1. Introduction
The dynamics of cavitation bubble expansion and collapse in proximity to boundaries has been extensively studied since the phenomenon (Silberrad Reference Silberrad1912; Benjamin & Ellis Reference Benjamin and Ellis1966; Lauterborn & Kurz Reference Lauterborn and Kurz2010; Wang et al. Reference Wang, Wang, Zeng and Wang2023) was observed on the backside of propeller blades of ships. Rich fluid mechanics has been revealed for bubbles collapsing near a wide range of boundaries, including rigid walls (Lauterborn & Bolle Reference Lauterborn and Bolle1975; Zhang et al. Reference Zhang, Li, Xu, Pei, Li and Liu2024a
), free surfaces (Chahine Reference Chahine1982), elastic boundaries (Brujan et al. Reference Brujan, Nahen, Schmidt and Vogel2001), ice sheets (Cui et al. Reference Cui, Zhang, Wang and Khoo2018) and porous plates (Andrews et al. Reference Andrews, Rivas and Peters2023), to name a few. For a wall-proximal cavitation bubble, its distal surface deforms and develops into a microjet that pierces its proximal surface, impacts the wall and then spreads along it (Philipp & Lauterborn Reference Philipp and Lauterborn1998). Depending on the stand-off distance
$\gamma$
, the microjet may take the form of either a regular jet, with a speed of the order of
$50\,\mathrm{m\,s}^{-1}$
, or a needle jet, with an extremely small radius and a velocity of the order of
$500\,\mathrm{m\,s}^{-1}$
. The large variation in jet size and speed leads to a broad range of impact pressures (Gonzalez-Avila et al. Reference Gonzalez-Avila, Denner and Ohl2021) and wall shear stresses (Zeng et al. Reference Zeng, Gonzalez-Avila, Dijkink, Koukouvinis, Gavaises and Ohl2018). These flows can be undesirable because of their destructive effects on materials and structures (Philipp & Lauterborn Reference Philipp and Lauterborn1998), but they can also be exploited in applications such as surface cleaning (Reuter & Mettin Reference Reuter and Mettin2016), surface processing (Ijiri et al. Reference Ijiri, Shimonishi, Nakagawa and Yoshimura2018; Zhang et al. Reference Zhang, Wei, Wang, Qiu and Zhang2024b
), lithotripsy and drug delivery.
Recently, growing attention has been devoted to cavitation bubble dynamics near complex rigid geometries, particularly those involving multiple boundaries with different shapes, such as tubes (Wang et al. Reference Wang, Wang, Zhang and Stride2019), thin gaps (Gonzalez-Avila et al. Reference Gonzalez-Avila, Klaseboer, Khoo and Ohl2011; Zeng et al. Reference Zeng, Gonzalez-Avila and Ohl2020), porous plates (Andrews et al. Reference Andrews, Rivas and Peters2023) and gas-entrapping micro-textured surfaces (Gonzalez-Avila et al. Reference Gonzalez-Avila, Nguyen, Arunachalam, Domingues, Mishra and Ohl2020; Robles et al. Reference Robles, Gonzalez-Parra, Cuando-Espitia and Aguilar2022). A notable effect is that the microjet direction is governed by the combined influence of the individual stand-off distances to the surrounding boundaries. The Kelvin impulse (Blake et al. Reference Blake, Leppinen and Wang2015) and inviscid potential flow theory predict (Obreschkow et al. Reference Obreschkow, Tinguely, Dorsaz, Kobel, de Bosset and Farhat2011; Supponen et al. Reference Supponen, Obreschkow, Tinguely, Kobel, Dorsaz and Farhat2016; Li et al. Reference Li, Zhang, Cui, Li and Liu2023) that, near a simple boundary, the jet direction aligns with the pressure gradient at maximum bubble expansion. By contrast, for a cavitation bubble oscillating near a curved rigid surface, the pressure gradient may point either towards or away from the boundary. Its sign depends on both the boundary curvature and stand-off distance, since both influence the bubble deformation (Tomita et al. Reference Tomita, Robinson, Tong and Blake2002). In geometries with multiple boundaries, the pressure gradient can be decomposed into contributions associated with each individual boundary (Brujan et al. 2018, Reference Brujan, Noda, Ishigami, Ogasawara and Takahira2022). The jet direction is therefore determined by the competition among these contributions and is thus a function of the corresponding individual stand-off distances.
Moreover, multiple boundaries can introduce qualitatively new jetting dynamics. A cavitation bubble may split into a pair of toroidal bubbles when it is located near two boundaries whose induced pressure gradients act in opposite directions. Such a configuration may arise, for example, between two rigid walls forming a narrow gap (Chahine Reference Chahine1982; Ogasawara et al. Reference Ogasawara, Tsubota, Seki, Shigaki and Takahira2015; Zeng et al. Reference Zeng, Gonzalez-Avila and Ohl2020) or in a system consisting of a curved rigid wall and a free surface (Li et al. Reference Li, Khoo, Zhang and Wang2018). In this case, bubble splitting is driven by an annular flow towards the bubble centre, which, upon impact, is redirected into two opposite directions. Beyond bubble splitting, specific rigid geometries can also generate unusual microjets, such as jets impacting a distant wall (Zeng et al. Reference Zeng, Gonzalez-Avila and Ohl2020), slow jets with a large diameter (Brujan et al. Reference Brujan, Zhang, Liu, Ogasawara and Takahira2022) and ultrafast yet slender jets (Zeng et al. Reference Zeng, Zhang, Tan, An and Ohl2024).
Cavitation bubbles can act as micropumps in confined rigid channels (Dijkink & Ohl Reference Dijkink and Ohl2008) because they are able to drive liquid motion through volume oscillations and microjet impact. Two distinct pumping mechanisms have been identified. The first relies on generating an asymmetric flow by means of a cavitation bubble inside a tubular conduit. Yuan & Prosperetti (Reference Yuan and Prosperetti1999) proposed a simplified model for the expansion and collapse of a cavitation bubble in a finite-length narrow tube open at both ends. They showed that the bubble generates a net flow directed away from the nearest tube end. Subsequent experimental (Torniainen et al. Reference Torniainen, Govyadinov, Markel and Kornilovitch2012), numerical (Ory et al. Reference Ory, Yuan, Prosperetti, Popinet and Zaleski2000) and theoretical (Sun et al. Reference Sun, Can, Dijkink, Lohse and Prosperetti2009) studies further demonstrated the potential of vapour bubbles as actuators in microfluidic devices.
The second mechanism arises near a perforated plate. In this case, a through-hole with a small diameter only weakly alters the overall jet dynamics, so that the microjet induced by a cavitation bubble still propagates towards the perforated wall and thereby drives liquid through the hole. This idea was first proposed by Khoo et al. (Reference Khoo, Klaseboer and Hung2005), who used a boundary integral method to analyse the flow induced by a bubble oscillating near a rigid wall containing a small through-hole, and was later confirmed experimentally by the same group (Lew et al. Reference Lew, Klaseboer and Khoo2007; Karri et al. Reference Karri, Pillai, Klaseboer, Ohl and Khoo2011). More recent studies have revisited this problem and revealed additional underlying fluid mechanics. For a cavitation bubble expanding and collapsing above a perforated plate, Reese et al. (Reference Reese, Schädel, Reuter and Ohl2022) showed that a reverse flow develops during expansion and degrades the overall pumping performance. By contrast, introducing a curved inlet can funnel the liquid and thereby enhance pumping (Agrež et al. Reference Agrež, Zevnik, Lokar, Dular and Petkovšek2025). For bubbles oscillating near a much longer tube, the reverse flow disappears, while the jet velocity and structure vary with the nucleation position (Ren et al. Reference Ren, Li, Xu, Wakata, Liu, Sun, Zuo and Liu2022). More recently, volume-of-fluid simulations by Heidary et al. (Reference Heidary, Fan, Mojra and Ohl2024) demonstrated that for a regular jet, the jet width is controlled jointly by the inner diameter and stand-off distance, whereas the jet speed is primarily governed by the stand-off distance. These studies collectively highlight that, within confined geometries, jet morphology and transport efficiency are highly sensitive to geometric parameters, yet a unified physical description of how multiple geometric constraints simultaneously control jet formation remains incomplete.
Different jet-formation mechanisms have also been identified in separate cavitation problems. For bubbles collapsing near smooth rigid surfaces, jetting typically occurs in two regimes depending on the stand-off distance. At relatively large stand-off distances (
$\gamma \gtrsim 0.2$
), a regular jet forms with typical velocities of
$O(50\,\mathrm{m\,s}^{-1})$
, driven by a moderate pressure buildup near the distal bubble surface. By contrast, at small stand-off distances (
$\gamma \lesssim 0.2$
), a fast, thin jet can form with velocities exceeding
$500\,\mathrm{m\,s}^{-1}$
(Lechner et al. Reference Lechner, Lauterborn, Koch and Mettin2020; Reuter & Ohl Reference Reuter and Ohl2021). In this regime, the bubble undergoes a more complex deformation: a surface kink develops near the lower interface during collapse and promotes annular flow. The convergence of this annular flow at the symmetry axis leads to either strong pressure focusing or bubble splitting, producing a much larger pressure amplification and resulting in jet velocities that can even reach up to
$2000\,\mathrm{m\,s}^{-1}$
in some cases (Lechner et al. Reference Lechner, Lauterborn, Koch and Mettin2020; Reuter et al. Reference Reuter, Deiter and Ohl2022). Importantly, for a smooth rigid surface, the kink remains localised and its position changes little during collapse.
In more complex confined geometries, however, the mechanism can differ qualitatively. Recent studies show that a protruding rigid boundary with a small outer diameter can strongly modify bubble-edge deformation and collapse dynamics, enabling the generation and manipulation of an interfacial disturbance that reshapes the bubble during collapse and promotes fast-jet formation (Koch et al. Reference Koch, Rosselló, Lechner, Lauterborn and Mettin2021; Xiang et al. Reference Xiang, Ren, Yang, Liu, Zuo and Liu2024). By contrast, rigid confinement combined with a free surface can suppress jet formation entirely and instead promote a more symmetric collapse with enhanced pressure emission under certain geometric conditions (Sun et al. Reference Sun, Fan, Yao, Wang and Ohl2025). These observations suggest that, unlike the fast jet near a smooth rigid wall, which is associated with a localised kink and annular-flow focusing, fast-jet formation in capillary-like geometries may involve an interfacial disturbance that either strengthens the annular flow and induces bubble splitting or propagates towards the distal surface and converges there.
Despite these advances, a key open question remains: how do multiple geometric constraints simultaneously regulate the competition between vertical collapse, annular flow and interfacial-wave dynamics, and thereby determine whether jet formation occurs via regular collapse or via fast-jet mechanisms involving interfacial focusing and splitting? In particular, a unified description linking these mechanisms across a wide parameter space is still lacking. Motivated by this, we investigate cavitation bubble dynamics near the exit of a long capillary, focusing on how the stand-off distance and the inner and outer capillary radii jointly control bubble collapse, jet morphology and jet velocity. We emphasise a distinctive fast-jet mechanism in capillary-like confinement, where an interfacial wave propagates along the bubble surface and converges at the distal side, producing strong flow focusing and leading to fast-jet formation. This mechanism differs fundamentally from that of the fast needle jet generated near a smooth rigid boundary at nearly zero stand-off distance, where the interfacial kink remains localised and does not propagate along the interface.
