3.1. The drop impact and partial deposition regimes

The drop impact regime yields weaker yet similar drop dynamics as those reported in the work of Biasiori-Poulanges et al. (Reference Biasiori-Poulanges, Bokman, Baumann and Supponen2022), and can be observed in the test case displayed in figure 3, which shows both numerical simulations and experimental radiographs. The initial bubble radius is $r_{b,0}=1.20$ mm and the drop-to-bubble size ratio is $\varepsilon =r_{d,0}/r_{b,0}=0.37$. The bubble centre is located 5.34 mm from the shock wave source and the drop rests near the proximal bubble side, 0.76 mm from the air shell's centre. The shock wave parameters upon contact with the GED are $p_{max} = 62.45$ MPa, $j = 2.79$ Pa s and $M=1.06$. Excellent agreement between the numerics and the experiment is observed if not for a slight over-expansion of the bubble at later stages in the numerical simulation.

Figure 3.

Image sequence of an example of shock–GED interaction within the drop impact regime ($We_{d} = 1940,\ Re_{d} = 11\ 000$). The upper halves display the numerically computed volume fractions of air and drop water in blue and black, respectively. The lower halves show corresponding experimental radiographs. The dimensionless time of the bubble, $t/(r_{b,0}\sqrt {\rho / p_{b,0}})$, starting from the shock contact with the GED, is indicated in the bottom-left corner. The inset figure shows the normalised bubble pressure map $p_{b}/p_{0}$.

The shock-induced collapse of the air bubble can be observed in frames 1–4 of figure 3 and is followed directly by a rebounding expansion phase in the subsequent frames. The drop's location on the bubble axis, very close to the proximal interface of the bubble, results in an instantaneous impact during the bubble contraction. A necking region then starts radially expanding from the contact point between the drop and the surrounding liquid, driving the latter into the air bubble in the form of a radially and axially propagating sheet jet, as can be observed on the second and third frames of figure 3. Such a sheet jet has been attributed to the strong water-hammer shock acting on the bubble cusp (Lesser & Field Reference Lesser and Field1983; Biasiori-Poulanges et al. Reference Biasiori-Poulanges, Bokman, Baumann and Supponen2022). However, the outer liquid impacts the drop displayed in figure 3 at an experimentally recorded impact speed of $u_{i,exp} = 12.62\ \textrm {m}\ \textrm {s}^{-1}$, close to the value of $u_{i,sim} = 10.98\ \textrm {m} \textrm {s}^{-1}$ retrieved from the numerical simulation along the axis shared between the bubble and drop. Here, the impact speed is defined as the relative velocity of the bubble interface, where the drop speed is typically negligible, being one to two orders of magnitude smaller. At such speeds, the drop is impacted at a Mach number in air of $M=0.04$, much lower than the limit of $M\geq 1$ at which compressibility effects should become relevant (Cheng, Sun & Gordillo Reference Cheng, Sun and Gordillo2022), resulting in a possible different driving mechanism for the sheet jet dynamics than the water hammer.

The splashing mechanism observed in figure 3 displays striking similarities with Worthington's first instantaneous photographs of a drop impacting a liquid surface (Worthington Reference Worthington1908) taken from A study of splashes in the early stages of necking, sheet jet inception and beyond. The Weber number, $We = 2\rho r_{d,0} u_{i}^{2} / \gamma$, describing the inertial to surface tension force balance, and the Reynolds number, $Re = 2\rho r_{d,0} u_{i} / \mu$, denoting the inertial to viscous force balance, are $We_{d} = 1940$ and $Re_{d} = 11\ 000$ for the drop in the experiment. The splash of a liquid drop on a flat liquid surface at such high Weber and Reynolds numbers adopts the shape of a crown as the sheet jet grows (Engel Reference Engel1966; Prosperetti & Oguz Reference Prosperetti and Oguz1993; Rein Reference Rein1996; Wang & Chen Reference Wang and Chen2000; Thoroddsen Reference Thoroddsen2002; Yarin Reference Yarin2006; Wang et al. Reference Wang, Liu, Bayeul-Lainé, Murphy, Katz and Coutier-Delgosha2023). This shape cannot be resolved herein; however, its consequences can be observed starting from the fourth frame of figure 3. As the thickened rim of the crown becomes unstable to the Rayleigh–Plateau instability, it forms a spray of drops which cannot be resolved individually in the radiographs, but can be observed upon clustering between the sheet jet and the bubble surface as shades of grey. In the simulation, the spray of small drops ejected from the crown tip can be seen stretching radially until it reaches the bubble interface, as highlighted on the fourth frame of figure 3. At the same moment, the sheet jet tip is redirected upstream after having impacted the bubble surface. However, due to the lower resolution of the sheet jet tip at the impact point, redirection of part of the jet downstream cannot be excluded. This has been observed in the splashing of drops on liquid surfaces Thoroddsen et al. (Reference Thoroddsen, Thoraval, Takehara and Etoh2011) and in the sheet jet cascade mechanism reported in violent shock-induced GED collapses (Biasiori-Poulanges et al. Reference Biasiori-Poulanges, Bokman, Baumann and Supponen2022). The sheet jet motion creates a distinctive region visible in the top corner of the fifth frame of figure 3. The inset displays the normalised pressure field, $p_{b}/p_{0}$, within the bubble, where a pressure drop of approximately 15 % is caused by fluid rotation. This rotating region likely contributes to the ejection of drops from the crown rim by vortex shedding (Jerome et al. Reference Jerome, Marty, Matas, Zaleski and Hoepffner2013). After ejection, the impact of these drops against the bubble wall can be observed in frames 5 and 6 where they disrupt the initially smooth air–water interface. This is especially visible in the highlighted region of the experimental part of the sixth frame of figure 3. Note that slight surface instabilities are also visible in all numerical simulations. However they are found to be grid dependent and are considered a numerical artifact with no incidence on the present work. Note that the bubble cusp located at the base of the sheet jet is visible in the radiographs as a vertical line starting from the fifth frame of figure 3, where the x-rays integrate all phase discontinuities along the beam path. It is, however, not visible in the two-dimensional representation of the axisymmetric numerical simulation.

In frames 7–9, the rim of the crown proceeds towards the central axis of the GED, and ends up separating the bubble into two large volumes of air. A central ‘twin’ jet forms, travelling both upstream and downstream and eventually piercing both sides of the bubble as revealed by the simulations. Although part of the drop is entrained by the sheet jet, its main core barely moves as it gets released locally during the overall process of bubble compression. The air is simply transferred to the distal side of the drop as it coalesces with the surrounding liquid. The central jet travelling upstream impacts the drop once more, increasing its deformation and dissemination. Note that a small fraction of drop contents is entrained by the sheet jet, which suggests that the splashing process contributes to the dissemination of the drop.

