3.1. Bubble collapse-based approach

The formation of a liquid jet following the passage of a shock wave has been observed only in cases of strong nonlinear bubble collapse (Bokman et al. Reference Bokman, Biasiori-Poulanges, Meyer and Supponen2023), typical of inertial cavitation, rather than during the linear oscillations that occur in so-called ‘stable cavitation’ (Petit et al. Reference Petit, Bohren, Gaud, Bussat, Arditi, Yan, Tranquart and Allémann2015) and driven by weaker shocks. This suggests that the onset of jetting depends on the nature of the bubble collapse. It is therefore insightful to examine the spatiotemporal evolution of the jet tip during the bubble collapse (i.e. before the appearance of the jet), in figure 4(a). The jet tip location is plotted with respect to time for five different shock wave impulses. The temporal evolution of the jet tip adopts a convex distribution for $J/(r_{0}^{3}\sqrt {p_{b,0}\rho }) \gt 5$ , and a concave distribution for $J/(r_{0}^{3}\sqrt {p_{b,0}\rho }) \lt 5$ . In between, the temporal evolution of the jet tip is linear. These results suggest that the jet’s motion follows a quadratic evolution over time:

(3.1) \begin{equation} z_{j}(t) = a_{j} t^{2} + u_{j,0} t + z_{j,0}, \end{equation}

where, $a_{j}$ , is the effective mean and constant acceleration of the jet, $u_{j,0}$ , the initial velocity jump induced by the shock wave and $z_{j,0}$ the initial location of the jet. The acceleration of the jet as well as its initial velocity, induced by the impulsive effect of the shock wave impact, can be extracted from the radiographs. However, to minimise the uncertainty propagation from the first and second derivative of the jet tip’s displacement, both values are fitted to the trajectory of the jet and verified to be well within the experimental uncertainty. The best fit of (3.1) is performed for the acceleration and jump velocity for each case, assuming $z_{j,0}=0$ at the moment the shock impacts the proximal side of the bubble, and displayed as solid curves in figure 4(a). All convex curves correspond to bubbles exhibiting a stable jet that pierces their distal side and are characterised by a positive acceleration of the jet tip $a_{j} \gt 0$ . The pressure driving corresponding to a linear curve is close to the (previously mentioned) empirical jetting limit, and unsurprisingly, the acceleration is close to zero and the jet exhibits an unstable behaviour, which will be discussed later. For a negative acceleration, no jet is observed. For each curve, the pressure impulse and jet tip acceleration are displayed in dimensionless form using the bubble’s characteristic momentum and acceleration, $p_{b,0}/\rho r_0$ , respectively.

Figure 4. (a) Time evolution of the jet tip location during the bubble collapse. The dimensionless impulse and acceleration of the jet are indicated in brackets for each curve $[J/(r_{0}^{3}\sqrt {p_{b,0}\rho }),a_{j}/(p_{b,0} /\rho r_0)]$ . (b) Dimensionless acceleration of the jet tip during the collapse phase of the bubble against the dimensionless pressure impulse of the shock wave applied to the bubble surface. The dashed dark line indicates the jetting limit for $a_{j}/(p_{b,0} /\rho r_0) \approx 0$ and the grey curve shows the scaling law in (3.4).

The results are presented in figure 4(b), where the jet tip acceleration, $ a_j$ , is plotted against the pressure impulse. For cases where stable bubble jetting occurs, the acceleration is positive, with values in the range $ a_{j}/(p_{b,0}/\rho r_0) = 0$ – $3.1$ . At the jetting threshold, where jets are unstable, the dimensionless acceleration approaches zero, while for non-jetting bubbles, it is negative, ranging from $ a_{j}/(p_{b,0}/\rho r_0) = -0.3$ – $0$ . As expected, a positive acceleration is necessary to generate sufficient momentum for the proximal side of the bubble to evolve into a stable jet, overcoming the ‘pulling’ force from the rebound of the gas phase and the surface tension effects. The acceleration is expected to show a quadratic dependence on the pressure impulse, consistent with the inverse relationship between the impulsive mean jet speed and the impulsive collapse time (Bokman et al. Reference Bokman, Biasiori-Poulanges, Meyer and Supponen2023), $ u_j \propto \tau _{c}^{-1}$ , given that the collapse time inversely scales with the impulse, $ \tau _{c} \propto J^{-1}$ , and acceleration is the first derivative of velocity. This relationship can be obtained by assuming that the jet acceleration scales with the mean jet speed, $ u_j = C r_{0}/\tau _{c}^{-1}$ , where $C=1.43$ is a fitted constant, leading upon dimensional considerations to

(3.2) \begin{equation} a_j = \frac {C r_0}{\tau _{c}^{2}}, \end{equation}

where the impulsive collapse time is defined as (Bokman et al. Reference Bokman, Biasiori-Poulanges, Meyer and Supponen2023)

(3.3) \begin{equation} \frac {\tau _c}{r_0\sqrt {\rho /p_{b,0}}} = \frac {8\pi }{5} \left ( \frac {J}{r_{0}^{3}\sqrt {p_{b,0}\rho }} \right )^{-1}\!. \end{equation}

Substituting this into (3.2) yields

(3.4) \begin{equation} \frac {a_j}{p_{b,0}/\rho r_0} = C \frac {25}{64\pi ^{2}} \left ( \frac {J}{r_{0}^{3}\sqrt {p_{b,0}\rho }} \right )^{2}\!. \end{equation}

The equation describes the amplitude of the mean acceleration of the jet as a function of the pressure impulse and is found to scale well with that found for jetting bubbles (i.e. $J/(r_{0}^{3}\sqrt {p_{b,0}\rho }) \geqslant 5$ ) as indicated by the grey curve displayed in figure 4(b). This scaling is, however, only valid for bubbles that produce a jet.