2. Methodology
2.1. Experiment
Experimental set-up and definition of the key parameters. In each experiment, the high-speed camera and the pulsed laser are synchronised using a delay generator.
Figure 1 shows the experimental configuration. The test section consists of a
$4\times 4\times 5\,\mathrm{cm}^3$
cuvette filled with degassed ultrapure water (resistivity:
$18.25\,\mathrm{M\Omega \,cm}$
). A quartz capillary of inner radius
$R_i$
, outer radius
$R_o$
and length
$L=10\,\mathrm{cm}$
is positioned at the centre of the cuvette. The upper exit of the capillary is placed above the free surface of the liquid. Both the upper end of the capillary and the bottom of the cuvette are therefore sufficiently far from the region of interest (
$L \gg R_{\textit{max}} \approx 520\,{\unicode[Arial]{x03BC}} \mathrm{m}$
) that their effects on the experiment are negligible. In the experiments, the inner radius
$R_i$
is varied from 0.1 to 0.25 mm, and the outer radius
$R_o$
from 0.2 to 0.35 mm.
A Q-switched Nd:YAG laser (Litron Nano SG 150-10; wavelength, 532 nm; pulse duration, 5 ns) is first expanded and collimated by a beam expander consisting of a plano-concave lens and a plano-convex lens, and is then focused by a
$4\times$
objective (Olympus UPLFLN4X, NA = 0.13, WD = 17 mm) at a distance
$d$
from the capillary exit. The laser energy is precisely controlled in the range 0.1–40 mJ using an attenuator placed between the beam expander and the objective, thereby ensuring highly repeatable cavitation nucleation at the focal point. In all experiments, the laser beam is carefully aligned parallel to the capillary axis and focused on the capillary centreline, i.e. at
$(x,y,z)=(0,-d,0)$
, to improve the sphericity of bubble nucleation and ensure axisymmetric flow. For the experimental results, only cases satisfying the off-centre criterion
$\Delta r/R_{\textit{max}}\lt 0.06$
are retained, where
$\Delta r$
is the horizontal distance from the bubble centre to the capillary centreline (
$r=0$
). Under this condition, the bubble dynamics and the induced flow can be reasonably approximated as axisymmetric.
A high-speed camera (Photron FASTCAM Nova S9) equipped with a macro lens (LAOWA 100 mm F2.8) records the bubble dynamics from the side, with illumination provided by an LED-powered fibre optic light source. The camera operates at 200 000 frames per second, corresponding to an interframe time of 5
${\unicode[Arial]{x03BC}}$
s, with an exposure time of 1.05
${\unicode[Arial]{x03BC}}$
s. The spatial resolution is 10.3
${\unicode[Arial]{x03BC}}$
m, and the imaging configuration provides a field of view of
$1.32\times 1.32\,\mathrm{mm}^2$
, which is sufficient to capture the dynamics of a bubble with a maximum radius of
$R_{\textit{max}} \approx 510\,{\unicode[Arial]{x03BC}} \mathrm{m}$
as well as the associated flow details. The laser and the high-speed camera are synchronised using a delay generator.
2.2. Numerical simulation
Here, we model the laser-induced cavitation bubble as a gas bubble with an initially high pressure within the finite-volume framework. This approach has been validated in a series of numerical studies for simulating the first oscillation cycle over a wide range of fluid domains by several groups (Koch et al. Reference Koch, Lechner, Reuter, Köhler, Mettin and Lauterborn2016; Zeng et al. Reference Zeng, Gonzalez-Avila and Ohl2020; Bußmann et al. Reference Bußmann, Riahi, Gökce, Adami, Barcikowski and Adams2023). The gas inside the bubble is assumed to follow a polytropic equation of state, while the surrounding water is treated as a compressible Tait liquid. The equations of state for both phases share a unified form:
\begin{align} p=(p_0+B)\left(\frac {\rho }{\rho _0}\right)^{\varGamma }-B , \end{align}
where
$\rho _0$
and
$p_0$
are the reference density and pressure;
$\varGamma$
is the specific heat ratio for the gas or the Tait exponent for the liquid; and B is the Tait pressure. We choose the following values: for the bubble,
$\rho _0=1.2$
kg m
$^{-3}$
,
$p_0=101\,325\,$
Pa,
$B=0$
and
$\varGamma =1.4$
; while for the liquid,
$\rho _0=998.2$
kg m
$^{-3}$
,
$p_0=101\,325\,$
Pa,
$B=304.6\,$
MPa and
$\varGamma =7.15$
.
To capture bubble dynamics, an indicator fraction function
$\alpha$
representing the volume fraction of the water is defined, whose motion follows the transport equation:
\begin{align} \frac {\partial \alpha }{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\alpha \boldsymbol{u})+\boldsymbol{\nabla }\boldsymbol{\cdot }(\alpha (1-\alpha )\boldsymbol{U_r})=\alpha (1-\alpha )\left (\frac {\psi _g}{\rho _g} -\frac {\psi _l}{\rho _l} \right ) \frac {{\rm D}p}{{\rm D}t}+\alpha \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} , \end{align}
where
$t$
and
$\boldsymbol{u}$
are time and flow velocity, respectively. The equation involves compressibilities
$\psi ={\rm D}\rho /{\rm D}p$
of both phases modelled based on their equation of state, where
$\rho$
and
$p$
are the fluid density and flow pressure, respectively. In the compressible volume-of-fluid method, an artificial compressible term
$\boldsymbol{U_r}$
is introduced to tune the sharpness of the bubble interface (Rusche Reference Rusche2003), which is obtained from the velocity difference between two phases in the transit region.
Since heat and mass transfer are negligible (Zeng et al. Reference Zeng, Gonzalez-Avila and Ohl2020), the governing equations for the flow are the three-dimensional Navier–Stokes equations:
\begin{align} \frac {\partial \rho }{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\rho \boldsymbol{u})=0 , \\[-28pt] \nonumber \end{align}
\begin{align} \frac {\partial \rho \boldsymbol{u}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\rho \boldsymbol{uu})=-\boldsymbol{\nabla }p +\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\tau } + \boldsymbol{f_\delta } , \\[0pt] \nonumber \end{align}
where
$\boldsymbol{\tau }=\mu [\boldsymbol{\nabla }\boldsymbol{u}+\boldsymbol{\nabla }\boldsymbol{u}^T-({2}/{3})(\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u})\boldsymbol{I}]$
is the viscous stress tensor for Newtonian fluid, which is calculated based on the dynamic viscosity
$\mu$
and the identity tensor
$\boldsymbol{I}$
. Here
$\boldsymbol{f_\delta }$
is the surface tension source term obtained from the continuous-surface-force (Brackbill et al. Reference Brackbill, Kothe and Zemach1992) method.
The above equations are solved in an axisymmetric domain using OpenFOAM (Weller et al. Reference Weller, Tabor, Jasak and Fureby1998; Greenshields & Weller Reference Greenshields and Weller2022). The computational domain has an overall size of
$20\times 20\,\mathrm{mm}^2$
and is recursively refined to a grid spacing of
$\Delta x = 2\,{\unicode[Arial]{x03BC}} \mathrm{m}$
in the region of interest, i.e. around the bubble and the capillary exit. The total number of cells is approximately
$10^6$
, which is sufficient to ensure convergence of the flow induced by bubbles with a maximum radius of
$R_{\textit{max}}\approx 497\,{\unicode[Arial]{x03BC}} \mathrm{m}$
.
A no-slip boundary condition is imposed on the rigid capillary wall. The ambient pressure,
$p_\infty = 101\,325\,\mathrm{Pa}$
, is prescribed at the top boundary of the domain. For the remaining boundaries, a vanishing gradient condition is applied to the velocity and stress fields, allowing both inflow and outflow and thereby minimising the influence of the outer boundaries on the simulated bubble dynamics.
In this study, although the pulse energy of the Nd:YAG laser is kept constant, the energy absorbed by the deionised water varies slightly because the optical loss depends on the focal position relative to the capillary. As a result, the equivalent maximum bubble radius ranges from 480 to 520
${\unicode[Arial]{x03BC}}$
m. In the simulations, the initial bubble radius is fixed at
$R_0 = 30\,{\unicode[Arial]{x03BC}} \mathrm{m}$
at
$t=0$
, while the initial internal pressure is determined for each case by matching the experimentally observed bubble dynamics in the comparisons below.
Throughout the study, we use three dimensionless parameters to account for the geometric effect on the bubble dynamics, which are the normalised inner radius
$R_i^*=R_i/R_{\textit{max}}$
and outer radius
$R_o^*=R_o/R_{\textit{max}}$
of the capillary, and the stand-off distance from the exit of the capillary
$\gamma = d/R_{\textit{max}}$
. The ranges of parameters are listed in table 1.
Ranges of parameters in the study.
Table 1. Long description
A table with three columns and four rows comparing simulation and experiment ranges of parameters. The columns are labeled Parameter, Simulation range, and Experiment range. The parameters listed are R max, R o over R max, R i over R max, and gamma. R max is given in micrometers; the other three parameters are dimensionless. For R max, the simulation range is 490-520 micrometers and the experiment range is 480-520 micrometers. For R o over R max, the simulation range is 0.31-20.3 and the experiment range is 0.40-0.71. For R i over R max, the simulation range is 0-1.0 and the experiment range is 0.20-0.51. For gamma, the simulation range is 0.10-1.22 and the experiment range is 0.19-1.21.
3. Results
3.1. Bubble dynamics
Bubble dynamics from the experiments (shadowgraph images) and simulations (red contours) for two representative cases. (a) Experiment:
$R_{\textit{max}}=506.2\pm 10.3\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$\gamma =1.19\pm 0.02$
,
$R_i^*=0.30\pm 0.01$
and
$R_o^*=0.50\pm 0.01$
; simulation:
$R_{\textit{max}}=516.37\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$\gamma =1.17$
,
$R_i^*=0.30$
and
$R_o^*=0.49$
. (b) Experiment:
$R_{\textit{max}}=482.4\pm 10.3\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$\gamma =0.69\pm 0.01$
,
$R_i^*=0.31\pm 0.01$
and
$R_o^*=0.52\pm 0.01$
; simulation:
$R_{\textit{max}}=493.62\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$\gamma =0.70$
,
$R_i^*=0.31$
and
$R_o^*=0.51$
. The scale bar is 0.5 mm, and time is given in
${\unicode[Arial]{x03BC}}$
s. Note that the time instants in the experiments (black) and simulations (red) differ slightly.