The partial deposition regime is shown in figure 4, where the numerical simulation, indicating the volume fraction of air in grey and drop in black, is displayed in the upper half of the image and the corresponding experimental radiograph is displayed in the lower half. Here, the highly deformed initial shape of the bubble in the experiment makes a direct comparison of the entire gas-encapsulated drop irrelevant. The liquid jet (not shown here) forms at the point of highest interfacial curvature in the experiment, which differs from the axisymmetric simulation results, where the bubble is a sphere. The full experimental and simulated GED dynamics are available in the supplementary movies are available at https://doi.org/10.1017/jfm.2024.1124. Since the drop dynamics take place at the distal side of the bubble, figure 4 solely focuses on that part.

Figure 4.

Image sequence of an example of shock–GED interaction within the partial deposition regime ($We_{d} = 16,\ Re_{d} = 777$). The upper halves display the numerically computed volume fractions of air and drop water in grey and black, respectively. The lower halves show corresponding experimental radiographs. The dimensionless time, $t/(r_{b,0}\sqrt {\rho / p_{b,0}})$, starting from the shock contact with the GED, is indicated in the bottom-left corner.

The initial size of the bubble is $r_{b,0}=1.14$ mm and the drop-to-bubble size ratio is $\varepsilon =0.24$. The bubble centre is located 5.67 mm from the shock wave origin and the drop centre 0.82 mm downstream from the bubble centre. The shock wave parameters are $p_{max}=55.93$ MPa, $j=2.50$ Pa s and $M=1.06$. The drop contacts the outer liquid at a relatively low experimentally recorded speed of $u_{i, exp}=1.45\ \textrm {m}\ \textrm {s}^{-1}$, close to that extracted from the numerical simulation of $u_{i, sim}=1.65\ \textrm {m}\ \textrm {s}^{-1}$. Since the drop is located near the distal side of the bubble, the closest interface is the slowest part of the bubble during its asymmetrical collapse. The Weber and Reynolds numbers associated with the drop upon impact in the experiment are $We_{d}=16$ and $Re_{d}=777$.

Here, the physical problem is otherwise similar to that presented in the drop impact regime, except for the weaker inertia as suggested by the dimensionless numbers which are two orders of magnitude lower. In particular, the associated Weber number suggests capillary effects, while viscosity is still expected to play a minor role due to the large Reynolds number. The drop dynamics observed in figure 4 display strong similarities with the deposition and partial coalescence of drops, reported in an abundant number of works (Charles & Mason Reference Charles and Mason1960; Blanchette & Bigioni Reference Blanchette and Bigioni2009; Kavehpour Reference Kavehpour2015; Kulkarni et al. Reference Kulkarni, Lolla, Tamvada, Shirdade and Anand2021). However, although the drop pinch-off process is governed by surface tension, the underlying mechanism here appears fundamentally different. Indeed, the partial coalescence of drops impacting flat liquid surfaces at very low speeds and the resulting ejection of daughter droplets have been attributed to the motion of capillary waves on the drop surface, generated by the formation of a neck at the drop–liquid interface in the early stages of coalescence (Blanchette & Bigioni Reference Blanchette and Bigioni2006). The momentum of sufficiently strong waves has been reported to distort the drop significantly when converging towards its summit, stretching the drop away from the liquid surface and inducing a collapse in the direction perpendicular to the stretching, which results in the production of a daughter droplet. Here, the deformation of the drop does not appear to be driven by capillary waves, but rather by the motion of the bubble interface following its interaction with the shock wave.

Once the bubble collapse initiates, the air film between the drop and the outer liquid becomes thinner. As soon as both liquids contact, the deformation of the drop and the coalescence begin. Similarly to the drop impact regime, a radially expanding necking region can be observed at the liquid–liquid interface, as highlighted in the second frame of figure 4. The early stage of the drop dynamics follows the ones reported during partial coalescence processes up to the end of the collapse of the air bubble surrounding the drop. Once the bubble rebounds, the reversed motion of its interface coupled with the capillary force linking the drop to the outer liquid deforms the drop into a pear-like shape, as observed in the third frame. Unlike in previously reported partial coalescence processes, capillary waves are absent and the drop deformation solely takes place at the proximal side of the drop, while its distal side remains at its initial location. The oscillatory motion of the bubble interface, as it pulls on the drop during the bubble expansion, drives the collapse of the newly formed neck by stretching the drop, as observed in the two last frames of figure 4. In the current work, however, daughter droplets rarely fully detach due to the second shock wave caused by the collapse of the laser-induced vapour bubble driving a second collapse of the air bubble. This often happens before a daughter droplet has formed completely making the drop dynamics difficult to follow after that point.

The early stage of the drop impact and partial deposition regimes are, in fact, different inertial regimes of drops being impacted by a liquid surface, which is fairly well understood and can be described through scaling laws. The spreading radius of the necking region between the drop and the outer liquid has previously been reported to scale with time as $r_{n}/r_{d,0} \propto \sqrt {2u_{i}t/r_{d,0}}$ (Josserand & Zaleski Reference Josserand and Zaleski2003), where $r_{d,0}/u_{i}$ is the characteristic impact time of the drop, based on the truncated sphere approximation and solely on geometric considerations. However, such approximations do not respect the continuity equation. Riboux & Gordillo (Reference Riboux and Gordillo2014) instead found using Wagner's theory (Wagner Reference Wagner1932),