3.2. Acceleration-induced interfacial instability-based approach

The acceleration of the proximal side of the bubble during collapse is a key factor in determining whether a jet will form. Acceleration-driven interfacial instabilities, such as the Rayleigh–Taylor and Richtmyer–Meshkov instabilities, which are classically associated with sustained and impulsive acceleration, respectively, may play a central role in this process. However, due to the complex interplay between the shock wave, bubble collapse and jet formation, it remains unclear to the authors whether the jet originates from one of these instabilities alone or from a more intricate coupling between multiple mechanisms.

The shock wave in itself could potentially induce the Richtmyer–Meshkov instability. However, the pulse generated in the present configuration is of very short duration with respect to the jetting dynamics and is immediately followed by an exponential decay as observed in figure 1(a), yielding an effective pressure profile characteristic of a blast wave. This inversion of the pressure gradient after the peak of the shock wave may rapidly reverse the vorticity deposited by the initial shock front, potentially cancelling the Richtmyer–Meshkov growth in its earliest stages. Moreover, blast-wave-driven interfacial instability studies have postulated that the deceleration caused by the rarefaction could induce Rayleigh–Taylor growth coupled to Richtmyer–Meshkov effects (Kuranz et al. Reference Kuranz2009; Drake Reference Drake2011). Assessing which of these competing effects dominates is difficult, as they occur within the first few frames of the experiment.

Additionally, the passage of the shock wave causes the bubble to collapse and rebound, effectively accelerating its interface. The stability of bubbles to non-spherical disturbances has been analysed using a model analogous to the spherical Rayleigh–Taylor instability (Birkhoff Reference Birkhoff1954; Plesset & Mitchell Reference Plesset and Mitchell1956). One common approach to modelling the dynamics of non-spherical bubbles involves expanding the interface perturbation in terms of spherical harmonics. While this method offers certain advantages, Plesset & Prosperetti (Reference Plesset and Prosperetti1977) noted that it is neither the only approach nor the most practical in all situations. Although mathematically elegant, this method becomes challenging to apply in the context of bubbles interacting with laser-induced shock waves, due to the inherently non-uniform and asymmetric nature of the shock-induced collapse. These complexities make it difficult to define an appropriate initial interface shape and velocity distribution, as well as to track the interface acceleration throughout the collapse. In this model, the mean radius, its velocity and acceleration are typically modelled using Rayleigh–Plesset or Keller–Miksis-type equations. Such equations indicate that during the shock passage and the early stage of collapse, the interface acceleration is directed inward, toward the bubble centre. Before the bubble reaches its minimum volume, the acceleration is directed outward, toward the surrounding liquid. This configuration renders the interface most susceptible to Rayleigh–Taylor instability (Brennen Reference Brennen2014). After its collapse, the bubble re-expands, and toward its maximum size, the direction of acceleration reverses once more, in the direction of the bubble centre, thereby stabilising the interface. A comprehensive analysis of this dynamic Rayleigh–Taylor instability behaviour lies beyond the scope of the present work.

This study presents a brief attempt to model the jet tip dynamics as the growth of an initial simple perturbation, through a Richtmyer–Meshkov instability initiated by the shock wave. Due to the complexity of the underlying dynamics, simplified linearised models are employed. More accurate approaches, such as Layzer-type models (Layzer Reference Layzer1955), based on potential flow theory could, in principle, offer a more precise approximation of the evolving jet tip (Zhang Reference Zhang1998; Mikaelian Reference Mikaelian2014; Zhou Reference Zhou2017). However, such models also fall outside the scope of the present analysis and are reserved for future investigation. The linearised models adopted here serve as foundational tools for providing a simplified interpretation of jet formation in bubbles subjected to laser-induced shock waves.

A single-mode perturbation in the context of acceleration-driven interfacial instabilities refers to the study of instability growth that originates from a disturbance or perturbation with a single, well-defined wavelength or frequency at the interface between two fluids of different densities. For a single-mode perturbation, this growth manifests as a regular, predictable pattern, such as a single sinusoidal wave that becomes progressively amplified over time. The study of acceleration-driven interfacial instabilities with single-mode perturbations provides a simpler framework to understand the fundamental mechanisms of instability growth, including the development of characteristic ‘spikes’ (penetrations of the heavier fluid into the lighter fluid) and ‘bubbles’ (penetrations of the lighter fluid into the heavier fluid). Here, considering the jet (or spike) to arise from a perturbation, $\eta$ , the perturbation growth rate can be computed based on a simple approach presented in figure 5(a) and first proposed by Haas & Sturtevant (Reference Haas and Sturtevant1987). For simplicity, the bubble interface is assumed to be a quasi-single-mode perturbation (Liang et al. Reference Liang, Zhai, Ding and Luo2019). The real part of the perturbation is initially described as

