gaseous layer surrounding it is thick enough to be resolved by the resolution of the mesh used in the simulations. It disappears when the layer thickness falls below the resolution limit, and where the jet-induced protrusion collapses such that its wall touches the jet flow. However, even when the gaseous layer disappears, the jet can still be seen in the velocity field via the magnitude of the flow field given in the simulation plots.

Figure 13.

Dynamics of an anti-phase bubble pair. Simulation parameters are $R_{max1}=1~\text{mm}$ , $R_{max2}=1.04~\text{mm}$ and ${\it\gamma}=1.25$ . (a) Bubble shapes; and (b–e) selected frames from the simulation with pressure and velocity fields. To be compared with experimental results from figure 6 with $R_{max1}\approx 1~\text{mm}$ and ${\it\gamma}\approx 1.25$ . The numerical peak jet velocities of bubble 1 and bubble 2 are about $35~\text{m}~\text{s}^{-1}$ and $16~\text{m}~\text{s}^{-1}$ , respectively.

Figure 14.

Dynamics of an anti-phase bubble pair. Simulation parameters are $R_{max1}=1~\text{mm}$ , $R_{max2}=1.04~\text{mm}$ and ${\it\gamma}=1.0$ . (a) Bubble shapes; and (b–e) selected frames from the simulation with pressure and velocity fields. To be compared with experimental results from figure 7 with $R_{max1}\approx 1~\text{mm}$ and ${\it\gamma}\approx 1.0$ . The numerical peak jet velocities of bubble 1 and bubble 2 are both about $30~\text{m}~\text{s}^{-1}$ .

Figure 15.

Dynamics of an anti-phase bubble pair. Simulation parameters are $R_{max1}=1~\text{mm}$ , $R_{max2}=1.04~\text{mm}$ and ${\it\gamma}=0.8$ . (a) Bubble shapes; and (b–g) selected frames from the simulation with pressure and velocity fields. To be compared with experimental results from figure 8 with $R_{max1,2}\approx 1~\text{mm}$ and ${\it\gamma}=0.75$ . The numerical peak jet velocity of bubble 2 is about $150~\text{m}~\text{s}^{-1}$ reached at about $70~{\rm\mu}\text{s}$ . The average jet velocity of bubble 2 is about $80~\text{m}~\text{s}^{-1}$ averaged over $30~{\rm\mu}\text{s}$ (from 70 to $100~{\rm\mu}\text{s}$ ).

Figure 16.

Dynamics of an anti-phase bubble pair. Simulation parameters are $R_{max1}=2.1~\text{mm}$ , $R_{max2}=2.09~\text{mm}$ and ${\it\gamma}=0.714$ . Initial conditions for bubble 1: radius 2.1 mm, interior pressure 0.01 bar, wall velocity $0~\text{m}~\text{s}^{-1}$ . Initial conditions for bubble 2: radius 0.3 mm, interior pressure 100 bar, wall velocity $0~\text{m}~\text{s}^{-1}$ . (a) Bubble shapes; and (b–g) selected frames from the simulation with pressure and velocity fields. To be compared with experimental results from figure 9 with $R_{max1}\approx 2.1~\text{mm}$ and ${\it\gamma}\approx 0.715$ . The numerical peak jet velocity of bubble 2 is about $100~\text{m}~\text{s}^{-1}$ reached at $140~{\rm\mu}\text{s}$ . The average jet velocity of bubble 2 is about $75~\text{m}~\text{s}^{-1}$ averaged over $70~{\rm\mu}\text{s}$ (from 140 to $210~{\rm\mu}\text{s}$ ).

At a relative distance of ${\it\gamma}=0.8$ (figure 15) bubble interaction gets strong and special. Bubble 1 collapses via a bean shape to a torus bubble. Bubble 2 expands with a pronounced elongation deep into bubble 1, thereby attaining the shape of a light bulb. During the expansion of bubble 2 the elongation develops into a constricting neck that already closes upon the start of collapse of bubble 2, leaving its gas remains between the torus of bubble 1. From the constriction region of the neck, a fast jet is formed upwards into the main body of bubble 2. It propagates with a maximum velocity of about $150~\text{m}~\text{s}^{-1}$ and deeply penetrates into the liquid behind the opposite bubble wall. Bubble 1 attains an almost quadratic shape after collapse. The whole sequence of events well models the experiment of figure 8 for a similar value of ${\it\gamma}=0.75$ . The pressure and velocity fields make the involved bubble dynamics more understandable. At $t=50~{\rm\mu}\text{s}$ , the liquid flow into the neck region squeezes the neck to contraction and the neck of bubble 2 is surrounded by a ring of higher pressure. At $60~{\rm\mu}\text{s}$ , the collapse of bubble 1 has produced a strongly confined region of high pressure below the elongation of bubble 2, together with a high-pressure ring that appears as two points in the cross-section presentation of the bubbles. These pressures combine to one region of higher pressure at the base of the jet of bubble 2 accelerating the jet upwards by inflow of liquid ( $80~{\rm\mu}\text{s}$ ). High pressures also appear at the tip of the jet of bubble 2, both when it first hits the opposite bubble wall and later upon pushing liquid ahead. From 50 to $80~{\rm\mu}\text{s}$ very high velocities are observed in the gas of bubble 2 in front of the emerging jet.