(3.1)\begin{equation} \frac{r_{n}}{r_{d,0}}=\sqrt{\frac{3u_{i}t}{r_{d,0}}}, \end{equation}

to be a more accurate formulation, while still neglecting viscosity or air entrapment. The model is valid until the necking radius reaches the drop's initial size, $r_{n}/r_{d,0} \leq 1$, although good agreement has also been reported at later times (Josserand & Zaleski Reference Josserand and Zaleski2003; Howland et al. Reference Howland, Antkowiak, Castrejón-Pita, Howison, Oliver, Style and Castrejón-Pita2016). Figure 5 shows the temporal evolution of the spreading radius of the necking region recorded in the experiment and simulation of the drop impact and partial deposition regimes against the model given by (3.1). Excellent agreement is found between the numerical simulations and the model for the entire range of validity of the drop impact regime. In the partial deposition regime, however, the neck radius diverges from the model at $u_{i}t/r_{d,0} = 0.18$, decreasing as the air bubble expands and stretches the drop, as observed in the three last frames of figure 4. The experiment follows the same trends as the numerical simulations, consistent with the excellent agreement reported in figures 3 and 4. The slightly higher values are likely caused by the drop being deformed and already pressed against the bubble interface at the moment of the impact in the drop impact regime, while the simulation is initiated with both the drop and bubble being modelled as spheres. They are, however, well within the experimental uncertainty. A fitted version of (3.1), $r_{n}/r_{d,0}=C\sqrt {3u_{i}(t-t_{\mu })/r_{d,0}}$, has been suggested (Jian et al. Reference Jian, Josserand, Popinet, Ray and Zaleski2018; Wang et al. Reference Wang, Liu, Bayeul-Lainé, Murphy, Katz and Coutier-Delgosha2023), where $C$ and $t_{\mu }$ are fitting parameters representing a scaling constant and the time delay correcting for viscous and cushioning effects, respectively. The fitted parameters found for the experiment and simulation are $C_{exp}=1.03$, $C_{sim}=1.02$, $t_{\mu,exp}=-1.78\ \mathrm {\mu }$s and $t_{\mu,sim}=0\ \mathrm {\mu }$s for the drop impact regime and $C_{exp}=1.11$, $C_{sim}=1.06$ and $t_{\mu,exp}=-1.83\ \mathrm {\mu }$s, $t_{\mu,sim}=0.53\ \mathrm {\mu }$s for the partial deposition regime. The scaling constants are for all cases close to one, which supports the modelling of the necking region. The very short time delays suggest that air cushioning and viscous effects are negligible. The negative values are likely caused by the necking region having begun forming at an earlier time in the experiment, but the exact timing of impact cannot be resolved at the current spatiotemporal resolution. The fitted curves are not displayed in figure 5 to maintain visual clarity since they are so close to the uncorrected model.

Figure 5.

Temporal evolution of the necking radius for the drop impact ($We_{d} = 1940$) and partial deposition ($We_{d} = 16$) regimes. All quantities are normalised to the drop's initial radius, $r_{d,0}$, the drop impact speed, $u_{i}$ and the drop characteristic time, $r_{d,0}/u_{i}$.

In the drop impact regime, the necking region eventually evolves into a sheet jet starting from the second frame of figure 3, which corresponds to $u_{i}t/r_{d,0}=0.51$. Since the sheet jet entrains a small fraction of drop contents with it, understanding and predicting the way it spreads within the air bubble is of interest. The model given by (3.1) agrees well with the temporal evolution of the spreading radius of the sheet jet beyond the limit of $r_{n}/r_{d,0} \leq 1$, up to $u_{i}t/r_{d,0}=0.69$ as displayed in figure 6(a), where $r_{s}=r_{n}$ before the apparition of the sheet jet. Past that time, the spreading radius of the sheet jet converges to a more or less constant value of $r_{s}/r_{d,0} = 1.5$ up to $u_{i}t/r_{d,0}=2.75$, corresponding to the second row of images in figure 3. This is likely caused by the sheet jet spraying daughter droplets from its tip as it becomes unstable starting from $u_{i}t/r_{d,0} \approx 1$. Note that a whole-process theory for drop impact, such as that proposed by Bisighini et al. (Reference Bisighini, Cossali, Tropea and Roisman2010) for the spreading of the drop crater, does not capture the spreading of the sheet jet because of the shock-induced bubble oscillation. Indeed, during the bubble collapse and expansion, Bisighini et al.'s (Reference Bisighini, Cossali, Tropea and Roisman2010) theory over- and under-estimates the sheet jet spreading, respectively. The radial speed of the neck and early sheet jet can be estimated by differentiating (3.1) with respect to time, yielding

(3.2)\begin{equation} \frac{\dot{r}_{n}}{u_{i}}=\frac{3}{2\sqrt{3u_{i}t/r_{d,0}}}. \end{equation}

This provides a good approximation of the spreading speed of the neck at an early stage as displayed in figure 6(b). However, the model soon overestimates the recorded radial speed as the radial evolution of the sheet jet halts. The radial deceleration occurs prior to the onset of instabilities at the sheet jet tip which eject daughter droplets and is likely the main cause of them. The maximum speed is observed during the expansion of the necking region, with $\dot {r}_{s} = 28.02\ \textrm {m}\ \textrm {s}^{-1}$. Past $u_{i}t/r_{d,0} = 2.75$, the sheet jet radius starts decreasing and the radial speed becomes negative as the sheet jet tip converges back towards the central axis of the GED, as observed in the last row of frames of figure 3.

Figure 6.

Temporal evolution of the crown's (a) radius, (b) radial speed, (c) height and (d) speed along the GED axis ($We_{d} = 1940, Re_{d} = 11\ 000$). All quantities are normalised to the drop's initial radius, $r_{d,0}$, the drop impact speed, $u_{i}$ and the drop characteristic time, $r_{d,0}/u_{i}$.

The height of the splash, $h_{s}$, measured from the drop proximal side, and speed at which it travels downstream are measured and displayed in figures 6(c) and 6(d), respectively. Prior to trackable sheet jet formation, $h_{s}$ is measured as the distance of the cusp of the bubble to the initial position of the drop's proximal side. One notices a slight acceleration phase during the necking stage up to sheet jet formation, where the sheet jet reaches a maximum speed of $\dot {h}_{s}=33.44\ \textrm {m}\ \textrm {s}^{-1}$ at $u_{i}t/r_{d,0} = 0.36$. The splash then grows at a more or less constant speed of $\dot {h}_{s,exp}\approx 15.72\ \textrm {m}\ \textrm {s}^{-1}$ and $\dot {h}_{s,sim}\approx 15.31\ \textrm {m}\ \textrm {s}^{-1}$ until it reaches the distal side of the bubble at $u_{i}t/r_{d,0} \approx 2.8$. Quantitative analysis beyond that point becomes arduous in the experiment due to the strong deformations of the bubble interface.

3.2. The jet impact regime

The third regime of shock-induced GED release, named the jet impact regime and reported for the first time herein, is shown in figure 7. Despite the considerable initial deformation of the bubble at the time of shock wave impact in the experiment, the numerical simulation agrees reasonably well with the experiment.

Figure 7.

Image sequence of an example of shock–GED interaction within the jet impact regime ($We_{d} = 2810$, $We_{j} = 1580$, $Re_{d} = 9730$, $Re_{j} = 5450$). The upper halves display the numerically computed volume fractions of air and drop water in red and black, respectively. The lower halves show corresponding experimental radiographs. The dimensionless time, $t/(r_{b,0}\sqrt {\rho / p_{b,0}})$, starting from the shock contact with the GED, is indicated in the bottom-left corner.