(3.5) \begin{align} \eta (x,0,0) &= {\textrm{Re}} \left [ \eta _{0}\exp { \left ( ik_{x}x \right )} \right ] = \eta _{0}\cos (k_{x}x), \nonumber\\ \eta (0,y,0) &= {\textrm{Re}} \left [\eta _{0}\exp {\left ( ik_{y}y \right )} \right ] = \eta _{0}\cos (k_{y}y), \end{align}

in $x$ and $y$ , respectively. Haas & Sturtevant (Reference Haas and Sturtevant1987) approximated the bubble interface by expressing the initial amplitude, $\eta _{0}$ , and the wavenumber, $k$ , of the perturbation as functions of the bubble radius,

(3.6) \begin{align} \eta _{0} &= r_{0}, \nonumber\\ k_{x} &= \frac {1}{r_0}, \end{align}

Figure 5. (a) Schematic drawing of the quasi-single-mode approximation of the bubble, defining the initial amplitude and wavelength of the initial perturbation. The dashed and full curves display the approximation proposed by Haas & Sturtevant (Reference Haas and Sturtevant1987) and in the present work, respectively. (b) Time evolution of the jet speed for ten jetting bubbles driven by different shock waves. The jet speed and time are made dimensionless by the wavenumber, $k$ , and impulsive perturbation growth rate in the sense of Richtmyer and Meshkov, $\dot {\eta }_{\textit{imp}}$ , and (3.9) is displayed in grey. (c) Comparison between a single test case and (3.11). The inset images show the bubble at its minimum size and when the jet reaches its distal side for the dimensionless times corresponding to the vertical dashed and dotted line, respectively. (d) Comparison between the theoretical prediction of the jet shape from (3.11) and the experiment for different non-dimensional times $k|\dot {\eta }_{\textit{imp}}|t$ . The initial bubble shape is indicated as a dotted circle.

as illustrated by the dashed sine wave in figure 5(a). These values are believed to offer a good approximation of the overall bubble dynamics in cases involving strong, sustained shock waves, as in the original study by Haas & Sturtevant (Reference Haas and Sturtevant1987). However, this approximation tends to overestimate the resulting jet speed (Haas & Sturtevant Reference Haas and Sturtevant1987) and appears to be inadequate for capturing jets generated by laser-induced impulsive shock waves. Although the reason for this discrepancy remains unclear, the authors believe it may stem from the coupling between bubble collapse and jet formation. The angle $\theta$ used to define the jetting region, taken here as $\theta \in [-\pi /6, \pi /6]$ , is selected empirically based on visual observations of the angular extent over which the jet structure develops consistently in experiments. Accordingly, the initial amplitude, $\eta _{0}$ , and wavenumber, $k$ , of the perturbation are defined as

(3.7) \begin{align} \eta _{0} &= r_{0}[1-\cos (\pi /6)] = \frac {(2 - \sqrt {3})r_0}{2}, \nonumber\\ &\,\,k_{x} = \frac {2\pi }{\lambda _{x}} = \frac {2\pi }{4r_0 \sin (\pi /6)} = \frac {\pi }{r_0}, \end{align}

where $\lambda$ is the single-mode wavelength. The problem is three-dimensional, as shown in figure 5(a), and the initial amplitude of the perturbation is taken to be identical in both the $x$ and $y$ directions; the wavenumbers satisfy $k_{x} = k_{y}$ . Thus, the total wavenumber is given by $k = \sqrt {k_{x}^2 + k_{y}^2} = \sqrt {2} / r_{0}$ for Haas & Sturtevant (Reference Haas and Sturtevant1987) and $k = \sqrt {2}\pi / r_{0}$ in the present work. These values of $\eta _{0}$ and $k$ therefore represent the magnitudes of the initial wave perturbation in three dimensions (Zhang & Sohn Reference Zhang and Sohn1999; Zhou et al. Reference Zhou2021).

Many models have been proposed to describe the perturbation growth rate in the sense of Richtmyer and Meshkov, $\dot {\eta }$ , over time, starting from Richtmyer (Reference Richtmyer1954) who suggested the impulsive growth rate of the perturbation to be constant over time:

(3.8) \begin{equation} \dot {\eta } = \dot {\eta }_{\textit{imp}} = Ak\eta _{0}u_{j,0}, \end{equation}

where $u_{j,0}$ is the velocity jump after the incident shock passage and $A$ , $k$ and $\eta _{0}$ are the post-shock quantities considered to be equal to the pre-shock conditions because of the impulsive and short nature of the laser-induced shock wave. An expansion of the flow equations to second order yields a linear relationship between the growth rate of the perturbation and time (Haan Reference Haan1991; Alon et al. Reference Alon, Hecht, Ofer and Shvarts1995; Zhang & Sohn Reference Zhang and Sohn1996), in accordance with experimental observations displayed in figure 5(b), where the grey line is given by

(3.9) \begin{equation} u_{j} = \dot {\eta }(t) = \lvert \dot {\eta }_{\textit{imp}} \rvert \left (1 + k\lvert \dot {\eta }_{\textit{imp}} \rvert t \right )\!. \end{equation}

Here, the speed and time are normalised to the wavenumber of the perturbation, $k$ , and the magnitude of the impulsive growth rate of the perturbation in the sense of Richtmyer and Meshkov, $\lvert \dot {\eta }_{\textit{imp}} \rvert$ (Zhang & Sohn Reference Zhang and Sohn1996; Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010). An excellent agreement is found between the measured early jet speed and the predicted growth rate of the perturbation through the entire collapse of the bubble for ten different bubble–shock wave experiments as displayed in figure 5(b). In these cases, the bubble dynamics and jet are inertially driven with Weber numbers higher than $100$ for both the bubble and the jet, so surface tension effects are negligible. Once the bubble has collapsed and starts re-expanding at $k \lvert \dot {\eta }_{\textit{imp}} \rvert t \gt 2.10$ , the jet fully forms and travels within the bubble, its speed decaying over time.