To begin, we compare two representative scenarios of bubble dynamics from the experiments and simulations in figure 2. In both cases, the agreement between experiment and simulation is very good. Figure 2(a) shows the results for a bubble nucleated at a distance of
$d\approx 602\,{\unicode[Arial]{x03BC}}$
m from the capillary exit, with inner and outer radii of
$R_i=150\,{\unicode[Arial]{x03BC}}$
m and
$R_o=250\,{\unicode[Arial]{x03BC}}$
m, respectively. In the experiment,
$R_{\textit{max}}\approx 506\,{\unicode[Arial]{x03BC}}$
m and
$\gamma \approx 1.19$
, while the simulated bubble is slightly larger, with
$R_{\textit{max}}\approx 516\,{\unicode[Arial]{x03BC}}$
m and
$\gamma \approx 1.17$
. During expansion, the bubble remains nearly spherical, similar to the expansion of a bubble near a rigid surface. During collapse, however, the liquid near the proximal bubble surface flows inward towards the axis, and a relatively strong annular flow develops, deforming the bubble into a dripping-drop shape (see
$t=90\,{\unicode[Arial]{x03BC}}$
s). Although the proximal surface attains a high curvature, it does not develop into a microjet that pierces the distal surface at later times. Instead, the distal surface, moving upward, generates a strong localised flow. Eventually, the bubble produces an upward microjet directed towards the capillary.
The results for a bubble nucleated closer to the capillary exit, at
$d=333\,{\unicode[Arial]{x03BC}}$
m, are shown in figure 2(b). In the experiment, the maximum bubble radius decreases to
$R_{\textit{max}}\approx 482.4\,{\unicode[Arial]{x03BC}}$
m, likely because more laser energy is lost in the more confined geometry, while in the simulation
$R_{\textit{max}}\approx 493\,{\unicode[Arial]{x03BC}}$
m. As the bubble is brought closer to the capillary exit, its proximal surface undergoes stronger deformation during expansion. The simulation reveals a wavy interface with humps near both the centre and the edge at
$t=45\,{\unicode[Arial]{x03BC}}$
s, whereas the experimental shadowgraph images do not resolve the internal interface details clearly enough for a direct comparison. During the early collapse stage, the distal surface contracts approximately spherically, while the proximal surface shrinks radially towards the axis
$r=0$
. The combined effect of these two modes of deformation generates an interfacial wave that originates near the proximal edge and propagates towards the lowest point of the bubble. Upon collision of this interfacial wave, a high-pressure region develops and accelerates the interface into a fast upward jet.
Overall, we identify two types of microjets based on their formation mechanism: a regular jet, driven primarily by the vertical collapse of the distal bubble surface, and a wave-driven jet, generated by surface-wave convergence and the associated local pressure amplification. In the present parameter range, regular jets typically have lower maximum velocities, whereas wave-driven jets generally reach much higher velocities.
3.2. Microjet formation
Before analysing the jet-formation mechanism, we define two characteristic velocities: the maximum velocity
$U_{\textit{max}}$
and the impact velocity
$U_{\textit{imp}}$
. The maximum velocity
$U_{\textit{max}}$
is defined as the highest jet velocity attained during the entire jetting process. The impact velocity
$U_{\textit{imp}}$
quantifies the jet speed when the microjet reaches a fixed geometric reference position. This reference position is taken as the axial location of the proximal (upper) bubble interface immediately before jet formation. The jet velocity is evaluated when the jet tip reaches this reference position. This definition provides a consistent measure of jet strength across different regimes. In wave-driven jet cases, the proximal interface remains nearly stationary prior to jet formation, ensuring a well-defined reference position. In regular-jet cases, the same definition enables consistent comparison despite the more gradual interface motion. Physically,
$U_{\textit{imp}}$
characterises the jet strength after it has traversed the bubble interior and reached the proximal side of the bubble, and thus serves as a useful measure for comparing the liquid transport capability of different jetting regimes.
Pressure (left-hand side of each panel) and velocity magnitude (right-hand side of each panel) during microjet formation for a bubble with
$R_{\textit{max}} = 516.38\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$R_i^* = 0.29$
,
$R_o^* = 0.48$
and
$\gamma = 1.16$
. The bubble interface is indicated by the red-coloured solid line, and the black arrows denote only the flow direction. The grey rectangular block marks the capillary cross-section. Each image shows a region 1.6 mm wide and 1.4 mm high.
Figure 3. Long description
Panel A: The first panel shows a combination of pressure and velocity magnitude graphs during microjet formation. The left side of the panel displays pressure in bars, while the right side shows velocity magnitude in meters per second. The bubble interface is marked by a red solid line, and black arrows indicate flow direction. The grey rectangular block represents the capillary cross-section. The region shown is 1.6 mm wide and 1.4 mm high. Panel B: The second panel continues to show the pressure and velocity magnitude at a later time point. The pressure and velocity magnitude scales are adjusted to show higher values. The bubble interface and flow direction are similarly marked. Panel C: The third panel shows further progression in time with adjusted scales for pressure and velocity magnitude. The bubble interface and flow direction are consistent with previous panels. Panel D: The fourth panel shows continued evolution with higher pressure and velocity magnitude values. The bubble interface and flow direction are marked as before. Panel E: The fifth panel shows the pressure and velocity magnitude at a later stage with even higher values. The bubble interface and flow direction are consistent. Panel F: The sixth panel shows further progression with adjusted scales for pressure and velocity magnitude. The bubble interface and flow direction are marked. Panel G: The seventh panel shows continued evolution with higher pressure and velocity magnitude values. The bubble interface and flow direction are consistent. Panel H: The eighth panel shows the pressure and velocity magnitude at a later stage with very high values. The bubble interface and flow direction are marked. Panel I: The ninth panel shows the final stage of the process with lower pressure and velocity magnitude values than in the previous panel. The bubble interface and flow direction are consistent with previous panels.
Next, we have a closer look into the jet-formation process for the two types of microjets. Figure 3 shows the detailed numerical results for the bubble in figure 2(a), corresponding to a relatively large stand-off distance of
$\gamma =1.16$
. During expansion, the capillary not only slows the upward flow and flattens the proximal bubble interface (see
$t=45\,{\unicode[Arial]{x03BC}}$
s), thereby forming a liquid gap of thickness about
$100\,{\unicode[Arial]{x03BC}}$
m, but also modifies the pressure-gradient distribution along the bubble interface. The pressure gradient is weakest near the proximal interface and relatively stronger over the rest of the interface. As a result, the top surface remains nearly stationary, whereas the other parts of the interface collapse inward towards the bubble centre with a relatively strong flow. This deforms the bubble into a cone-like shape during the late stage of collapse (see
$t=90\,{\unicode[Arial]{x03BC}}$
s).
Once the pressure gradient at the proximal interface becomes sufficiently strong, the conical top starts to collapse and develops into a concave shape. The bubble then becomes elongated, while a much higher pressure, about 2.8 MPa, builds up near the bottom at
$t=96.25\,{\unicode[Arial]{x03BC}}$
s. This pressure further increases to 17 MPa and eventually accelerates the bottom surface into an upward microjet with a velocity of approximately 400 m s−1 within 0.5
${\unicode[Arial]{x03BC}}$
s. The upward microjet pierces the proximal interface at
$t=97.25\,{\unicode[Arial]{x03BC}}$
s and then slows down to 365 m s−1.
Spatio-temporal distribution of the axial velocity magnitude for a bubble with
$R_{\textit{max}} = 516.38\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$R_i^* = 0.29$
,
$R_o^* = 0.48$
and
$\gamma = 1.16$
. Blue and red indicate downward and upward velocity along the axis, respectively. The black dashed and solid lines denote the proximal and distal bubble interfaces on the axis
$r=0$
, respectively. Owing to the wide velocity range, a logarithmic scale is used. The temporal and spatial resolutions are 0.1 μs and 2
${\unicode[Arial]{x03BC}}$
m, respectively.
Figure 4 provides a detailed map of the spatio-temporal evolution of the axial velocity of the liquid phase for a bubble with
$\gamma =1.16$
,
$R_i^* = 0.29$
and
$R_o^* = 0.48$
. Negative velocity values (directed downward) are shown in blue, whereas positive values (directed towards the capillary) are shown in red. The velocity is plotted on a base-10 logarithmic scale to cover its wide dynamic range. To avoid zero velocities in the logarithmic plot, a threshold velocity of 0.1 m s−1 is set for velocities smaller than this value.
During the early expansion stage, the proximal and distal interfaces both move at velocities of about 100 m s−1, but in opposite directions. During the subsequent late expansion and early collapse stages, the flow velocity throughout the domain decreases to the order of 10 m s−1. For the distal surface, the velocity changes sign immediately after maximum expansion at
$t\approx 45\,{\unicode[Arial]{x03BC}}$
s, indicating that the surface starts to retract upward as soon as the bubble reaches its maximum size. By contrast, this reversal is delayed until about
$t=70\,{\unicode[Arial]{x03BC}}$
s for the proximal surface, which remains nearly stationary because of the weak pressure gradient before the conical shape develops. The proximal surface begins to move downward significantly only after
$t=80\,{\unicode[Arial]{x03BC}}$
s, and continues accelerating downward until reaching a velocity of 132 m s−1 just before colliding with the distal surface.
Meanwhile, the distal surface starts to move upward immediately after maximum expansion at
$t\approx 45\,{\unicode[Arial]{x03BC}}$
s, initially at a relatively low speed, but it accelerates strongly during the final stage of collapse and reaches a velocity of about 400 m s−1. This produces an upward microjet that drives liquid into the capillary.
Pressure (left-hand side of each panel) and velocity magnitude (right-hand side of each panel) during microjet formation for a bubble with
$R_{\textit{max}} = 499.91\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$R_i^* = 0.30$
,
$R_o^* = 0.50$
and
$\gamma = 0.8$
. The bubble interface is indicated by the red-coloured solid line, and the black arrows denote only the flow direction. The grey rectangular block marks the capillary cross-section. Each image shows a region 1.6 mm wide and 1.4 mm high.
Figure 5. Long description
Panel A: The first panel shows a contour plot with pressure on the left and velocity magnitude on the right when the bubble reaches its maximum radius. The bubble interface is marked by a red line, and black arrows indicate flow direction. The grey rectangular block represents the capillary cross-section. The plot spans a region 1.6 mm wide and 1.4 mm high. Panel B: The second panel shows a similar contour plot at a later time, highlighting the annular edge. Panel C: The third panel depicts the dimple ring with corresponding pressure and velocity fields. Panel D: The fourth panel shows the development of a surface wave. Panel E: The fifth panel illustrates the surface wave stage. Panel F: The sixth panel captures the further progression of the surface wave. Panel G: The seventh panel shows the bubble interface and flow direction as the microjet becomes more pronounced. Panel H: The eighth panel depicts the bubble interface and flow direction during microjet propagation through the bubble. Panel I: The ninth panel shows the microjet penetrating the proximal bubble surface and entering the capillary with detailed pressure and velocity fields.