In this regime, the properties of the shock wave and the GED are very similar to those presented in the previous regimes. The initial bubble radius is $r_{b,0}=1.20$ mm and the drop-to-bubble size ratio is $\varepsilon =0.19$. The bubble centre is located 5.83 mm from the shock wave source. The shock wave parameters are $p_{max}=54.45$ MPa, $j=2.44$ Pa s and $M = 1.06$. The shock-induced bubble dynamics in frames 1–3 are unaffected by the small drop located 0.26 mm downstream from the centre of the bubble, and thus avoiding any early contact with the bubble wall. As a result, the proximal side of the bubble has sufficient space to evolve into a liquid jet which can be observed in the following frames. Such high-speed jets are expected after the interaction of bubbles with shock waves of substantial pressure impulse (Philipp et al. Reference Philipp, Delius, Scheffczyk, Vogel and Lauterborn1993; Ohl & Ikink Reference Ohl and Ikink2003; Bokman et al. Reference Bokman, Biasiori-Poulanges, Meyer and Supponen2023b). Here, jetting is likely favoured by weak surface tension effects associated with the large size of the bubbles, as well as the additional contribution of the flow induced by the growth of the laser-induced vapour bubble. This might explain the discrepancies between the observed and computed speeds of the jet since the vapour bubble dynamics is slightly different in the experiment and simulation as displayed in figure 2(b). In the fourth radiograph, the jet impacts the liquid drop at a speed of $u_{i, exp}=21.02\ \textrm {m} \textrm {s}^{-1}$, which is faster than in the numerical simulation, $u_{i, sim}=16.96\ \textrm {m} \textrm {s}^{-1}$. To the best of the authors’ knowledge, only a few studies have investigated the impact of liquid jets with drops (Quetzeri-Santiago & Rivas Reference Quetzeri-Santiago and Rivas2023; Mulbah et al. Reference Mulbah, Kang, Song and Yin2024). Previous investigations on binary drops (Qian & Law Reference Qian and Law1997; Pan & Suga Reference Pan and Suga2005; Pan, Chou & Tseng Reference Pan, Chou and Tseng2009; Roisman et al. Reference Roisman, Planchette, Lorenceau and Brenn2012; Cimpeanu & Moore Reference Cimpeanu and Moore2018), however, show strong similarities with the dynamical process observed in the jet impact regime. One can simplify the liquid jet impact with a drop as an unequal-sized drop collision (Ganti, Khare & Bravo Reference Ganti, Khare and Bravo2020), where an almost-cylinder-shaped drop impacts a larger spherical liquid drop. The present analogy is restricted to initial jet radii smaller or equal to the radii of the drops they are impacting, $r_{j,0} \leq r_{d,0}$. The smaller size of the jet with respect to the drop induces drop deformations upon impact in the shape of a thin sheet jet opening to the opposite direction of the travelling jet, i.e. upstream. Here, the experimental Weber and Reynolds number computed for the drop and jet are $We_{d} = 2810$, $We_{j} = 1580$ and $Re_{d} = 9730$, $Re_{j} = 5450$, which are in the same order of magnitude as the inertia-driven impact in the drop impact regime. At the early stage, a necking region forms, starting from the fourth frame in figure 7, which evolves into a sheet jet that gives a bag-like shape to the drop. As the bag spreads within the air bubble, it adopts an inflection point near its rim in the numerical simulation, with the sheet jet tip pointing perpendicularly to the jet axis, while in the experiment, the rim of the bag points upstream, parallel to the jet axis. This is particularly visible in the sixth frame of figure 7. The higher impact velocity in the experiment certainly contributes to this discrepancy, however other factors such as the assumption of two-dimensional axisymmetry in the simulation might also play a role. Due to the high momentum of the jet compared with the small liquid mass of the drop, the jet continues to travel downstream, displacing and deforming the drop located on its path. At first, the drop mainly deforms while maintaining its distal side at the same location, with most of its mass being pushed to the sides of the jet. This can be observed as the spreading radius of the sheet jet expands and the drop thickness along the jet axis reduces, as seen in frames 4–6 of figure 7. Once the drop has adopted a fine bag-like structure with a large surface area, its distal side starts moving with the jet towards the back of the bubble. The drop moves faster in the experiment than in the numerical simulation, which can easily be explained by the greater experimental impact speed. On the two last frames of figure 7, the bag has become so thin that it is arduous to track both in the experiment and numerical simulation. Finally, the drop impacts the distal bubble surface and mixes with the surrounding liquid, as seen in the last frame of figure 7, provoking a slight increase in pressure at the impact point. The drop rim also experiences instabilities turning it into a finger-like structure from which some liquid is ejected in the form of a spray of daughter droplets. This is very similar to the crown instability reported in the drop impact regime.

The spreading of the necking region and early formation of the sheet jet is quantified in figure 8(a). Here quantities are non-dimensionalised to the initial radius of the jet tip at the moment of impact, $r_{j,0}$, instead of the initial drop radius. The scaling of quantities by the jet's initial radius, where $r_{j,0}= 0.13$ mm, is herein used because of the smaller jet size with respect to the drop, in which it becomes the impactor and the drop becomes the impacted. This new scaling is further justified by the simulated dynamics of drops of different sizes ($\varepsilon =0.19$ and $\varepsilon =0.36$) resting at the same location in a 1.20-mm bubble approximately collapsing on a single curve. The initial jet radius and impact speed for the larger drop are $r_{j,0}= 0.15$ mm and $u_{i}= 14.03\ \textrm {m}\ \textrm {s}^{-1}$, respectively. Unsurprisingly, the spreading of the necking region displays once more a square root scaling to the dimensionless time. However, a better agreement through

(3.3)\begin{equation} \frac{r_{n}}{r_{j,0}}=\sqrt{\frac{2u_{i}t}{r_{j,0}}}, \end{equation}

compared with the corrected factor used in (3.1), is shown in figure 8(a). This is likely caused by the cylindrical shape of the impacting jet and, in fact, a similar scaling has previously been reported for the splashing of a jet on a thin liquid film (Bokman et al. Reference Bokman, Biasiori-Poulanges, Lukić, Bourquard, Meyer, Rack and Supponen2023a). A best fit of $r_{n}/r_{j,0}=C\sqrt {2u_{i}(t-t_{\mu })/r_{j,0}}$, where $C$ and $t_{\mu }$ are fitting parameters, to the experimental case and both numerical simulations yield $C_{exp}=0.92$, $C_{sim(\varepsilon =0.19)}=0.97$, $C_{sim(\varepsilon =0.36)}=1.00$ and $t_{\mu }=0\ \mathrm {\mu }$s for all. The scaling constants are close to one for both simulations, and larger for the experiments, which is expected considering the large uncertainty at such small scales. The absence of a time delay suggests once more the absence of air cushioning and viscous effects.