Only the early-stage dynamics of the perturbation are described by (3.9). To account for the later, nonlinear evolution of the perturbation, Li & Zhang (Reference Li and Zhang1997) introduced a three-dimensional nonlinear model that captures the perturbation growth rate of the Richtmyer–Meshkov instability

(3.10) \begin{equation} u_{j} = \dot {\eta }(t) = \frac {\lvert \dot {\eta }_{\textit{imp}} \rvert }{1 + D_{0}\eta _{0}k^{2} \lvert \dot {\eta }_{\textit{imp}} \rvert D_{1}t + {\textrm{max}}(0, k^{2}\eta _{0}^{2}D_{1}^{2} - D_{2})k^{2}\lvert \dot {\eta }_{\textit{imp}} \rvert ^{2}t^{2}}, \end{equation}

where $D_{0}=1$ for a system with no phase inversion and $D_{0}=-1$ for a system with phase inversion, i.e. negative amplitude is interpreted as a phase inversion (Yang et al. Reference Yang, Zhang and Sharp1994). This nonlinear equation is obtained by developing the initial perturbation using a Taylor’s series up to order three, followed by Padé approximants, which lead to the coefficients $D_{1}=0.54$ and $D_{2}=0.16$ for the Atwood number of the present work (Li & Zhang Reference Li and Zhang1997). Here, the quasi-single-mode perturbation assumption leads to $k\eta _{0} = 0.60$ , which means that the second-order term in (3.10) is equal to zero and yields inconsistent results with regards to the jet speed evolution displayed in figure 5(b). However, a heuristic approach (Li & Zhang Reference Li and Zhang1997; Sadot et al. Reference Sadot, Erez, Alon, Oron, Levin, Erez, Ben-Dor and Shvarts1998) suggests that the jet speed is adequately modelled as the growth rate of a Richtmyer–Meshkov perturbation by

(3.11) \begin{equation} u_{j} = \dot {\eta }(t) = \frac {\lvert \dot {\eta }_{\textit{imp}} \rvert }{1 - E_{1}\eta _{0}k^{2} \lvert \dot {\eta }_{\textit{imp}} \rvert t + E_{2}\eta _{0}^{2}k^{4}\lvert \dot {\eta }_{\textit{imp}} \rvert ^{2}t^{2}}, \end{equation}

where $E_{1} = 0.87 \pm 0.04$ and $E_{2} = 0.27 \pm 0.03$ are fitted scaling constants found to be fairly consistent over the range of shock driving studied herein. The heuristic model matches the experimental data well during the bubble collapse, indicated by the vertical dashed line at $k \lvert \dot {\eta }_{\textit{imp}} \rvert t = 2.10$ , where the jet speed grows linearly and during the expansion of the bubble all the way up to the moment the jet contacts the distal bubble side, indicated by the vertical dotted line at $k \lvert \dot {\eta }_{\textit{imp}} \rvert t \gt 4.20$ , although it tends to underestimate the peak jet speed. For the range of experiments investigated herein, a single scaling constant can be used by setting $E_2 = \eta _{0}^{2}k^{2}E_1^{2} \approx 0.27$ , building on the relationship between the exponents used in (3.10).

Based on the heuristic approach proposed here (3.10), an approximation of the jet shape can be attempted by considering the full single-mode form $\eta = \eta _{0} \cos (kx)$ , rather than only its centre value $\eta = \eta _{0}$ . The resulting theoretical jet shapes are compared with experimental observations in figure 5(d). Before the jet adopts a concave profile and penetrates into the bubble, only the jet tip aligns with experimental data, as seen for dimensionless times up to $k |\dot {\eta }_{\textit{imp}}| t = 2.33$ . However, once the jet begins to penetrate the bubble, the approximation yields a reasonable match within the limits of the linear framework employed. The remainder of the bubble interface, which is influenced by collapse and rebound dynamics, cannot be captured by this model. To improve accuracy, this approach should be combined with the Rayleigh–Plesset or Keller–Miksis equations to capture the radial dynamics, along with a model for predicting the translational motion of the bubble (Ohl Reference Ohl2002). Even so, spurious cusps are expected to appear at the junction between the bubble collapse solution and the jet shape described by (3.10), as previously noted by Dear et al. (Reference Dear, Field and Walton1988), who also attempted to approximate cavity collapse under shock using a different linear framework.