Figure 5 shows the computed flow details for a smaller stand-off distance,
$\gamma =0.8$
. After nucleation, the bubble expands rapidly and reaches its maximum radius at
$t = 45\,{\unicode[Arial]{x03BC}} \mathrm{s}$
. During expansion, the proximal surface deforms strongly as it approaches the capillary. The liquid between the bubble and the capillary is driven either into the capillary or laterally outward. Nevertheless, a thin liquid film remains, with a thickness of approximately
$30\,{\unicode[Arial]{x03BC}}$
m. At this stage, the proximal surface is no longer smooth, but instead exhibits a wavy shape, with the centre point at
$r=0$
protruding slightly in the positive
$y$
direction and a hump forming near the annular edge.
A vortex ring forms near the proximal edge of the bubble as the expanding flow approaches the outer corner of the capillary orifice. This vortex ring initiates an interfacial wave that travels from the proximal edge towards the distal surface during collapse. By contrast, the proximal edge itself moves inward only slowly, and the proximal surface remains nearly at the same vertical position. As the interfacial wave propagates, a high hydrodynamic pressure develops behind the wave front and increases in magnitude as the bubble continues to shrink. At
$t=92.84\,{\unicode[Arial]{x03BC}} \mathrm{s}$
, the interfacial wave converges on the axis
$r=0$
, producing a pressure of 260 MPa that drives the distal surface upward and forms a microjet with a transient velocity of 1300 m s−1. The jet velocity then decreases to about 450 m s−1 when it pierces the proximal surface and enters the capillary at
$t=93.5\,{\unicode[Arial]{x03BC}} \mathrm{s}$
.
In addition to the interfacial wave initiated from the humped annular edge, the annular flow near the capillary exit becomes significant during collapse, shaping the proximal part of the bubble into a dimple ring, as indicated by the red arrows. This dimple ring moves further inward towards the axis
$r=0$
and may cause an additional bubble-splitting event (see figure 15
a).
Evolution of the azimuthal vorticity
$\omega _\theta$
(left) and axial velocity
$U_y$
(right) during expansion and collapse of a bubble with
$R_{\textit{max}} = 499.91\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$R_i^* = 0.30$
,
$R_o^* = 0.50$
and
$\gamma = 0.8$
. The red contour marks the bubble interface, and the grey blocks represent the capillary walls. Each snapshot spans a region 1.0 mm wide and 0.8 mm high.
Figure 6 further illustrates the generation and propagation of the interfacial wave-like deformation. During the early expansion stage (
$t=5\,{\unicode[Arial]{x03BC}}$
s), the expanding flow generates strong vorticity near the capillary edge as a result of flow past a rigid object. As the proximal interface approaches the capillary, this flow structure persists and promotes the formation of a kink at the bubble edge by
$t=40\,{\unicode[Arial]{x03BC}}$
s. At this stage, the liquid above the bubble edge and near the capillary begins to move downward, while the liquid adjacent to the edge still moves upward, so that the resulting velocity imbalance amplifies the interfacial deformation. When the bubble starts to collapse, the central part of the proximal interface (
$|x|\lt R_o$
) remains nearly stationary because the local pressure gradient is weak, whereas the outer part (
$|x|\gt R_o$
) is driven downward by the collapsing flow. During the early collapse stage (
$t=50\,{\unicode[Arial]{x03BC}}$
s), the wavy proximal interface induces vorticity of opposite sign, which further increases the amplitude of the initial kink. In the subsequent collapse, the downward flow drives the wave peaks towards the distal interface. Finally, the distal wave peak converges near the axis and triggers the formation of the wave-driven fast microjet.
Left axis: temporal evolution of the positions of the proximal interface, distal interface and jet tip. Right axis: temporal evolution of the jet velocity.
Figure 7. Long description
A line graph with the horizontal axis labeled as time (t) in microseconds (μs) ranging from 80 to 94 and the left vertical axis labeled as Y in millimeters (mm) ranging from -0.8 to 0.2. The right vertical axis is labeled as U in meters per second (m/s) with labelled ticks from 0 to 1000. The graph includes four data lines: Jet interface in blue, Proximal interface in orange, Distal interface in green, and Jet velocity in cyan. The Jet interface and Jet velocity lines show a significant increase around 92.83 microseconds, indicating jet formation; jet impact occurs at approximately 93.4 microseconds. The Proximal interface remains nearly stationary before jet impact, while the Distal interface moves upward before jet formation and suddenly shifts downward when the tip splits. An inset zooms in on the time range from 92.75 to 92.90 microseconds, highlighting detailed changes in the Jet interface, Distal interface, and Jet velocity during this period.
Figure 7 shows the evolution of the positions of the proximal interface, distal interface and jet tip during and after jet formation. During bubble collapse, the distal interface, which initially coincides with the jet tip, moves gradually upward with an increasing but still relatively low velocity. At
$t\approx 92.83\,{\unicode[Arial]{x03BC}}$
s, the distal interface suddenly shifts downward as the tip splits, producing a fast microjet with a velocity of the order of
$10^3\,\mathrm{m\,s}^{-1}$
and a tiny daughter bubble moving downward. The jet tip at a velocity of approximately 100 m s−1 accelerates within only 30 ns to a velocity of 1300 m s−1, and then decelerates to about 1000 m s−1 over approximately 0.3
${\unicode[Arial]{x03BC}}$
s, when the jet tip is about
$100\,{\unicode[Arial]{x03BC}}$
m from the proximal interface. At
$t\approx 93.4\,{\unicode[Arial]{x03BC}}$
s, the jet impacts the proximal interface, after which both the jet tip and the proximal surface slow to about 600 m s−1 at
$t=94\,{\unicode[Arial]{x03BC}}$
s. The jet tip and distal interface do not coincide because the liquid jet is sufficiently fast for the gas-cushion effect to become significant, thereby entrapping vapour and leading to a discontinuous liquid jet.
Magnified view of the microjet development in figure 5. The image width is
$100\,{\unicode[Arial]{x03BC}} \mathrm{m}$
, and the vertical extent ranges from
$-0.55$
to
$-0.35\,\mathrm{mm}$
.
Figure 8 provides a magnified view of the convergence of the interfacial wave. Before the interfacial wave collides on the axis, the annular collapsing flow is faster than the upward collapsing flow, although both are already very rapid. At
$t=92.82\,{\unicode[Arial]{x03BC}}$
s, the annular flow stretches the tip into an elongated shape and generates a high local pressure of 1000 bar. Within 10 ns, this high pressure drives the annular collapsing flow towards the axis
$r=0$
. As a result, the tip splits at
$t=92.823\,{\unicode[Arial]{x03BC}}$
s, leaving a tiny residual bubble below. The shock wave generated by this splitting, together with the collapse of the residual bubble, further amplifies the local pressure to 2600 bar. This produces an acceleration of approximately
$\Delta p/(\rho \delta )\approx 6\times 10^{10}\,\mathrm{m\,s^{-2}}$
, increasing the jet velocity from 100 to 1300 m s−1 within 20 ns, from
$t=92.82\,{\unicode[Arial]{x03BC}}$
s to
$t=92.84\,{\unicode[Arial]{x03BC}}$
s (see figures 7 and 8).
Magnified view of the jet penetrating the bubble in figure 5. The image width is
$360\,{\unicode[Arial]{x03BC}} \mathrm{m}$
, and the vertical extent ranges from
$-0.6$
to
$0.2\,\mathrm{mm}$
.
Figure 9. Long description
Panel A: A cross-sectional view showing the newly formed microjet with pressure and velocity fields. The vertical axis ranges from -0.6 to 0.2 millimeters. The pressure scale ranges from 0 to 700 bar, and the velocity scale ranges from 0 to 1300 meters per second. The bubble is outlined in red, and arrows indicate the flow direction. Panel B: A cross-sectional view at 93.1 microseconds showing the microjet propagating through the bubble with pressure and velocity fields. The vertical axis ranges from -0.6 to 0.2 millimeters. The pressure scale ranges from 0 to 700 bar, and the velocity scale ranges from 0 to 1300 meters per second. The bubble is outlined in red, and arrows indicate the flow direction. Panel C: A cross-sectional view at 93.3 microseconds showing the microjet approaching the proximal bubble interface with pressure and velocity fields. The vertical axis ranges from -0.6 to 0.2 millimeters. The pressure scale ranges from 0 to 700 bar, and the velocity scale ranges from 0 to 1300 meters per second. The bubble is outlined in red, and arrows indicate the flow direction. Panel D: A cross-sectional view at 93.5 microseconds showing the microjet after impact on the proximal bubble interface with pressure and velocity fields. The vertical axis ranges from -0.6 to 0.2 millimeters. The pressure scale ranges from 0 to 700 bar, and the velocity scale ranges from 0 to 1300 meters per second. The bubble is outlined in red, and arrows indicate the flow direction. Panel E: A cross-sectional view at 93.7 microseconds showing the vapour jet advancing ahead of the liquid jet with pressure and velocity fields. The vertical axis ranges from -0.6 to 0.2 millimeters. The pressure scale ranges from 0 to 700 bar, and the velocity scale ranges from 0 to 1300 meters per second. The bubble is outlined in red, and arrows indicate the flow direction. Panel F: A cross-sectional view at 93.9 microseconds showing the jet propagating further into the capillary with pressure and velocity fields. The vertical axis ranges from -0.6 to 0.2 millimeters. The pressure scale ranges from 0 to 700 bar, and the velocity scale ranges from 0 to 1300 meters per second. The bubble is outlined in red, and arrows indicate the flow direction.
Once formed, however, this fast jet decelerates rapidly. Figure 9 illustrates the propagation of the microjet from its formation to its penetration of the upper bubble surface. The simulation resolution of 2
${\unicode[Arial]{x03BC}}$
m is not sufficient to fully resolve the propagation of such a thin microjet. Nevertheless, the flow details shown in figure 9 remain physically reasonable. Although the jet speed decreases to about
$900\,\mathrm{m\,s}^{-1}$
before reaching the proximal surface, the jet remains remarkably focused. The radius of the microjet base, initially approximately
$10\,{\unicode[Arial]{x03BC}}$
m at
$t=92.89\,{\unicode[Arial]{x03BC}}$
s, increases by about a factor of three to
$30\,{\unicode[Arial]{x03BC}}$
m at
$t=93.4\,{\unicode[Arial]{x03BC}}$
s.
After the microjet impacts the proximal bubble interface, a vapour jet forms ahead of the liquid jet and generates a high pressure of approximately 200 bar in front of it. The shock wave transmitted from the previous tip-splitting event at the distal surface also accelerates the gas near the distal tip locally to 1500 m s−1. However, this highly energetic gas dissipates its kinetic energy rapidly, so that the internal gas velocity within the bubble decreases to the order of 100 m s−1, which is typical of the gas-phase collapsing velocity during jet formation.