Figure 8.

Temporal evolution of (a) the necking radius, the drop's (b) opening radius, (c) radial speed, (d) location and (e) speed along the GED axis from the first instant the jet contacts the drop. All quantities are normalised to the jet's initial radius, $r_{j,0}$, the drop impact speed, $u_{i}$ and the jet and drop characteristic times, $r_{j,0}/u_{i}$ and $r_{d,0}/u_{i}$. The drop-to-bubble size ratio, $\varepsilon$, is indicated for a 1.20-mm bubble.

In the context of payload delivery, the deformation and displacement of the drop are relevant. The spreading radius of the drop rim, $r_{d}$, is substantially smaller in the experiment than in the simulations, as shown in figure 8(b), and is expected from the differences in shape observed in figure 7. A square root scaling with the dimensionless time is obvious and, interestingly, the experimental spreading radius agrees with the model given by (3.3), also shown in figure 8(b). The speed at which the drop rim opens, $\dot {r}_{d}$, can be estimated from the spreading of the neck, assuming $r_{d}=r_{n}$. Differentiating (3.3) with respect to time yields

(3.4)\begin{equation} \frac{\dot{r}_{n}}{u_{i}}=\frac{1}{\sqrt{4u_{i}t/r_{j,0}}}. \end{equation}

The model agrees reasonably well with the experimentally observed radial speeds and slightly underestimates the numerical results, as observed in figure 8(c). At later stages, the agreement between the experimental speed and numerical speed improves for $\varepsilon =0.19$, where $\dot {r}_{d, exp} = 4.21\ \textrm {m}\ \textrm {s}^{-1}$ and $\dot {r}_{d, sim(\varepsilon =0.19)} = 4.24\ \textrm {m}\ \textrm {s}^{-1}$ at $u_{i}t/r_{j,0}\approx 19$. The radial speed for the larger drop at a similar dimensionless time is $\dot {r}_{d, sim(\varepsilon =0.36)} = 1.19\ \textrm {m}\ \textrm {s}^{-1}$.

The displacement of the drop along the jet axis is another parameter of interest in the context of payload delivery. Before being set in motion, the distal side of the liquid drop stays immobile for $u_{i}t/r_{d,0} \lesssim 2$, where $r_{d,0}/u_{i}$ is the characteristic time of the drop and $u_{i}t/r_{d,0} = 0$ is the time at which the jet contacts the drop. Here, Buckingham–Pi theory is employed to find an equation to describe the displacement of the distal point on the drop surface, $x_{d}$, once it is set in motion by the jet. The theorem yields three different groups based on the main quantities of interest, mainly ${\rm \pi} _{1} = x_{d}/r_{d,0}$, ${\rm \pi} _{2} = r_{j,0}/r_{d,0}$ and ${\rm \pi} _{3} = u_{i}t/r_{d,0}$, which can be expressed as ${\rm \pi} _{1} = f({\rm \pi} _{2},{\rm \pi} _{3})$ in the form of

(3.5) \begin{equation} \frac{x_{d}}{r_{d,0}} = D \left(\frac{r_{j,0}}{r_{d,0}}\right)^{\alpha_{1}} \left[\frac{u_{i}(t-t_{x_{d}})}{r_{d,0}} \right]^{\alpha_{2}}, \end{equation}

where $D$, $\alpha _{1}$ and $\alpha _{2}$ are scaling constants to be fitted. A best fit from the first instant the distal point is set in motion, $t_{x_{d}}$, corresponding to $t_{x_{d}, exp}=33.44\ \mathrm {\mu }$s in the experiment and $t_{x_{d}, sim(\varepsilon =0.19)}=36.96\ \mathrm {\mu }$s and $t_{x_{d}, sim(\varepsilon =0.36)}=73.92\ \mathrm {\mu }$s in the numerical simulations, until $u_{i}t/r_{d,0}=10$ yields $D_{exp} = 0.13$, $D_{sim(\varepsilon =0.19)} = 0.14$ and $D_{sim(\varepsilon =0.36)} = 0.12$ and $\alpha _{2} = 1.5$ for all data sets. This yield the following scaling law

(3.6)\begin{equation} \frac{x_{d}}{r_{j,0}} = D \left[\frac{u_{i}(t-t_{x_{d}})}{r_{d,0}} \right]^{3/2}, \end{equation}

which adequately predicts the displacement of the distal side of the drop as shown in figure 8(d). Based on (3.6), an estimation of the speed at which the drop comes into contact with the bubble distal side can be drawn by simple time differentiation:

(3.7)\begin{equation} \frac{\dot{x}_{d}}{u_{i}} = D \left(\frac{r_{j,0}}{r_{d,0}}\right) \left[\frac{u_{i}(t-t_{x_{d}})}{r_{d,0}} \right]^{1/2}. \end{equation}

Figure 8(e) shows the time evolution of the speed of the drop's distal side. A relatively good agreement with the model in (3.7) using the fitted parameters $A$ is obtained for each case. The terminal speed at which the drop travels right before impacting the distal bubble interface is $\dot {x}_{d, exp} = 7.36\ \textrm {m} \textrm {s}^{-1}$ in the experiment, $\dot {x}_{d, sim(\varepsilon =0.19)} = 5.94\ \textrm {m}\ \textrm {s}^{-1}$ in the corresponding simulation and $\dot {x}_{d, sim(\varepsilon =0.36)} = 2.76\ \textrm {m}\ \textrm {s}^{-1}$ for the larger drop. Despite a non-negligible displacement of the part of the drop directly located on the jet's path towards the back of the bubble, the entire drop's centre of mass, computed from the numerical simulations, does not evolve significantly as observed in figure 8(d). This is mainly caused by the lateral displacement of the drop rim perpendicular to the jet counterbalancing the drop displacement towards the distal bubble side.

Note that some regimes can also be combined. For example, a drop located on the GED axis, close to the distal side might first coalesce and then be impacted by a high-speed liquid jet. This leads to a more complex type of drop release which is outside of the scope of the current work.