3.3. Jet breakup and end-pinching

The shock-induced bubble jets evolve into a narrow bell-shaped structure during their propagation within the air bubble. Their shape can quickly be approximated by a quasi-cylindrical semifinite liquid ligament, whose dynamics is governed by inertial, surface tension and viscous forces. The velocity of the shock-induced bubble jets decreases with a decreasing shock wave impulse. Close to the jetting limit, capillary forces begin to overcome inertial ones, causing the jets to become unstable, which may cause the jet tip to break up into a small droplet as observed in figure 6(a). This capillary-driven instability for semifinite ligaments and their breakup dynamics, also known as end-pinching, are displayed in figure 6(b). At the tip of the jet, the inertial pressure of the liquid cannot balance the increase in pressure caused by the curvature of the interface, causing the ligament to retract at a constant speed, called Taylor–Culick speed:

(3.12) \begin{equation} u_{{TC}} = \sqrt {\frac {\gamma }{\rho r_{j}}}, \end{equation}

Figure 6. (a) Image sequence of the dynamics of an air bubble interacting with a shock wave travelling from left to right and having a pressure impulse of $J/(r_{0}^{3}\sqrt {p_{b,0}\rho }) = 7.87$ . The subsequent end-pinching breakup of the liquid jet under capillary forces is observed. The normalised rescaled time, $\tilde {t}u_{j}(\tilde {t}=0)/r_{j}$ , is indicated on each frame where $\tilde {t} = t-t_{j} = 0$ is the time at which the bubble becomes concave and a jet forms ( $t_{j}=18$   $\unicode{x03BC} {\textrm{s}}$ ) and $r_{j}$ and $u_{j}(\tilde {t}=0)$ the jet radius and initial jet speed. The jet’s Weber number is $\textit{We}_{j}=46$ . (b) Image sequence focusing on the end-pinching phenomenon of a larger shocked air bubble for a pressure impulse of $J/(r_{0}^{3}\sqrt {p_{b,0}\rho }) = 5.11$ . The jet’s Weber number is $\textit{We}_{j}=33$ . The jet forms $t_{j}=36$ $\unicode{x03BC} {\textrm{s}}$ after the passage of the shock wave. The scale bar is the same for both image sequences.

where $r_{j}$ is the radius of the cylindrical jet. This speed corresponds to ligaments at rest and is therefore relative to the jet motion here. The retraction of the jet due to capillary forces results in liquid accumulation in the form of a bulge at the tip, as observed in the third frame of figure 6(b). Eventually, the back flow creates a neck at the base of the bulge which collapses under the pressure jump associated with its curvature, leading to the separation of a drop at the jet’s tip, as shown in the fourth and fifth frames of figure 6(b). Finally, the newly created drop travels towards the distal bubble side at a constant speed, while the end of the jet becomes unstable once more and a new protuberance associated with the end-pinching starts developing.

Figure 7. (a) Spatiotemporal evolution of the jet tip’s location, normalised to the jet radius, from the moment the jet forms, $\tilde {t} = t - t_{j}$ , normalised to the inertial time scale, $r_{j}/u_{j}(\tilde {t}=0)$ . The drop and jet are displayed as open and closed markers, respectively. The solid, dashed and dotted curves correspond to (3.13) for $\textit{We}_{j} \gt 1$ , (3.14) for $\textit{We}_{j}\lt 1$ and the evolution of the ejected drop position derived from the Taylor–Culick speed (3.12) relative to the jet speed at the drop pinch-off, $u_{j}(\tilde {t}_{p}) - u_{{TC}}$ , respectively. The inset figure displays the ejected drop volume with respect to (3.15), representing the liquid volume that accumulates at the jet tip because of capillary forces. (b) Superimposition of the bubble jet contours corresponding to figure 7(a), illustrating selfsimilarity of their shape while in the inertia-capillary regime, for different dimensionless times.

The spatiotemporal evolution of the jet is displayed in figure 7(a) from the time at which the bubble becomes concave and a jet forms, $t_{j}$ . The time and position of the jet tip are rescaled as $\tilde {t} = t - t_{j} = 0$ and $\tilde {x}_{j} = z_{j}(t) - z_{j}(t_{j}) = 0$ , respectively, and the initial jet speed $u_{j}(\tilde {t}=0)$ can be evaluated. The Weber numbers, $\textit{We}_{j} = 2\rho r_{j} u_{j}^{2}(\tilde {t}=0) / \gamma$ , are found to be in the $16$ – $84$ range. Upon spatial and temporal normalisation to the jet radius, $r_{j}$ , and jet inertial characteristic time, $r_{j}/u_{j}(\tilde {t}=0)$ , self-similarity is observed and the jet position is found to scale with time as

(3.13) \begin{equation} \dfrac {\tilde {x}_{j}}{r_{j}} = F \left ( \dfrac {\tilde {t}u_{j}(\tilde {t}=0)}{r_{j}}\right )^{2/3}\! ,\end{equation}

as indicated by the grey curve in figure 7(a), where $F=1.82$ is an empirically determined scaling constant. A good agreement is found between all tests and (3.13), which describes the time evolution of inertia- and capillary-driven jets, with the exception of the test at $\textit{We}_{j} = 16$ , which diverges from the self-similar solution at $\tilde {t}u_{j}(\tilde {t}=0)/r_{j} = 10$ . This is further illustrated by the superimposed bubble jet shapes shown in figure 7(b), corresponding to the tests in figure 7(a). While some variation in the bubble contours near the jet base arises from differences in the stages of collapse or expansion, the jet contours remain self-similar at each dimensionless time, except for minor discrepancies caused by liquid accumulation at the jet tip shortly before end-pinching, as well as the case with $\textit{We}_{j} = 16$ , which begins to deviate from $\tilde {t}u_{j}(\tilde {t}=0)/r_{j} = 10$ . Indeed, as the jet evolves, its speed decreases because of viscous and capillary effects, which results in the jet’s Weber number decreasing, eventually dropping below one, and capillary effects eventually becoming dominant. At that moment, non-dimensionalisation of the time must be switched from the inertial time scale to the capillary time scale $\sqrt {\rho r_{j}^{3}/\gamma }$ ,