Spatio-temporal distribution of the axial velocity of the liquid phase for a bubble with
$R_{\textit{max}} = 499.91\,{\unicode[Arial]{x03BC}} \mathrm{m}$
,
$R_i^* = 0.30$
,
$R_o^* = 0.50$
and
$\gamma = 0.8$
. Blue and red indicate downward and upward velocity along the axis, respectively. The black dashed and solid lines denote the proximal and distal bubble interfaces on the axis
$r=0$
, respectively. Owing to the wide velocity range, a logarithmic scale is used. The temporal and spatial resolutions are 0.1
${\unicode[Arial]{x03BC}}$
s and 2
${\unicode[Arial]{x03BC}}$
m, respectively.
Figure 10 shows the spatio-temporal distribution of the axial velocity of the liquid phase along the axis
$r=0$
. The proximal surface initially moves upward with a rapidly decreasing velocity during the early expansion stage,
$0\lt t\lt 20\,{\unicode[Arial]{x03BC}}$
s, and then remains nearly stationary at
$y\approx -20\,{\unicode[Arial]{x03BC}}$
m until it is pierced by the microjet at about
$t=93.3\,{\unicode[Arial]{x03BC}}$
s. By contrast, the distal surface follows an evolution similar to that in figure 4, moving downward during expansion and upward during collapse. Here, however, the trajectory of the distal surface is much steeper, indicating the formation of a much faster microjet.
Magnified spatio-temporal distributions of the axial velocity
$U$
(in both gas and liquid phases) along the axis
$r=0$
, corresponding to figure 10. (a) The early expansion stage (
$0\lt t\lt 1\,{\unicode[Arial]{x03BC}}$
s), where arrows
$a$
and
$b$
indicate the shock wave and the bubble interface, respectively. (b) The jet-formation and impact stage (
$92.6\lt t\lt 93.6\,{\unicode[Arial]{x03BC}}$
s), where arrows
$c$
,
$d$
and
$e$
indicate the shock wave, the wave-driven jet and the post-impact jet penetrating into the capillary, respectively. Red and blue denote upward and downward velocities, respectively.
We now examine the expansion and collapse at a finer temporal resolution. Figure 11 shows the velocity distribution during the first microsecond after bubble nucleation. Once the bubble is nucleated at
$y=-400\,{\unicode[Arial]{x03BC}}$
m, the explosive expansion generates a blast wave that drives the surrounding liquid radially outward. The trajectory of the shock wave is clearly captured by tracking the location of the first velocity maximum, as indicated by arrow
$a$
. Similarly, the bubble interface can also be identified, as indicated by arrow
$b$
.
The map of jet formation and impact in figure 11
$(b)$
is even more complex, since tip splitting and the generation of multiple shock waves occur within a very short time interval of 0.5
${\unicode[Arial]{x03BC}}$
s. During collapse, both the gas and the liquid in the near-bubble region move upward until the bubble tip splits at
$t=92.83\,{\unicode[Arial]{x03BC}}$
s. Immediately after splitting, a strong shock wave is released near
$y=-450\,{\unicode[Arial]{x03BC}}$
m. As this shock wave propagates spherically, it decelerates the liquid for
$y\lt -450\,{\unicode[Arial]{x03BC}}$
m, and its trajectory is captured by the low-velocity region indicated by arrow
$c$
. By contrast, the shock wave accelerates both the liquid and the interface for
$y\gt -450\,{\unicode[Arial]{x03BC}}$
m. The enormous pressure gradient associated with this shock wave drives the interface into a thin fast jet with a velocity of approximately 1200 m s−1, as indicated by arrow
$d$
. However, this microjet rapidly decelerates to around 1000 m s−1 within 0.2
${\unicode[Arial]{x03BC}}$
s because of the large drag exerted by the surrounding gas. After penetrating the proximal bubble surface, the microjet still retains a velocity of about 400 m s−1 and enters the capillary (see arrow
$e$
in the region
$y\gt 0$
).
4. Discussion
4.1. Jet regime
Phase diagram and maximum jet velocity distribution in the
$(R_o^*,\gamma )$
parameter space for a fixed inner radius
$R_i^*=0.30$
. (a) Regime diagram of the two jet morphologies. Experimental and numerical results are shown by filled and open symbols, respectively. Circle symbols denote regular jets and diamond symbols denote wave-driven jets. (b) Distribution of the maximum jet velocity
$U_{\textit{max}}$
. The solid line in both panels represents the fitted boundary
$\gamma = 1.08 - 0.65(R_o^* - 0.20)^{2.4}$
separating regular and wave-driven jet regimes. Both panels use a logarithmic scale for the horizontal axis and a linear scale for the vertical axis.
Figure 12. Long description
Panel A: A scatter plot shows the regime diagram of two jet morphologies. The horizontal axis is labeled R_o* with a logarithmic scale, and the vertical axis is labeled γ with a linear scale. The plot includes filled and open symbols representing experimental and numerical results, respectively. Circle symbols denote regular jets, and diamond symbols denote wave-driven jets. A solid line represents the fitted boundary separating regular and wave-driven jet regimes. Panel B: A contour plot displays the distribution of the maximum jet velocity U_max. The horizontal axis is labeled R_o* with a logarithmic scale, and the vertical axis is labeled γ with a linear scale. The color scale on the right indicates the velocity in meters per second, ranging from 50 to 2000. The solid line in both panels represents the fitted boundary separating regular and wave-driven jet regimes.
The above results indicate that microjet formation is governed by two geometric factors: the stand-off distance
$\gamma$
, which determines the liquid-film dynamics and the vertical collapse of the bubble, and the normalised outer radius of the capillary,
$R_o^*$
, which affects the annular flow. Figure 12 summarises the distribution of the microjet regimes on a parameter map spanned by
$\gamma$
on the vertical axis and
$R_o^*$
on the horizontal axis, based on both experimental and numerical data.
Here, we distinguish two types of microjets. The first is a regular jet, which is primarily induced by the vertical collapse of the bubble interface farthest from the capillary, as shown in figure 2(a). Its formation mechanism is similar to that of the typical microjet generated by a cavitation bubble near a rigid wall. Under some conditions, however, the capillary boundaries enhance the annular flow and deform the bubble into a conical shape, so that the pressure built up near the distal interface becomes higher and therefore produces a faster microjet than the typical microjet from a collapsing bubble near a smooth rigid wall.
The second type is referred to as a wave-driven jet, since it forms through the convergence of the interfacial wave on the distal bubble surface. This mechanism results from the combined effect of the vertical collapse and the convergence of an interfacial wave travelling from the proximal surface to the distal surface. Depending on the location and intensity of the shock wave released by the splitting event, the resulting microjet can reach velocities as high as 2000 m s−1 in the simulation.
Figure 12(a) shows the regime diagram of jet morphology in the
$(R_o^*,\gamma )$
parameter space, distinguishing regular and wave-driven jets. The results indicate that wave-driven jets occur for relatively small
$\gamma$
and moderate-to-small
$R_o^*$
, where confinement effects from the capillary are strong. The critical stand-off distance for fast-jet formation is approximately
$\gamma \approx 1.05$
for the smallest
$R_o^*$
considered, and it decreases with increasing
$R_o^*$
. This trend can be attributed to the fact that, for larger capillary radii
$R_o^*$
, stronger confinement (i.e. smaller
$\gamma$
) is required to sufficiently deform the proximal interface and generate an interfacial kink, which then evolves into a propagating interfacial disturbance. For
$\gamma \approx 0.2$
, the transition between wave-driven and regular jets is still observed near
$R_o^* \approx 1.4$
.
This fast-jet mechanism of the wave-driven jet differs fundamentally from that of bubble collapse near a flat rigid surface, corresponding to the limiting case
$R_o^* \rightarrow \infty$
and
$R_i^* \rightarrow 0$
, where fast needle jets form for
$\gamma \lesssim 0.2$
(Lechner et al. Reference Lechner, Lauterborn, Koch and Mettin2020; Reuter & Ohl Reference Reuter and Ohl2021; Reuter et al. Reference Reuter, Deiter and Ohl2022). There, the interfacial kink arises intrinsically from the boundary-layer flow and remains spatially localised during collapse. In the present capillary geometry, by contrast, the deformation evolves into a propagating interfacial disturbance that travels towards the distal surface. The difference between these two mechanisms is discussed in detail in § 4.4.
Figure 12(b) shows the distribution of the maximum jet velocity
$U_{\textit{max}}$
over the same parameter space. The largest velocities occur at relatively small
$R_o^*$
and intermediate
$\gamma$
, slightly below the transition between the two types of jets. Across the explored parameter range, wave-driven jets consistently exhibit velocities exceeding 400 m s−1. However, regular jets at small
$R_o^*$
but relatively large
$\gamma$
can also reach velocities above 400 m s−1. This is attributed to enhanced pressure buildup near an elongated distal interface, similar to jet enhancement induced by an additional rigid boundary (Zeng et al. Reference Zeng, Zhang, Tan, An and Ohl2024).
Scatter plots of the (a) maximum and (b) impact velocities of the microjet on the parameter plane spanned by
$\gamma$
on the horizontal axis and
$R_o^*$
is distinguished by colour.
Figure 13. Long description
Two scatter plots illustrate the relationship between microjet velocities and parameters. Panel A: A scatter plot shows the maximum velocity (Umax) of the microjet in meters per second on the vertical axis against the parameter gamma (γ) on the horizontal axis. Different colors represent different values of Ro* (0.40, 0.51, 0.61, 0.71). Panel B: Another scatter plot displays the impact velocity (Uimp) of the microjet in meters per second on the vertical axis against the parameter gamma (γ) on the horizontal axis. Similar to Panel A, different colors represent different values of Ro* (0.40, 0.51, 0.61, 0.71).
The scatter plots of jet velocity are shown in figure 13 as functions of the normalised outer radius of the capillary,
$R_o^*$
, and the stand-off distance,
$\gamma$
. For all values of
$R_o^*$
, both
$U_{\textit{max}}$
and
$U_{\textit{imp}}$
first increase and then decrease with increasing
$\gamma$
.
The maximum value of
$U_{\textit{max}}$
occurs at
$R_o^*=0.40$
and
$\gamma =0.91$
, where both the interfacial wave motion and the vertical collapsing flow are relatively strong. This condition also lies near the transition to the wave-driven jet regime, where the pressure generated by the vertical collapse and the convergence of the interfacial wave superpose, producing a driving pressure as high as 2500 bar, as shown in figure 8. As
$R_o^*$
increases, the surface-wave convergence becomes less violent, and the resulting jet becomes slower.
For high-speed microjets, the velocity decreases immediately after jet formation. The drag force depends nonlinearly on both the jet shape and its speed. Despite this complicated relation, we find that the maximum impact velocity occurs at
$R_o^*=0.51$
and
$\gamma =0.8$
.
4.2. Influence of the outer radius
$R_o^*$
Comparison of bubble dynamics for different
$R_o^*$
at fixed
$\gamma \approx 0.8$
and
$R_i^*=0.30$
. The dashed line indicates the capillary exit.