3.3. Off-centre collision

Only axisymmetric cases have thus far been investigated, as they allow for a direct comparison with the numerical simulations, which are kept axisymmetric to minimise computational cost. However, in practice, the initial location of a drop within the bubble is random at the moment the shock wave impacts the GED, and the interfacial collisions are often asymmetric and off-centre. In the drop impact and jet impact regimes, the dynamics are affected by off-centre collisions, while in the partial deposition regime their effect is minor. The drop's eccentricity with respect to the impact is quantified by the impact parameter, $\psi _{d} = y_{d,0}/r_{d,0}$, which is defined as the distance of the drop's centre to the central axis of the bubble normalised to the initial radius of the drop.

In the drop impact regime, off-centre collisions induce asymmetrical sheet jets, as observed in the example of figure 9(a). The impact parameter here is $\psi _{d} = 0.42$ and the initial drop-to-bubble size ratio is $\varepsilon =0.40$. The shock wave inducing the GED dynamics has a peak pressure, pressure impulse and Mach number of $p_{max}=53.38$ MPa, $j=2.39$ Pa s and $M=1.06$, respectively. The asymmetric sheet jet dynamics shows strong similarities with the single-sided splashing reported in oblique drop impacts on liquid surfaces (Gielen et al. Reference Gielen, Sleutel, Benschop, Riepen, Voronina, Visser, Lohse, Snoeijer, Versluis and Gelderblom2017). In the present study, the bubble collapse is asymmetrical, with the proximal side collapsing first and faster than its distal side. The part of the drop located on the bubble axis, where the bubble collapse speed is maximal, is impacted by the bubble wall at a higher velocity than the lower part of the drop, which results in a more violent splashing on the upper side of the drop. Despite the asymmetric sheet jets, in essence, the splashing dynamics stay fundamentally the same as observed in the axisymmetric drop impact regime.

Figure 9.

Radiographs showing both the initial location of a drop in a bubble and the off-centre (a) drop impact ($We_{d} = 768$, $Re_{d} = 7180$) and (b) jet impact ($We_{d} = 631$, $We_{j} = 253$, $Re_{d} = 14\ 600$, $Re_{j} = 5860$) regimes. The dimensionless time, $t/(r_{b,0}\sqrt {\rho / p_{b,0}})$, starting from the first contact with the surrounding liquid, drop and jet impact parameter, $\psi _{d}$, and, $\psi _{j}$, and jet deflection angle, $\theta _{j}$, are indicated. The central axis of the bubble and drop centre are shown as a dotted line and dark circle, respectively. (c) Jet deflection angle evolution as a function of the impact parameter.

The off-centre impact dynamics of a drop located in the jet impact regime is shown in figure 9(b). The impact parameter here is slightly modified to account for the thickness of the impacting jet, and is defined as $\psi _{j} = y_{d,0}/(r_{j,0}+r_{d,0}) = 0.58$. The initial drop-to-bubble size ratio is $\varepsilon =0.41$ and the shock wave has a peak pressure, pressure impulse and Mach number of $p_{max}=57.36$ MPa, $j=2.57$ Pa s and $M=1.06$, respectively. The off-centre jet impact on the drop results in the jet being deflected from its original course. Here, the deflection angle of the jet, $\theta _{j}$, can be estimated solely based on the impact parameter, $\psi _{j}$, as displayed in figure 9(c). Results indicate that upon impact, the jet deflection angle is dictated by the local inclination of the drop surface at the contact point with respect to the central drop axis, and can be expressed as

(3.8)\begin{equation} \theta_{j} = \arccos{ ( \psi_{j} )} = \arccos{ \left( \frac{y_{d,0}}{r_{d,0}+r_{j,0}} \right)}. \end{equation}

A good agreement between the model and the experimentally observed deflection angles is found. This suggests a certain degree of rigidity of the liquid drop upon contact. In fact, the drop acts as an unmovable rigid surface which reorients the jet completely in the first instants after the impact. At a later stage, drop deformation comes into play, which is why only the tip of the jet is deflected by $\theta _{j}$ and the rest of the drop adopts a more complex shape. The tip of the jet, which is likely mixed with drop contents, finally impacts the bubble interface and mixes with the surrounding liquid.

3.4. Drop dynamics regime map

The two most important parameters dictating the drop release process in GEDs are the initial location of the drop within the bubble and the shock wave energy that results in different collapse speeds. The different regimes are mapped in figure 10(a) based on the initial location of the drop within the bubble at the moment the shock wave arrives. The shock parameters are kept approximately constant with a mean shock wave peak pressure and pressure impulse of $p_{max} = 59.11 \pm 13.22$ MPa and $j = 2.65 \pm 0.6$ Pa s, respectively. The $x$–$y$-coordinates of the initial location of the drop are normalised to the bubble's initial radius, $r_{b,0}$. The drop-to-bubble size ratio, $\varepsilon$, is also provided, showing that in most cases, the drop size is 40 % of the bubble size and is only smaller in some isolated cases. The consistent drop-to-bubble size ratio is a result of the high repeatability of the GED production.

Figure 10.

(a) Mapping of the initial position of the drop within the bubble, normalised to the bubble's initial radius, $r_{b,0}$, as it interacts with a shock wave of pressure impulse $j = 2.65 \pm 0.6$ Pa s. The drop release regimes, namely the drop impact regime (blue), partial deposition regime (grey) and jet impact regime (red), are indicated for a range of drop-to-bubble size ratios, $\varepsilon = r_{d,0}/r_{b,0}$, which is qualitatively emphasised by the marker size. The dashed circle shows the minimum equivalent size reached by the bubble. (b) The local curvature of the bubble at the impact location, $\mathcal {K}_{b}$, normalised to the drop's curvature, $\mathcal {K}_{d}$, and the drop's Weber and Reynolds numbers, delimit a regime map for all three drop release regimes. Quantities are collected at the moment of impact. The vertical and horizontal limits are $We^{1/2}Re^{1/4} = 76$ and $\mathcal {K}_{b}/\mathcal {K}_{d} = 1$, respectively.