(3.14) \begin{equation} \dfrac {\tilde {x}_{j}}{r_{j}} = F \left ( \dfrac {\tilde {t}\sqrt {\gamma }}{\sqrt {\rho r_{j}^{3}}} \right )^{2/3}, \end{equation}

which captures well the deviation of the jet position as observed by the good agreement between the dashed curve and corresponding test case at $\textit{We}_{j}=16$ shown in figure 7(a). Note that the scaling constant is the same as in (3.13). The time at which the tip of the jet pinches off as a drop, $\tilde {t}_{p}$ is marked by the transition of the full markers into hollow markers, indicating the position of the jet tip and distal side of the drop, respectively. The gap between both markers, at the moment the drop detaches, corresponds roughly to the drop diameter. From that moment, the newly formed jet tip is tracked simultaneously to the drop location and can be observed as a full marker below each hollow marker. The drop is ejected at a constant speed equal to the jet speed at the moment the drop detaches and travels within the air bubble before impacting the distal bubble side, as observed in the last frame of figure 6(b). Once the drop detaches, the retraction at the newly formed jet tip starts forming another bulge of liquid, whose evolution is estimated by subtracting the Taylor–Culick speed from the speed of the jet at the moment the drop detaches $u_{j}(\tilde {t}_{p}) - u_{{TC}}$ . This relative speed yields the slope of the drop position corresponding to the dotted lines in figure 7(a). Since the retraction of the jet tip is at the origin of the end-pinching breakup of the jet, the volume of the drop ejected from the jet depends on the jet size and retraction speed, but is independent of the jet speed. The drop volume, $V_{d}$ , is in fact equal to the portion of the jet which has retracted up to the pinch-off time, $\tilde {t}_{p}$ , which can be expressed as

(3.15) \begin{equation} V_{d} = \tilde {t}_{p} u_{{TC}} r_{j}^{2} \pi , \end{equation}

where the jet is approximated as a cylinder of cross-sectional area $r_{j}^{2}\pi$ . A good correlation between the experiment and (3.15) is observed in the inset of figure 7(a).

3.4. Water-entry of the liquid jet

The impact of objects such as solid spheres (May & Woodhull Reference May and Woodhull1948; Aristoff & Bush Reference Aristoff and Bush2009), liquid drops (Engel Reference Engel1966) or liquid jets (Soh, Khoo & Yuen Reference Soh, Khoo and Yuen2005; Qu et al. Reference Qu, Goharzadeh, Khezzar and Molki2013), normal to an initially quiescent free surface, typically forms a cavity that is driven deeper into the pool by the momentum of the object. Past a certain water-entry velocity, the long narrow cavity surrounding the object collapses from the sides, entrapping a bubble, which pinches off from the end of the cavity and starts journeying in the surrounding water (Oguz, Prosperetti & Kolaini Reference Oguz, Prosperetti and Kolaini1995). The entrapment of the bubble can be classified as a shallow seal or a deep seal, depending on the collapse location of the cavity. A shallow seal occurs when the cavity collapses near the free surface, while a deep seal happens when the collapse occurs midway between the tip of the cavity and the free surface. In their pioneering work on shock-induced microbubble dynamics, Ohl & Ikink (Reference Ohl and Ikink2003) have observed the appearance of small daughter bubbles next to the distal side of the gas bubbles right after the passage of a lithotripter shock wave. They have suggested that the water-entry of the shock-induced bubble jets at the distal bubble side, driven by the asymmetrical collapse of the bubble, entrains some gas, which separates from the main bubble. Kersten et al. (Reference Kersten, Ohl and Prosperetti2003) have proposed that the water-entry of shock-induced micrometre-sized bubble jets is similar to the canonical water-entry of millimetre-sized jets into a flat liquid pool. However, more recently, works on the liquid pool impact of micrometre-sized jets (Speirs et al. Reference Speirs, Pan, Belden and Truscott2018), still larger than those investigated in the present study, have highlighted some fundamental differences with respect to their millimetre-sized counterparts. Indeed, microjets are characterised by very small Bond numbers, $Bo = \rho g r_{j}^{2} / \gamma \sim O(10^{-3})$ , where $g$ is the gravitational acceleration, compared with millimetre-sized ones, $Bo \gt O(10^{0})$ , indicating that the dynamics of the cavity is not driven by hydrostatic pressure any more but by capillary forces.