A total entanglement of the bubbles occurs at ${\it\gamma}=0.714$ (figure 16) in good agreement with the experiment of figure 9. Now, bubble 2 mixes its elongation part with bubble 1, leaving it shattered into several torus layers. Nevertheless, the neck of bubble 2 closes in a similar fashion as for ${\it\gamma}=0.8$ in figure 15 and, therefore, a fast jet shooting through bubble 2 is produced that reaches a velocity of ${\approx}100~\text{m}~\text{s}^{-1}$ . The jet, however, has a peculiar shape, with its side lobes of bubble protrusions and a wavy roughness of the surrounding gas tube. The experiment also shows a disturbed jet, even more disturbed because of the different and less controllable symmetry conditions than in the simulations. Pressure and velocity fields are complicated. When bubble 2 penetrates into bubble 1 ( $50~{\rm\mu}\text{s}$ ), its inner gas pressure is higher than in bubble 1. This initiates a flow field of gas through the neck that finally results in a protrusion downwards along the axis ( $100~{\rm\mu}\text{s}$ ). Liquid flowing into the neck region leads to a ring-shaped higher-velocity region in the gas, where bubble 2 and bubble 1 meet, and to a higher-pressure ring in the liquid, both visible at $130~{\rm\mu}\text{s}$ . The closing neck together with a high-pressure field between the two bubbles drives a liquid jet up into bubble 2 ( $150~{\rm\mu}\text{s}$ ). The liquid jet is preceded by a fast torus-like gas jet flow that produces a ring-like protrusion sticking out of the upper part of bubble 2 ( $200~{\rm\mu}\text{s}$ ). The liquid jet of bubble 2 gets broader than in the case of ${\it\gamma}=0.8$ . A downward jet of bubble 1 is not directly visible, but a flow through bubble 1 is present due to momentum conservation.

Figure 17.

The averaged jet velocity of bubble 2 as a function of the relative initiation time, ${\it\tau}$ , and the relative distance, ${\it\gamma}$ , of micrometre-sized bubble pairs of equal size ( ${\it\rho}=1$ ). Initial conditions are $R_{1,2}=1~{\rm\mu}\text{m}$ with an internal pressure of 100 bar and bubble-wall velocity $v_{1,2}=0~\text{m}~\text{s}^{-1}$ , giving $R_{max1,2}=6.74~{\rm\mu}\text{m}$ at the static pressure in the liquid of 1 bar. The averaged jet velocity of bubble 2 as a function of ${\it\gamma}$ for an anti-phase ( ${\it\tau}=1$ ) equal-sized millimetre bubble pair ( ${\it\rho}=1$ , $R_{max1}=1~\text{mm}$ , $R_{max2}=1.04~\text{mm}$ ) is also given as the inserted bar. The initial conditions in this case are $R_{1}=1~\text{mm}$ with an internal pressure of 0.01 bar, $R_{2}=0.13~\text{mm}$ with an internal pressure of 100 bar and bubble-wall velocity $v_{1,2}=0~\text{m}~\text{s}^{-1}$ . Colour denotes the jetting direction: when the jet of bubble 2 is directed towards bubble 1, the jet velocity is defined as negative (blue range), while it is positive (green–yellow–red range) when the jet of bubble 2 is directed away from bubble 1.