The release of a drop located on the proximal side of the bubble takes place within the drop impact regime. That is because the proximal side of the bubble undergoes higher accelerations and speeds compared with the rest of the bubble as it collapses asymmetrically. This region appears to span along the entire $y/r_{b,0}$ domain as long as $x/r_{b,0} \leq -0.1$, where $(x/r_{b,0},y/r_{b,0}) = (0,0)$ is the bubble centre. In this region, a liquid jet does not have time to form and the drops are impacted by a relatively flat or even concave surface. The partial deposition regime instead occupies the annular region in the range 0.6–1.0 of the bubble's initial radius for $x/r_{b,0} > -0.1$. In this region close to the bubble wall, the interface contracts less violently than the proximal bubble side, where the jet forms. The mean of all the minimum equivalent bubble radii, $r_{min}=0.79r_{b,0}$, recorded from the bubble's observable area in the experiment after their contraction, is shown as a dashed line in figure 10(a). The drops located in the region between this circle and the one denoting the bubble's initial radius inevitably contact the outer liquid, initiating the partial coalescence regime. Larger drops may be located closer to the bubble centre and still coalesce. This is highlighted by the higher number of large drops near the bubble centre in comparison with the smaller drops located closer to the bubble wall. The jet impact regime only occurs in the central region of the GED where the jet has space to form and hits the drop before coalescence with other parts of the bubble wall can take place. The drops in this regime are located relatively close to the central bubble axis. This region is within the range $x/r_{b,0} = -0.1$ to $x/r_{b,0} = 0.5$ and $y/r_{b,0} = -0.5$ to $y/r_{b,0} = 0.5$.

The identified regime map in figure 10(a) is, however, rather specific to the shock type and bubble sizes used here. Below a certain shock pressure threshold, bubbles only respond through radial oscillations without producing any jets (Ohl & Ikink Reference Ohl and Ikink2003; Bokman et al. Reference Bokman, Biasiori-Poulanges, Meyer and Supponen2023b). Under such conditions, the regime map would look different, as the jet impact regime would be absent from the map. Similarly, smaller GEDs undergoing the same pressure driving as here may react more violently and collapse to smaller minimum radii leading to the absence of both the jet impact and partial deposition regimes. Therefore, a more general approach for regime identification is needed.

Although the main driver of the overall release process is the shock wave, the speed at which the drop is impacted by the surrounding liquid, $u_{i}$, is the actual variable dictating the drop release regime. More precisely, it is the competition between inertial, capillary and viscous forces that dictates the regime taking place. The drop impact and jet impact regimes are clearly inertia-driven, while in the partial deposition capillary effects play a much more important role. In drop impact theory, both the Weber and Reynolds numbers are leveraged to describe the competition between all three forces. Past studies have investigated the transition between the splashing and deposition of drops on flat surfaces and found that this happens for a given experimental parameter $K$ (Mundo, Sommerfeld & Tropea Reference Mundo, Sommerfeld and Tropea1995; Josserand & Zaleski Reference Josserand and Zaleski2003; Zhang, Zhang & Zheng Reference Zhang, Zhang and Zheng2008; Cimpeanu & Moore Reference Cimpeanu and Moore2018; Sykes et al. Reference Sykes, Fudge, Quetzeri-Santiago, Castrejón-Pita and Castrejón-Pita2022). Thoroddsen et al. (Reference Thoroddsen, Thoraval, Takehara and Etoh2011) defined the splashing parameter as a local Weber number, $We_{l} = K^{2}$, characterising the sheet jet, which is done by expressing the ejecta sheet thickness as $\delta _{s} \sim \sqrt {\mu 2r_{d,0} /(\rho u_{i})}$ and the ejecta velocity as $u_{s} \sim u_{i}\sqrt {Re}$. By substitution, the splashing parameter becomes

(3.9)\begin{equation} K = \sqrt{We_{l}} = \sqrt{\frac{\rho \delta_{s} u_{s}^{2}}{\gamma}} = We^{1/2}Re^{1/4}. \end{equation}

Here, $K_{c}$ is a critical number, the value of which has been found to vary in experiments between a few tens and a few hundreds (Josserand & Zaleski Reference Josserand and Zaleski2003; Cimpeanu & Moore Reference Cimpeanu and Moore2018; Sykes et al. Reference Sykes, Fudge, Quetzeri-Santiago, Castrejón-Pita and Castrejón-Pita2022). Over the wide range of experiments conducted herein, the transition between the splashing dynamics of the drop impact regime and coalescence witnessed within the partial deposition regime occurs for a critical number $56 < K_{c} < 96$, which is close to values reported previously (Mundo et al. Reference Mundo, Sommerfeld and Tropea1995).

Another general parameter playing a major role in defining the drop dynamics regime is the shape of the air bubble surface as it makes contact with the drop, may it be a flat interface or the tip of a jet. Here, the local curvature of the bubble surface, $\mathcal {K}_{b}$, is measured at the contact point with the drop and normalised to the drop's curvature, $\mathcal {K}_{d} = 1/r_{d,0}$. The local curvature is assessed from the inside of the bubble, meaning that a convex and concave interface contours yield positive and negative curvatures, respectively. Figure 10(b) displays the general regime map, where a clear demarcation between the different regimes is found. Clearly the Weber and Reynolds numbers play an important role in defining whether inertia will be driving strong dynamics such as the formation of a sheet jet or the deformation of the drop upon impact with the jet, as observed by the vertical limit set at $K_{c}=76$. The difference between the drop impact and jet impact regimes can be explained by the local curvature of the bubble interface-to-drop curvature ratio. A value larger than one ($\mathcal {K}_{b}/\mathcal {K}_{d} > 1$) suggests that the local radius of jet impacting the drop is smaller than the drop radius, inducing the jet impact regime, for which the drop adopts a bag-like shape. For values smaller than zero ($\mathcal {K}_{b}/\mathcal {K}_{d} < 0$), where zero means that the bubble interface is flat and negative values means the interface has not yet started becoming convex, the drop is released in the drop impact regime. Between zero and one ($0 < \mathcal {K}_{b}/\mathcal {K}_{d} < 1$), the bubble interface is convex and larger than the drop. The impact of the curved interface with the drop causes the drop to deform in a flat bag-like shape Sykes et al. (Reference Sykes, Fudge, Quetzeri-Santiago, Castrejón-Pita and Castrejón-Pita2022). This limit region is however hardly distinguishable in the radiographs at current resolutions. There are no experimental points in the top left side of the regime map, where jets must impact the drop at low speeds to enable coalescence between the two, which has not been witnessed experimentally.

3.5. Drop release

Finally, the implications of the identified regimes for shock-driven GED dynamics to the release of the drop within the surrounding liquid are investigated. The drop release is investigated numerically by tracking the volume of liquid initially within the drop, which can be traced by the colour applied to it. The assumption of axisymmetry in the numerical simulation is likely to influence the problem. Nonetheless, the good agreement between the experimental radiographs and numerical simulations so far suggests that the general dynamical behaviour of the drop release within the surrounding liquid must be well captured by the numerical simulation. Since the drop dissemination is a relatively slow process, the effects of an impulse on the drop dynamics is investigated. The impulse effectively comprises the combined effects of a laser-induced shock wave, the growth and collapse of a laser-induced vapour bubble and a collapse-induced shock wave.