Figure 8. Image sequences of the water entry of shock-induced bubble jets, where (a) the shock wave is characterised by a pressure impulse of $J/(r_{0}^{3}\sqrt {p_{b,0}\rho }) = 5.44$ , driving a cavity that stays attached to the main bubble and where (b) the shock wave is characterised by a pressure impulse of $J/(r_{0}^{3}\sqrt {p_{b,0}\rho }) = 8.62$ , driving a cavity that detaches in the shape of an ejected toroidal bubble. The normalised rescaled time, $\tilde {t} \Delta u_{i}/r_{j}$ , is indicated on each frame where $\tilde {t} = t-t_{i} = 0$ is the time at which the jet impacts the distal bubble side and is (a) $t_{i}=102$ $\unicode{x03BC} {\textrm{s}}$ and (b) $t_{i}=34$ $\unicode{x03BC} {\textrm{s}}$ ; $r_{j}$ and $\Delta u_{i}$ are the jet radius and velocity of the jet relative to the distal bubble side, respectively. The scale bar is the same for both image sequences.

Figure 8(a,b) shows two image sequences of the water-entry of a micrometre-sized jet at the distal side of the bubble, the formation of a cavity and the subsequent pinch-off of an ejected toroidal bubble for figure 8(b). Depending on the speed of the bubble jet, the water-entry at the distal side can happen right at the beginning of the expansion phase, or later, when the bubble comes close to reaching its maximum size. Therefore, the relative impact velocity of the jet $\Delta u_{i}$ is influenced by the bubble’s expansion velocity, $u_{b}$ , where $u_{b} \gt 0$ when the bubble is not close to its minimum or maximum size. The velocity of the jet relative to the distal bubble side at the moment of impact, $t_{i}$ , is defined as $\Delta u_{i} = u_{i} - u_{b}$ , where $u_{i}$ is the jet’s impact speed and $u_{b}$ the distal bubble wall velocity at that time. For figure 8(a), the relative jet speed at impact is $\Delta u_{i} =7.41$ ${\textrm{m}}$ $\textrm{s}^{-1}$ and the associated Weber number, $\textit{We}_{j} = \rho \Delta u_{i}^{2} r_{j}/\gamma = 15$ . For figure 8(b), the jet speed is $\Delta u_{i}=16.85$ $\textrm{m}$ $\textrm{s}^{-1}$ and the Weber number is $\textit{We}_{j} = 88$ . The bubble wall velocity is greater for a stronger collapse because the jet reaches the distal side during the bubble’s early expansion phase. In contrast, with milder driving, the bubble is close to reaching its maximum size, resulting in a bubble wall velocity close to zero. Immediately after impact, the bubble adopts a torus-like shape and a subsequent toroidal cavity forms, as highlighted in the second frame of figure 8(b). In figure 8(b), the pinch-off of a toroidal daughter bubble is observable due to the sufficiently high relative jet velocity at the moment of impact. The absence of a pinch-off for figure 8(a) is, however, not surprising, given the Weber number associated with this case, which is close to the value of $\textit{We}_{j} = 6.8$ , where Kersten et al. (Reference Kersten, Ohl and Prosperetti2003) did not observe pinch-off in their plunging jet experiments. The resulting daughter bubble is indeed toroidal as suggested by Kersten et al. (Reference Kersten, Ohl and Prosperetti2003).

Figure 9. Temporal evolution of the speed of the jet-driven cavity’s tip, $u_{j}$ , from the moment the jet impacts the distal bubble side, $\tilde {t} = t-t_{i}$ . The jet speed and time are normalised to the relative impact speed, $\Delta u_{i} = u_{i} - u_{b}$ and jet radius, $r_{j}$ . The grey horizontal line indicates $u_{j}/\Delta u_{i}=0.5$ and the dashed vertical lines show the time at which the toroidal bubble pinches off from the tip of the cavity, $\tilde {t}_{p}$ . The impulses are $J/r_0 \sqrt {p_{b,0}\rho } = 7.52$ (), $9.56 \: (\diamond )$ and $10.31 \: (\circ )$ . The inset figure displays the dimensionless pinch-off time against the jet’s Weber number, $\textit{We}_{j}$ , comparing with the model given by (3.16). Only the minimal and maximal uncertainty are displayed for visual clarity.

Figure 9 shows the velocity of the jet tip, which initially coincides with the tip of the cavity as it penetrates the surrounding liquid, and later evolves into the tip of the resulting toroidal bubble. The velocity is normalised to the relative impact velocity, $\Delta u_{i} = u_{i} - u_{b}$ , and the time by the characteristic time of the jet, $r_{j}/\Delta u_{i}$ . The reference frame for time is also modified, where $t=0$ corresponds to the moment the shock wave impacts the bubble and $\tilde {t} = t - t_{i} = 0$ to the moment the jet impacts the distal side of the bubble. The temporal evolution of the jet speed is shown for three different shock wave–bubble experiments, with the instant at which a toroidal bubble pinches off from the jet-driven cavity (as visually highlighted in the fourth frame of figure 8 b) being indicated. The dimensionless speed at which the cavity elongates has been reported to be $ u_{j} /\Delta u_{i} = 0.5$ for infinitely long millimetre-sized liquid jets (Oguz et al. Reference Oguz, Prosperetti and Kolaini1995; Speirs et al. Reference Speirs, Pan, Belden and Truscott2018), although Kersten et al. (Reference Kersten, Ohl and Prosperetti2003) reported values are in the $ u_{j} /\Delta u_{i} = 0.6$ – $0.9$ range. Here, after a first phase of deceleration lasting up to $\tilde {t}\Delta u_{i} / r_{j} = 25$ for all three test cases, the dimensionless growth speed of the cavity settles to a constant speed of $ u_{j} /\Delta u_{i} = 0.5$ , as highlighted by the horizontal grey line in figure 9. In fact, the mean and standard deviation for $N = 24$ tests yield a dimensionless speed of the cavity right before pinch-off of the daughter bubble of $u_{j}/ \Delta u_{i} = 0.51 \pm 0.06$ for the jets of finite length investigated herein. As the tip of the cavity pinches off, the toroidal bubble travels at a speed which slightly decreases, while staying close to $u_{j}/ \Delta u_{i} = 0.5$ for the duration of the experiment. Fluctuations in the speed of the ejected toroidal bubble are observed because of the deformation of the toroidal bubble. The dimensionless time at which the bubble pinches off from the tip of the cavity, $\tilde {t}_{p} \Delta u_{i} / r_{j}$ , grows as the shock wave intensity increases (see vertical lines). The dimensionless pinch-off time is plotted in the inset of figure 9 against the Weber number of the jet at the moment it impacts the distal bubble side. The pinch-off time includes the growth and collapse times of the cavity, which have previously been suggested to scale with $\textit{We}_{j}^{2}$ (Quetzeri-Santiago et al. Reference Quetzeri-Santiago, Hunter, Van Der Meer and Fernandez Rivas2021) and $\textit{We}_{j}$ (Kroeze et al. Reference Kroeze, Fernandez Rivas and Quetzeri-Santiago2024), respectively. Here, however, the experimental results indicate a linear dependency on $\textit{We}_{j}$ for both the growth and collapse time through the relation