In this section, we investigate the influence of the outer radius
$R_o^*$
on the bubble and microjet dynamics. Here, we fix the normalised inner radius at
$R_i^*=0.3$
and vary the capillary wall thickness. Figure 14 compares the bubble dynamics for four values of
$R_o^*$
, namely 0.30, 0.40, 0.81 and 1.31, shown in different colours. Interestingly, at the same stand-off distance
$\gamma =0.8$
, the bubbles exhibit similar distal-surface shapes and the associated vertical collapsing flows induced by the lower surface. This confirms that the stand-off distance has the dominant influence on the vertical collapsing flow, because it determines the pressure distribution near the distal interface, whereas the boundary geometry has only a minor effect.
By contrast, the proximal interface behaves differently. When the horizontal bubble radius exceeds the outer radius of the capillary (also see figure 5), the proximal interface develops a humped annular edge during expansion. This edge is located farther from the axis
$r=0$
and becomes smoother as
$R_o^*$
increases. Such a humped annular edge promotes the formation of an interfacial wave during collapse and thereby intensifies the microjet. Once
$R_o^*$
exceeds the horizontal bubble radius, the humped annular edge disappears and the proximal interface becomes smooth (see figure 14 for
$ R_o^*$
= 0.81 and 1.31). As a result, the interfacial wave either develops much later (
$R_o^*=0.81$
) or vanishes altogether (
$R_o^*=1.31$
).
Pressure and velocity fields during and after jet formation for three values of the normalised outer radius: (a)
$R_o^* = 0.31$
, (b)
$R_o^* = 0.40$
and (c)
$R_o^* = 1.31$
. For each snapshot, the left half shows pressure and the right half shows velocity magnitude; the red contour marks the bubble interface, and the black arrows indicate flow direction. The normalised inner radius and stand-off distance are fixed at
$R_i^* = 0.30$
and
$\gamma = 0.80$
. Each snapshot has a horizontal extent of 1 mm.
Next, we explain this trend by comparing three representative cases with
$R_o^*=0.31$
, 0.40 and 1.31. For smaller
$R_o^*$
, the humped annular edge is located closer to the axis
$r=0$
. As a result, the interfacial wave travels more slowly towards the distal surface and its convergence is relatively weaker. Meanwhile, the annular flow near the capillary exit becomes stronger as
$R_o^*$
decreases, shaping the proximal part of the bubble into a deeper dimple ring. This dimple ring converges on the axis
$r=0$
at a later time and leads to a secondary splitting of the upper part of the bubble. This secondary splitting generates a pressure of about 100 bar and drives the nearby interface upward into a microjet with a speed of about 700 m s−1 (see figure 15
a at
$t=93.5\,{\unicode[Arial]{x03BC}}$
s).
For
$R_o^*=0.40$
, the annular flow is weaker, and the dimple ring no longer converges on the axis to produce a secondary splitting before the jet impacts the proximal surface. Under this condition, the microjet forms when the interfacial wave converges at the distal surface of the bubble. The pressure built up in this case reaches about 2500 bar, producing a microjet with a maximum velocity of 1250 m s−1 and an impact velocity of 800 m s−1.
For
$R_o^*=1.31$
, the rigid boundary is sufficiently large to suppress the formation of the humped annular edge and the nearby vortex ring. The interfacial wave therefore disappears, and microjet formation is dominated by the vertical collapse of the distal surface; that is, the bubble returns to the regular-jet regime. Nevertheless, this regular jet still reaches a speed of about 200 m s−1, which is greater than the typical microjet velocity (about 80 m s−1) of a wall-proximal cavitation bubble at the same stand-off distance
$\gamma =0.8$
(Zeng et al. Reference Zeng, Zhang, Tan, An and Ohl2024).
In summary, before
$U$
reaches its peak value, the annular flow near the capillary exit becomes less prominent as
$R_o^*$
increases, while the secondary splitting is delayed and eventually disappears. These changes favour a stronger pressure buildup and therefore a faster microjet. When
$R_o^*$
is increased further, however, the interfacial wave converges earlier, so its collision velocity is lower. The resulting pressure buildup then decreases, and the microjet becomes slower. Once
$R_o^*\gt 1.3$
, the interfacial wave vanishes altogether, and the bubble generates a regular microjet.
Comparison of the motion of the bubble distal surface for six selected values of
$R_o^*$
at fixed
$\gamma \approx 0.8$
and
$R_i^* = 0.3$
. (a) Axial position of the distal surface as a function of time. (b) Velocity of the distal surface as a function of time.
Figure 16. Long description
Two line graphs depict the motion and velocity of a bubble’s distal surface over time for different values. Panel A: A line graph shows the axial position of the distal surface as a function of time. The x-axis represents time in microseconds (t (μs)), ranging from 84 to 94. The y-axis represents the axial position in micrometers (Y (μm)), ranging from -600 to 0. Multiple lines, each representing a different value of R_o*, show the position of the jet interface and the proximal interface over time. Panel B: Another line graph shows the velocity of the distal surface as a function of time. The x-axis represents time in microseconds (t (μs)), ranging from 84 to 94. The y-axis represents velocity in meters per second (U (m s^-1)), ranging from 10^1 to 10^3. Multiple lines, each representing a different value of R_o*, show the velocity of the distal surface over time.
Figure 16 compares the position and velocity of the distal surface for six different values of
$R_o^*$
. The results show that the distal surface undergoes two main stages of motion: a relatively slow upward motion before jet formation, followed by a rapid jetting stage after jet formation. Before
$t\approx 90\,{\unicode[Arial]{x03BC}}$
s, the distal surface evolves similarly in all cases, indicating that the dynamics is mainly governed by the vertical collapse flow. At later times, two distinct mechanisms emerge. In the regular jet regime, corresponding to
$R_o^*=0.91$
and 1.11, no convergence of the interfacial disturbance is involved, and the pressure buildup near the distal surface is mainly determined by the curvature of the distal surface, which decreases as
$R_o^*$
increases within this regime. As a result, the case
$R_o^*=0.91$
reaches a higher jet velocity of about 500 m s−1 and reaches the proximal surface earlier.
In the wave-driven jet regime, the distal surface undergoes rapid acceleration once the interfacial wave converges near the axis. As
$R_o^*$
increases within this regime, the convergence occurs earlier, although the resulting jet velocity varies non-monotonically with
$R_o^*$
. A local minimum in the distal-surface velocity is observed before the rapid acceleration stage, which is associated with the downward motion of the converging interfacial wave and the resulting temporary deceleration of the distal tip.
(a) The duration of the first oscillation cycle
$t_c$
versus
$R_o^*$
. (b) Jet-formation time
$t_{jet}$
as a function of
$R_o^*$
. (c) Non-dimensional jet-formation time
$t_{jet}/t_c$
. The stand-off distance and inner radius are fixed as
$\gamma \approx 0.8$
and
$R_i^* = 0.3$
. The transition between regular and wave-driven jets is around
$R_o^*=0.81$
.
Figure 17(a) shows the duration of the first oscillation cycle,
$t_c$
, as a function of
$R_o^*$
. The oscillation period increases with
$R_o^*$
, indicating that a larger outer radius slows the overall bubble collapse because of the stronger influence of the rigid boundary. Figure 17(b) shows the jet-formation time
$t_{jet}$
, defined here as the instant at which the jet reaches its maximum velocity. The results show that
$t_{jet}$
first decreases for
$0.3\lt R_o^*\lt 0.5$
, then varies only weakly over
$0.5\lt R_o^*\lt 0.9$
and finally increases again for
$R_o^*\geqslant 0.9$
. The corresponding normalised time
$t_{jet}/t_c$
is plotted in figure 17(c). Overall,
$t_{jet}/t_c$
decreases with increasing
$R_o^*$
, indicating that the jet forms earlier relative to the bubble-oscillation time scale.
In the wave-driven jet regime, the maximum value of
$U_{\textit{max}}$
occurs at
$R_o^*\approx 0.35$
, which also corresponds to a relatively large normalised jet-formation time. This indicates that a longer convergence time of the interfacial disturbance leads to stronger flow focusing and a larger pressure buildup at the distal surface, thereby producing a faster jet. In the regular-jet regime, by contrast, earlier normalised jet formation
$t_{jet}/t_c$
is associated with a weaker pressure buildup near the distal surface and therefore a lower jet velocity.
(a) Impact velocity
$U_{\textit{imp}}$
versus
$R_o^*$
for four stand-off distances:
$\gamma =0.8$
, 0.9, 1.0 and 1.2. (b) Axial position of the proximal bubble surface at impact,
$Y_{\textit{imp}}$
. The insets in (a) show three representative jet morphologies at impact, corresponding to the dashed rectangles of the same colour. The normalised inner radius is fixed at
$R_i^* = 0.303$
for all cases.
Figure 18. Long description
Two scatter plots depict the dynamics of cavitation bubble impact velocity and axial position. Panel A: A scatter plot shows the impact velocity (U_imp) in meters per second on the vertical axis versus the normalized outer radius (R_o*) on the horizontal axis. The plot includes data points for four stand-off distances: 0.8, 0.9, 1.0, and 1.2, represented by different shades of green. Insets within the plot display three representative jet morphologies at impact, corresponding to dashed rectangles of the same color. Panel B: Another scatter plot shows the axial position of the proximal bubble surface at impact (y_imp) in micrometers on the vertical axis versus the normalized outer radius (R_o*) on the horizontal axis. The data points are color-coded similarly to Panel A, representing the same stand-off distances.
Figure 18 compares the impact velocity for four stand-off distances:
$\gamma =0.8$
, 0.9, 1.0 and 1.2. The microjets can be grouped into three types based on the bubble shape at impact. The first is a regular jet with a relatively wide base, formed mainly by the vertical collapsing flow associated with the distal surface, as shown by the inset within the grey rectangle in figure 18(a). These jets have impact velocities below 300 m s−1, corresponding to the grey dashed region.
The second type is a secondary jet generated by the secondary splitting caused by the annular collapsing flow (green inset). These jets impact at velocities between 400 and 800 m s−1.
The third type, highlighted by the orange inset, is a thin and fast jet generated by the convergence of the interfacial wave on the distal surface of the bubble, with impact velocities ranging from 500 to 1500 m s−1. These jets are also accompanied by fast vapour jets entering the capillary. Both the second and third jet types are the wave-driven jets.
For the largest stand-off distance considered,
$\gamma =1.2$
, the capillary is too far from the bubble to induce an interfacial wave, so only the regular-jet regime is observed. The impact location,
$Y_{\textit{imp}}$
, increases with
$R_o^*$
from
$-520\,{\unicode[Arial]{x03BC}}$
m and approaches an approximately constant value of
$-120\,{\unicode[Arial]{x03BC}}$
m for
$R_o^*\gt 2$
.
When the stand-off distance is reduced to
$\gamma =1.0$
, all three jet types can occur. As
$R_o^*$
increases, the jet morphology changes from a secondary jet to a thin jet, and finally to a regular jet. A similar trend is observed for
$\gamma =1.0$
, 0.9 and 0.8 in that the impact location first shifts to positive values, indicating penetration into the capillary, and then decreases towards an approximately constant value as
$R_o^*$
increases.