Figures 11(a)–11(c) display the dissemination and mixing through time of drops in the drop impact,partial deposition and jet impact regimes, respectively. All three drops have an initial radius of 0.44 mm and are within bubbles of 1.2 mm radius. The shock driving conditions are selected to induce impact speeds close or equal to those within the examples presented in all three regimes. In each case, the shock wave comes from the left and the initial location of the drop and bubble are the same as the ones used for the test case displayed in figure 3 for the drop impact and partial deposition regimes, and the impact speeds are $u_{i}= 10.98\ \textrm {m} \textrm {s}^{-1}$ and $u_{i}= 2.94\ \textrm {m}\ \textrm {s}^{-1}$, respectively. In the jet impact regime, the drop and bubble are initialised at the same location as the test case displayed in figure 7 and the jet impacts the drop at $u_{i}= 14.31\ \textrm {m}\ \textrm {s}^{-1}$.

Figure 11.

Temporal evolution of the dissemination of a drop's volume fraction within the (a) drop impact ($We_{d} = 1940$, $Re_{d} = 11\ 000$), (b) partial deposition ($We_{d} = 104$, $Re_{d} = 2580$) and (c) jet impact ($We_{d} = 2460$, $We_{j} = 842$, $Re_{d} = 12500$, $Re_{j} = 4280$) regimes for dimensionless times of $u_{i}t/r_{d,0} = 0, 4, 8, 12, 16$ and 20. The initial location of the drop is indicated as a dashed circle on every frame. (d) Time evolution of the dissemination of the drop's mean density of distribution, $D$, within the surrounding liquid normalised to its initial state, $D_{0}$. The colour bars display the maximum spreading radius of the drop contents, normalised to the drop's initial radius.

The drop dissemination of the drop impact regime displayed in figure 11(a) is quantified by measuring its mean density of distribution (Stone & Stone Reference Stone and Stone2005) normalised to its initial value before contacting the surrounding liquid, $D/D_{0}$. The mean density of distribution is computed at each time step by averaging the volume fraction of drop contents by grid cell over the region of interest, while only considering cells for which the colour associated with the drop is non-zero. The time evolution of $D/D_{0}$ and dimensionless maximum spreading radius of drop contents, $r_{D,max}/r_{d,0}$, are shown in figure 11(d). Once the shock wave initiates the bubble collapse and the surrounding liquid impacts the drop, at $u_{i}t/r_{d,0}=0$, the major part of the drop quickly coalesces at the proximal bubble side. While splashing unfolds, drop contents are entrained to the distal bubble side by the sheet jet. In the second frame of figure 11(a), the mean density of distribution of the drop has already dropped to $0.46D_{0}$ thanks to the strong inertial driving of the splash. The splash transports the drop contents across the bubble, rapidly increasing the maximum spreading radius to $r_{D, max} \approx 5r_{d,0}$. Dissemination of the drop is further enhanced by the central jet that forms at the latest stage of splashing and travels both upstream and downstream. It deforms the main bulk of the drop upon impact and brings the mean density of distribution of the drop down to a value of $0.35D_{0}$. At $u_{i}t/r_{d,0}\approx 12$, the drop is impacted by the second shock wave emitted at the collapse of the vapour bubble. This induces a second contraction of the GED which briefly reconcentrates part of the drop contents, driving the mean density of distribution of the drop up to $0.40D_{0}$. Once the second shock wave has passed, the bubble relaxes and the main bulk of the drop is entrained by the vapour bubble-induced flow towards the origin of the shock wave. This increases the distance over which the drop disseminates to almost 12 times the initial drop radius. Parts of the drop contents can be seen clustering to the right, at the gas–liquid interface, on the two last frames of figure 11(a). At that point, the impulsive driving is over and the mean density of distribution of the drop has reached $0.29D_{0}$.

The partial deposition regime, displayed in figure 11(b), shows a much milder dissemination rate in figure 11(d). The bubble collapse is driven by a weaker impulse, which results in coalescence of the drop without splashing at the proximal bubble side, as soon as the bubble surface impacts the drop. The post-collapse expansion of the bubble interface induces stretching and necking of part of the drop. In the third and fourth frames of figure 11(b), one observes the formation of a daughter droplet. However, before the daughter droplet can detach from the main bulk of the drop, the second collapse of the bubble induces the total coalescence of the drop. Meanwhile, the main bulk of the drop is entrained by the vapour bubble-induced flow towards the shock wave origin, as observed in the last two frames of figure 11(b). Quantitatively, the dissemination of the drop is clearly not very efficient as the mean density of distribution of the drop slowly diminishes to $0.80D_{0}$ and the maximum spreading radius of the payload barely reaches $r_{D, max}= 2r_{d,0}$ at the end of the impulsive driving.

In the jet impact regime displayed in figure 11(c), the release dynamics of the drop are quite different from the ones previously introduced. The impact of the jet with the drop first deforms the drop before setting the entire bulk in motion at $u_{i}t/r_{d,0}\approx 8$. At this instant, the maximum spreading radius of the drop increases thanks to the liquid being pushed to the sides and the drop adopting a bag-like shape. As time passes, the bag becomes larger and thinner, inducing a steady dissemination of the drop until the mean density of distribution of the drop reaches $0.45D_{0}$ at $u_{i}t/r_{d,0}\approx 12$. The bag-shaped drop contacts the outer liquid during the second air bubble collapse. The increase in the mean density of distribution of the drop due to the bubble collapse-induced drop focusing is not as appreciable as in the drop impact case. The second shock wave, however, further improves the mixing of the drop to the surrounding, bringing the mean density of distribution of the drop down to $0.29D_{0}$. At that point, the impulsive release is over and a large amount of drop contents can be seen clustering at gas–liquid interfaces in the last frame of the image sequence of figure 11(c). This regime suggests a better distribution of the payload, although the maximum spreading radius of the drop is limited to approximately seven initial drop radii.

One observes a major difference in the mixing efficiency between the partial deposition regime and the two other regimes. The dissemination of the drop is clearly enhanced by inertia, and, unsurprisingly, higher Weber and Reynolds numbers provide a better framework for mixing in general. In the drop impact and jet impact regimes, the provided impulse is similar and the dissemination, although slower in the jet impact regime, ends up being the same. The advantage of the jet impact regime is that the distribution of the payload appears qualitatively more uniform and closer to the initial location of the GED, while in the drop impact regime most of it is entrained upstream towards the shock origin.