(3.16) \begin{equation} \frac {\tilde {t}_{p} \Delta u_{i}}{r_{j}} = G We_{j}, \end{equation}

where $G=0.37$ is a scaling constant fitted to the observations within the parameter range investigated herein.

The pinch-off dynamics following the impact of finite micrometre-sized jets on liquid surfaces is governed by surface tension effects. Therefore, owing to the linear relationship between the jet speed and pressure impulse (Bokman et al. Reference Bokman, Biasiori-Poulanges, Meyer and Supponen2023), the pinch-off time scales with the pressure impulse of the shock wave as

(3.17) \begin{equation} \tilde {t}_{p} \sim We_{j} \sim \Delta u_{i}^{2} \sim J^{2}. \end{equation}

Kroeze et al. (Reference Kroeze, Fernandez Rivas and Quetzeri-Santiago2024) have shown that for jets having a $\textit{We}_{j} \leqslant 150$ at impact and a uniform constant velocity from tip to tail, the cavity behaves in the deep seal regime, where the collapse of the cavity around the jet occurs approximately in the middle of the cavity, while Speirs et al. (Reference Speirs, Pan, Belden and Truscott2018) suggest a shallow seal for the same range of Weber number but Bond numbers in the $O(10^{-2})$ – $O(10^{-1})$ range. Solely deep seal behaviour is observed in the present work.

As the pinch-off occurs, the X-ray phase contrast images provide a clear quantifiable visual access to the structure of the ejected toroidal bubble. Experiments suggest the outer radius of the toroid to be three times larger than its inner radius. The toroidal bubble’s volume can easily be computed, assuming symmetry along the jet axis, by measuring the area of the lobe, $A_{\textit{lobe}}$ , and its average radius, $r_{\textit{lobe}}$ , which yields $V = 2\pi r_{\textit{lobe}}A_{\textit{lobe}}$ . Figure 10(a) displays the volume of the ejected toroidal bubble at the moment of pinch-off. The volume is in the nanolitre range for different shock wave drivings and suggests a linear relationship with the relative impact speed of the jet on the distal bubble side. A simple model can easily be derived by assuming that the cavity promptly converges to its constant growth speed of $u_{j} = \Delta u_{i} / 2$ , that the toroidal bubble pinches off at the centre of the deep cavity formed by the jet, and therefore, its length equals half of the cavity’s length $l_{e} = l_{c}/2$ , and that the outer radius of the cavity is $3r_{j}$ . These assumptions, combined with (3.16) and the scaling proposed in (3.17), yield the following relation relating the ejected volume with the shock wave impulse:

(3.18) \begin{equation} \frac {V}{r_{j}^{3}}=\frac {\pi \left [ (3r_{j})^{2} - r_{j}^{2} \right ]l_{c}}{2r_{j}^{3}} = \frac {\pi 8r_{j}^{2} \tilde {t}_{p} \Delta u_{i}}{4r_{j}^{3}} = G 2\pi We_{j} \sim G2\pi J^{2}. \end{equation}

The model is compared with experiments in figure 10(b) yielding a fairly good estimate of the dimensionless volume at the moment of pinch-off. However, the data are plagued by significant uncertainty caused by the very small jet radii, which are in the $16$ – $32$ $\unicode{x03BC} {\textrm{m}}$ range and therefore approaching resolution limits. Nevertheless, these findings suggest that the volume of the ejected gas can be controlled through the shock wave impulse, which could have implications in technologies such as needle-free injections of nanoaerosols (Hindle & Longest Reference Hindle and Longest2010).

Figure 10. (a) Ejected toroidal bubble volume, $V$ , against the relative impact speed of the jet upon water entry, $\Delta u_{i} = u_{i}-u_{b}$ . (b) Dimensionless volume of the ejected gas bubble against the jet’s Weber number upon impact. The grey line displays (3.18). Only the minimal and maximal uncertainty are shown for visual clarity.