To conclude,
$R_o^*$
mainly affects the bubble deformation and the flow near the capillary exit. When the bubble is located sufficiently far from the capillary, i.e. for
$\gamma \gt 1.2$
, the capillary is unable to induce a wavy deformation on the proximal bubble surface. Although annular flow may still be present, jet formation is then dominated by the vertical collapsing flow, resulting in regular jets with velocities below 300 m s−1.
For
$\gamma \lt 1.2$
, the bubble develops wavy deformations on its distal surface during expansion. Two types of surface motion may then arise, depending on
$R_o^*$
and
$\gamma$
. The first is an interfacial wave that originates near the proximal edge of the bubble, travels downward, and converges at the distal surface. The second is a dimple ring that remains near the proximal surface and moves towards the axis
$r=0$
under the action of the annular flow. Both types of surface motion can converge and induce splitting of the bubble surface, thereby generating a large local pressure that accelerates the nearby interface into a fast microjet.
For a smaller
$R_o^*$
, the annular flow during collapse is stronger, so the dimple ring converges with a higher annular velocity. This motion causes the splitting of the proximal part of the bubble and forms a microjet with a velocity in the range of 300–800 m s−1. As
$R_o^*$
increases, the annular flow weakens, and only the interfacial wave converges at the distal surface of the bubble. The microjet produced by this mechanism can reach velocities up to 1250 m s−1.
4.3. Influence of the inner radius
$R_i^*$
Comparison of bubble dynamics for different
$R_i^*$
at fixed
$\gamma \approx 0.8$
and
$R_o^*=0.51$
. The dashed line indicates the capillary exit.
Next, we discuss the influence of the normalised inner radius of the capillary,
$R_i^*$
. Here, we fix the normalised outer radius at
$R_o^*=0.51$
and vary the capillary-wall thickness accordingly. Figure 19 compares the bubble dynamics for four selected values of
$R_i^*$
, namely 0.45, 0.40, 0.30 and 0.10. At the same stand-off distance
$\gamma$
and outer radius
$R_o^*$
, the bubble-interface motions are nearly indistinguishable over the first 80
${\unicode[Arial]{x03BC}}$
s, except near the proximal part of the bubble; that is, the surface-wave motion and the overall interface shrinkage remain very similar in all cases.
The main difference lies in the shape of the proximal interface. For the smallest tube,
$R_i^*=0.10$
, the proximal interface remains relatively flat, whereas it gradually develops a wavy shape as
$R_i^*$
increases, since a wider tube allows stronger deformation. Figure 19 also shows that the interface position on the axis
$r=0$
is slightly higher for larger
$R_i^*$
. This difference has a noticeable effect only during the very late stage of collapse: as
$R_i^*$
decreases, the onset of jet formation is slightly delayed.
Pressure and velocity fields during and after jet formation for two values of the normalised inner radius: (a)
$R_i^* = 0.40$
and (b)
$R_i^* = 0.05$
. For each panel, the left half shows pressure and the right half shows velocity magnitude. The normalised outer radius and stand-off distance are fixed at
$R_o^* = 0.51$
and
$\gamma = 0.81$
. Each snapshot has a horizontal extent of 1 mm.
Figure 20. Long description
The image contains two sets of three panels each, labeled (a) and (b). Each panel shows a combination of pressure and velocity magnitude fields. Panel (a) has three sub-panels at 92.8, 93.0 and 98.0 microseconds, and Panel (b) has three sub-panels at 92.3, 93.0 and 98.0 microseconds. The left half of each sub-panel displays pressure in bars. The right half of each sub-panel shows velocity magnitude in meters per second. The horizontal axis for each sub-panel spans 1 millimeter. Arrows indicate the flow direction. Red contours outline the bubble interface. The pressure and velocity fields illustrate the dynamics of jet formation and the interaction with the surrounding fluid.
Figure 20 compares the flow details immediately after jet formation for two different inner radii:
$R_i^*=0.40$
and 0.05. The convergence of the interfacial wave occurs at about
$t=91.9\,{\unicode[Arial]{x03BC}}$
s for
$R_i^*=0.40$
and is delayed to approximately
$t=92.3\,{\unicode[Arial]{x03BC}}$
s for
$R_i^*=0.05$
. For the larger capillary, the bubble develops a convex distal surface, whereas in the smaller capillary the proximal surface remains nearly flat.
When the microjet impacts the proximal surface, the flow diverges slightly in the case of the convex interface, but remains more focused when the interface is flat. As a result, the jet impact velocity is higher for smaller
$R_i^*$
. After impact, the vapour jet can penetrate into the capillary for
$R_i^*=0.40$
. By contrast, for the smaller inner radius, the rigid wall at the capillary exit prevents the jet from propagating upward, so that it spreads along the wall instead.
(a) Impact velocity
$U_{\textit{imp}}$
versus
$R_i^*$
for four stand-off distances:
$\gamma =0.6$
, 0.8, 1.0 and 1.2. (b) Axial position of the proximal bubble surface at impact,
$Y_{\textit{imp}}$
. The normalised outer radius is fixed at
$R_o^* = 0.51$
for all cases.
Figure 21 compares the impact velocity and impact location for four stand-off distances:
$\gamma =0.6$
, 0.8, 1.0 and 1.2. For the largest stand-off distance,
$\gamma =1.2$
, only a regular microjet is formed, with a velocity of approximately 300 m s−1, since an interfacial wave cannot be initiated. As the bubble moves closer to the capillary, the impact velocity increases to around 600 m s−1 for
$\gamma =1.0$
. For
$\gamma =0.8$
, the jet velocity reaches about 900 m s−1 when
$R_i^*\leqslant 0.1$
and fluctuates around 800 m s−1 when
$R_i^*\gt 0.1$
. For
$\gamma =0.6$
,
$U_{\textit{imp}}$
decreases to about 600 m s−1, except for the case
$R_i^*=0.45$
, where it reaches 800 m s−1.
For the relatively large stand-off distances
$\gamma =1.2$
and 1.0, the impact location is nearly independent of
$R_i^*$
, with values of about
$-400\,{\unicode[Arial]{x03BC}}$
m and
$-20\,{\unicode[Arial]{x03BC}}$
m, respectively. For the smaller stand-off distances
$\gamma =0.8$
and 0.6, the proximal surface remains flat and stays very close to the capillary exit when
$R_i^*\leqslant 0.2$
. As the inner radius increases, the proximal surface is allowed to move into the capillary during expansion, while remaining at nearly the same position during collapse. As a result,
$Y_{\textit{imp}}$
increases with
$R_i^*$
. In addition, bubbles with smaller stand-off distances exhibit higher impact locations.
To conclude, the inner radius has only a minor influence on the jet-formation process, as it mainly causes a slight delay in the convergence time of the interfacial wave. The overall collapse dynamics remains dominated by the outer radius
$R_o^*$
. Nevertheless, the inner radius still plays an important role in modulating the flow: the impact velocity can be tuned within a limited range, while the impact location and the penetration of the vapour jet can be controlled through the inner radius.
4.4. Comparison with jetting cavitation bubbles near smooth rigid surface
Comparison of bubble dynamics at
$\gamma =0.1$
for three jetting scenarios. (a) Needle jet from a bubble collapsing near a smooth rigid surface, with
$R_i^*=0$
,
$R_o^*=20.2$
,
$t_{jet}^*=0.995$
. (b) Regular jet from a bubble near a capillary, with
$R_i^*=0.51$
,
$R_o^*=20.2$
,
$t_{jet}^*=0.965$
. (c) Wave-driven jet from a bubble near a capillary, with
$R_i^*=0.51$
,
$R_o^*=0.52$
,
$t_{jet}^*=0.916$
. The dashed lines mark the vertical position of the rigid wall or the capillary exit. The scale bar is 0.5 mm.
Figure 22. Long description
Panel A: Sequence of bubble shapes collapsing near a smooth rigid surface, labeled with time intervals in microseconds. The bubble deforms and collapses asymmetrically, forming a needle jet. Panel B: Sequence of bubble shapes collapsing near a capillary, labeled with time intervals in microseconds. The bubble deforms and collapses symmetrically, forming a regular jet. Panel C: Sequence of bubble shapes collapsing near a capillary, labeled with time intervals in microseconds. The bubble deforms and collapses asymmetrically, forming a wave-driven jet. Dashed lines indicate the vertical position of the rigid wall or capillary exit. Scale bar represents 0.5 mm.
Figure 22(a) shows a typical needle-jet formation for bubble collapse near a smooth rigid surface at
$\gamma =0.1$
. A kink forms at a vertical distance of about
$70\,{\unicode[Arial]{x03BC}}$
m from the rigid wall when the bubble starts to collapse. This kink moves towards the axis faster than the surrounding parts of the interface and eventually converges there, generating a needle jet with a velocity of about 800 m s−1 and a radius of
$5\,{\unicode[Arial]{x03BC}}$
m.
The bubble behaves differently when an inner radius of
$R_i^*=0.51$
is introduced, as shown in figure 22(b). The proximal interface is no longer smooth, but intrudes into the capillary with a convex shape. Meanwhile, the bubble edge is located farther from the boundary, forming a relatively thicker liquid layer than in figure 22(a). This allows the bubble edge to deform more strongly and develop a pronounced interfacial wave. However, because of the confinement imposed by the rigid boundary, the liquid above the proximal bubble interface cannot move downward. As a result, the interfacial wave shrinks towards the axis rather than propagating downward, and therefore fails to converge before the distal surface has already collapsed into a regular jet.
When the same inner radius
$R_i^*=0.51$
is maintained but the outer radius is reduced to
$R_o^*=0.52$
, the liquid above the proximal bubble interface moves downward during bubble collapse (see figure 6). This downward motion causes the interfacial wave to propagate downward as well and later converge at the axis, thereby forming a wave-driven jet, as shown in figure 22(c). This jet has a velocity of about 800 m s−1, comparable to that in figure 22(a), but a larger radius of approximately
$10\,{\unicode[Arial]{x03BC}}$
m.
The maximum jet velocity as a function of
$R_i^*$
for
$\gamma =0.1$
. Square, circle and diamond symbols represent needle jet, regular jet and wave-driven jet, respectively.
Figure 23 shows the variation of
$U_{\textit{max}}$
with
$R_i^*$
for
$\gamma =0.1$
. For the smooth-surface case, corresponding to
$R_i^*=0$
, the bubble produces a needle jet with a velocity of approximately 790 m s−1. As
$R_i^*$
increases to about 0.51, the jet transitions from a needle jet to a regular jet. By contrast, for the smaller-outer-radius case
$R_o^*=0.52$
, the jet becomes wave-driven. Overall,
$U_{\textit{max}}$
decreases with increasing
$R_i^*$
, but exhibits a local minimum near
$R_i^*\approx 0.51$
, which corresponds to the transition from the needle-jet regime to the regular-jet regime. At the same stand-off distance
$\gamma =0.1$
, the wave-driven jet attains a velocity comparable to that of the needle jet near a smooth rigid surface, indicating that capillary-like confinement provides a robust route for generating fast jets through geometric control, while avoiding the damage associated with shock-wave self-focusing during bubble collapse at near-zero stand-off distances (Reuter et al. Reference Reuter, Deiter and Ohl2022).