4.3. Maps of jet velocity as a function of parameters

Figure 17 shows the parameter map of the dependence of direction and averaged jet velocity of bubble 2 on the relative initiation time ${\it\tau}$ and the relative distance ${\it\gamma}$ of micrometre-sized bubble pairs of equal size ( ${\it\rho}=1$ ) with one example series of millimetre-sized bubble pairs for ${\it\tau}=1$ . The averaging of the jet velocity is done from the start of the indentation of the bubble to the contact of the jet with the opposite bubble wall. In the cases when the jet does not reach the opposite bubble wall, the stopping time is taken from the inversion of the flow field at the jet tip. When the jet of bubble 2 is directed towards bubble 1, the jet velocity is defined as negative. Twenty-five points ( ${\it\gamma},{\it\tau}$ ) have been selected and interpolated to obtain the diagram. They are mainly taken in the high-velocity regions, as only these are important for opto-injection. Owing to large errors in the lower-velocity regions, the jet velocities there are smoothed interpolations and not to be taken as exact values. When ${\it\tau}=0$ (in-phase bubble pair), two symmetric jets moving towards each other are induced. When there is a small difference in the initiation time of the bubbles, $0<{\it\tau}<0.4$ , the jet of bubble 2 will still be directed to bubble 1, but with a smaller velocity than at ${\it\tau}=0$ . When enlarging the generation time difference between the bubbles, $0.4<{\it\tau}<1.2$ , the jet of bubble 2 induced by the presence of bubble 1 will be pointing into bubble 2, away from bubble 1. Here, the jet direction is reversed and the strength of the jet is enhanced with increasing ${\it\tau}>0.4$ up to ${\it\tau}=1$ . The value ${\it\tau}\approx 0.4$ is also the line where bubble attraction (largest for ${\it\tau}=0$ ) changes to repulsion (largest for ${\it\tau}=1$ ). In the case of a millimetre-sized bubble pair, the highest jet velocity of the bubble pair with ${\it\rho}=1$ and ${\it\tau}=1$ occurs with bubble 2, when ${\it\gamma}\approx 0.8$ (see the horizontal bar in the figure). This agrees with the experimental results reported in figures 5–9. For a micrometre-sized bubble pair, on the other hand, the highest jet velocity is found for ${\it\gamma}\approx 0.7$ . This indicates that the change of the effects due to the liquid parameters, i.e. surface tension and viscosity, on the bubble dynamics is not negligible when the maximum bubble radius reduces from millimetre-sized to micrometre-sized. While for large cavities jet formation is insensitive to liquid surface tension (Antkowiak et al. Reference Antkowiak, Bremond, Le Dizès and Villermaux2007), surface tension leads to a larger pressure inside micrometre-sized bubbles and tends to counteract the formation of strongly curved convex surfaces, such as the tip of a jet. As a result, the highest jet velocity is now predicted for a slightly smaller ${\it\gamma}$ value, i.e. for a stronger interaction between the neighbouring bubbles.

Figure 18.

The averaged jet velocity of bubble 2 as a function of the relative bubble size, ${\it\rho}$ , and the relative distance, ${\it\gamma}$ , of micrometre-sized anti-phase bubble pairs ( ${\it\tau}=1$ ). Initial conditions are $R_{max1}=8~{\rm\mu}\text{m}$ with an inside gas pressure of 0.01 bar, $R_{2}$ varied from 4.47 to $8.2~{\rm\mu}\text{m}$ by choosing appropriate initial radii with an inside gas pressure of 100 bar and bubble-wall velocity $v_{1,2}=0~\text{m}~\text{s}^{-1}$ at the static pressure in the liquid of 1 bar. Colour denotes the jetting direction: when the jet of bubble 2 is directed towards bubble 1, the jet velocity is defined as negative (blue range), while it is positive (green–yellow–red range) when the jet of bubble 2 is directed away from bubble 1. Despite the fact that there are no negative velocities in the diagram due to ${\it\tau}=1$ , the same colour bar as in figure 17 is used for ease of comparison.

In practical applications, the distance $d$ of the focal spots and the time lag between the generation of the two bubbles is fixed, but the bubble sizes may vary as a result of pulse-to-pulse fluctuations of the laser energy. This will lead to fluctuations of all dimensionless parameters, ${\it\rho}$ , ${\it\gamma}$ and ${\it\tau}$ . The map in figure 17 reveals that small fluctuations in ${\it\gamma}$ and ${\it\tau}$ of the order of $\pm 15\,\%$ are not critical. When the bubble pair is generated with parameters in the optimum region (upper left region in the diagram of figure 17), the jetting behaviour stays sufficiently close to the optimum.

However, the influence of fluctuations of the relative bubble size, ${\it\rho}$ , still needs to be addressed. Figure 18 shows the parameter map of the dependence of direction and averaged jet velocity of bubble 2 on the relative bubble size, ${\it\rho}$ , and the relative distance, ${\it\gamma}$ , of micrometre-sized anti-phase bubble pairs ( ${\it\tau}=1$ , $R_{max1}=8~{\rm\mu}\text{m}$ ). Twenty points $({\it\gamma},{\it\rho})$ have been calculated and interpolated to obtain the diagram. Figure 18 can be considered as the extension of the line ${\it\tau}=1$ for micrometre-sized bubble pairs of the plane $\{({\it\rho}=1,{\it\gamma},{\it\tau})\}$ in figure 17 into the third dimension by ${\it\rho}$ , the relative bubble size, through the plane $\{({\it\rho},{\it\gamma},{\it\tau}=1)\}$ . There are two regions in figure 18 where the jet of bubble 2 attains high average velocities. One region is centred at ${\it\rho}\approx 0.97$ and ${\it\gamma}\approx 0.73$ , and a second region extends around ${\it\rho}\approx 0.65$ and ${\it\gamma}\approx 0.77$ .