The above comparison highlights a clear difference between the needle jet near a smooth rigid wall and the wave-driven jet near a capillary. For the needle jet, the motion of the bubble edge is strongly constrained by the nearby rigid wall and remains predominantly parallel to the rigid surface. As a result, the kink changes little in its vertical position towards the rigid surface during collapse. The jet forms at the relatively late non-dimensional time
$t_{jet}^*=0.995$
, indicating that the needle jet results primarily from annular-flow focusing of a localised interfacial kink rather than from the convergence of a propagating interfacial wave. By contrast, for a bubble near a capillary, a relatively thick liquid layer still separates the bubble edge from the boundary because liquid is squeezed out of the capillary by the convex proximal interface. This allows the bubble edge to deform more freely and generate an interfacial wave that propagates towards the distal surface and converges there at
$t_{jet}^*=0.916$
. The resulting fast jet is therefore fundamentally wave-driven and forms significantly earlier in the collapse cycle.
5. Conclusion
In this study, we systematically investigated microjet formation from the expansion and collapse of a cavitation bubble near capillary structures by combining ultrahigh-speed imaging with axisymmetric compressible volume-of-fluid simulations. The parameter space was explored by varying the normalised inner and outer radii,
$R_i^*$
and
$R_o^*$
, and the stand-off distance,
$\gamma$
.
We identify two types of microjets whose formation depends on the three geometric parameters
$\gamma$
,
$R_i^*$
and
$R_o^*$
. Their formation results from the competition between the vertical collapsing flow, which is mainly determined by
$\gamma$
, and the interaction between the interfacial wave and the annular converging flow, which are mainly controlled by
$R_o^*$
. The first type is a regular jet, which occurs when the interfacial wave either vanishes or fails to converge before the distal bubble surface develops into a jet under vertical collapse. This regime is typically found for large
$\gamma$
and large
$R_o^*$
. The second type is a wave-driven jet, which forms through the convergence of an interfacial wave initiated by capillary-induced deformation of the expanding bubble. In this case, a pressure of
$O(100\,\mathrm{MPa})$
can be generated by bubble splitting, producing a fast microjet. The jet velocity at formation is typically above 500 m s−1 and can reach up to 2000 m s−1 in the simulations.
A regime diagram summarising the distribution and velocity of these two jet types is presented in figure 12. Within the present parameter space, wave-driven jets can occur over a much wider range, namely
$\gamma \lt 1.1$
, than the needle jet from bubble collapse near a flat surface, where the corresponding condition is
$\gamma \leqslant 0.2$
. In the present capillary geometry, the interfacial wave is generated on the boundary-proximal surface through capillary-induced shaping of the expanding bubble, whereas for a flat surface, the interfacial kink arises from annular flow near the boundary-distal surface and remains spatially localised during collapse. As
$R_o^*$
increases, the bubble must be located closer to the capillary exit in order for the interfacial wave to form.
The three geometric parameters play different roles. The stand-off distance
$\gamma$
controls the vertical collapsing flow. The outer radius
$R_o^*$
governs the generation and motion of the interfacial wave and thereby strongly affects jet formation. For relatively small stand-off distances,
$\gamma \leqslant 1.1$
, wave-driven jets are obtained when
$R_o^*\leqslant 1.4$
, while the threshold value of
$\gamma$
decreases as
$R_o^*$
increases. The inner radius
$R_i^*$
has a comparatively weaker influence on jet formation, as it only slightly delays the convergence of the interfacial wave. Nevertheless, it plays an important role in the later jet dynamics by controlling the jet impact location and the penetration of the vapour jet into the capillary.
The wave-driven jet identified here differs fundamentally from the needle jet generated by a bubble collapsing near a smooth rigid surface, although the jet velocities may reach a similar magnitude. In the present capillary geometry, the wavy proximal interface formed during bubble expansion develops into a pronounced interfacial wave that propagates towards the distal bubble surface. The convergence of this wave at the axis
$r=0$
produces large pressure to accelerate the interface into a fast jet. By contrast, for a bubble near a smooth rigid surface, annular flow drives a localised interfacial kink that shrinks towards the axis without appreciable propagation along the interface, thereby producing the needle jet. As a result, the needle jet is restricted to a narrow range of stand-off distances,
$\gamma \leqslant 0.2$
, whereas the wave-driven jet can be generated over a much broader range,
$\gamma \lt 1.1$
. Its velocity and size can also be tuned by adjusting the three geometric parameters
$\gamma$
,
$R_o^*$
and
$R_i^*$
.
In summary, we have characterised both the experimental and numerical bubble dynamics and the associated microjet formation near a capillary over a range of geometric parameters. This detailed parametric study provides a physical basis for tuning the stand-off distance
$\gamma$
and the capillary radii
$R_o^*$
and
$R_i^*$
to achieve different jet velocities and impact characteristics, with potential implications for microfluidics, targeted cleaning, biomedical injection and other cavitation-based technologies.
Acknowledgements
The anonymous referees are acknowledged for their very helpful suggestions to improve the manuscript.
Funding
This research was partly funded by the National Natural Science Foundation of China under grant nos. 52525102, 12302313 and 52521009, and supported by the National Key Laboratory of Marine Engine Science and Technology through grant no. LAB-2024-07-WD.
Declaration of interests
The authors report no conflict of interest.
Appendix. Grid-sensitivity analysis
We perform a grid-sensitivity analysis of the bubble dynamics and jet-formation process using four grid spacings:
$\Delta x = 1.5$
, 2, 2.5 and 3
${\unicode[Arial]{x03BC}}$
m. The geometric parameters are
$R_i^* = 0.30$
,
$R_o^* = 0.50$
and
$\gamma = 0.8$
, corresponding to a fast-jet case. Figure 24(a) compares the evolution of the bubble-equivalent spherical radius. The results show that the radius evolution converges as the grid spacing is reduced below 2.5
${\unicode[Arial]{x03BC}}$
m. Figure 24(b) shows the bubble interface at two representative time instants. At
$t=90\,{\unicode[Arial]{x03BC}}$
s, the bubble develops an interfacial wave propagating in the negative
$y$
direction. As the grid spacing decreases, the surface-wave profile exhibits clear convergence. At
$t=93\,{\unicode[Arial]{x03BC}}$
s, after surface-wave convergence and fast-jet formation, both the bubble outline and the jet profile are nearly indistinguishable for
$\Delta x = 1.5$
and 2
${\unicode[Arial]{x03BC}}$
m. Considering both computational efficiency and numerical accuracy, we therefore adopt
$\Delta x = 2\,{\unicode[Arial]{x03BC}}$
m throughout this study.
Comparison of (a) the evolution of the bubble-equivalent spherical radius and (b) the bubble interface before (
$t=90\,{\unicode[Arial]{x03BC}}$
s) and after (
$t=93\,{\unicode[Arial]{x03BC}}$
s) interfacial-wave convergence for a wave-driven jet, using four grid spacings:
$\Delta x = 1.5$
, 2, 2.5 and 3
${\unicode[Arial]{x03BC}}$
m. The results for
$\Delta x = 1.5$
and 2
${\unicode[Arial]{x03BC}}$
m are nearly indistinguishable; therefore,
$\Delta x = 2\,{\unicode[Arial]{x03BC}}$
m is adopted in the present study.
Figure 24. Long description
Panel A: A line graph shows the evolution of the bubble-equivalent spherical radius over time. The x-axis represents time in microseconds (μs), ranging from 0 to 100 μs. The y-axis represents the equivalent radius in micrometers (μm), ranging from 0 to 500 μm. Four different grid spacings are represented by different colors: 1.5 μm (blue dashed line), 2 μm (orange solid line), 2.5 μm (green solid line), and 3 μm (red solid line). The inset zooms in on the time range from 90 to 97.5 μs, showing a detailed view of the radius evolution. Panel B: Two line graphs show the bubble interface before and after interfacial-wave convergence at 90 μs and 93 μs. The x-axis represents the X position in micrometers (μm), ranging from -200 to 200 μm. The y-axis represents the Y position in micrometers (μm), ranging from -600 to 100 μm. The same four grid spacings are represented by the same color scheme as in Panel A. The graphs illustrate the shape and position of the bubble interface at these specific times.
Rmax=506.2±10.3μm
γ=1.19±0.02
Ri∗=0.30±0.01
Ro∗=0.50±0.01
Rmax=516.37μm
γ=1.17
Ri∗=0.30
Ro∗=0.49
Rmax=482.4±10.3μm
γ=0.69±0.01
Ri∗=0.31±0.01
Ro∗=0.52±0.01
Rmax=493.62μm
γ=0.70
Ri∗=0.31
Ro∗=0.51
μ
Rmax=516.38μm
Ri∗=0.29
Ro∗=0.48
γ=1.16
Rmax=516.38μm
Ri∗=0.29
Ro∗=0.48
γ=1.16
r=0
μ
Rmax=499.91μm
Ri∗=0.30
Ro∗=0.50
γ=0.8
ωθ
Uy
Rmax=499.91μm
Ri∗=0.30
Ro∗=0.50
γ=0.8
100μm
−0.55
−0.35mm
360μm
−0.6
0.2mm
Rmax=499.91μm
Ri∗=0.30
Ro∗=0.50
γ=0.8
r=0
μ
μ
U
r=0
0
a
b
92.6
c
d
e
(Ro∗,γ)
Ri∗=0.30
Umax
γ=1.08−0.65(Ro∗−0.20)2.4
γ
Ro∗
Ro∗
γ≈0.8
Ri∗=0.30
Ro∗=0.31
Ro∗=0.40
Ro∗=1.31
Ri∗=0.30
γ=0.80
Ro∗
γ≈0.8
Ri∗=0.3
tc
Ro∗
tjet
Ro∗
tjet/tc
γ≈0.8
Ri∗=0.3
Ro∗=0.81
Uimp
Ro∗
γ=0.8
Yimp
Ri∗=0.303
Ri∗
γ≈0.8
Ro∗=0.51
Ri∗=0.40
Ri∗=0.05
Ro∗=0.51
γ=0.81
Uimp
Ri∗
γ=0.6
Yimp
Ro∗=0.51
γ=0.1
Ri∗=0
Ro∗=20.2
tjet∗=0.995
Ri∗=0.51
Ro∗=20.2
tjet∗=0.965
Ri∗=0.51
Ro∗=0.52
tjet∗=0.916
Ri∗
γ=0.1
t=90μ
t=93μ
Δx=1.5
μ
Δx=1.5
μ
Δx=2μ
