2.1. Theoretical model

On the theoretical side, we are considering a spherical, inertial, oscillating bubble located at the centre of a spherical droplet (referred to as fluid 1) that is surrounded by the host liquid (referred to as fluid 2). This is illustrated in figure 2(a). The bubble dynamic equation in our particular context was originally formulated by Raman et al. (Reference Raman, Rosselló, Reese and Ohl2022b). They reached this equation through a modification of a model for the bubble dynamics with a linear elastic shell surrounded by a viscous fluid, as initially described in Church (Reference Church1995). In order to improve clarity and facilitate a more intuitive comprehension of the equation, particularly for researchers who may not have extensive expertise in this field, we offer an alternative derivation based on the Laplace equation and the Bernoulli equation as follows.

Figure 2. Sketch illustrating the physical problem of bubble dynamics within an initially spherical droplet (referred to as fluid 1) surrounded by the host liquid (referred to as fluid 2). (a) The bubble is situated at the centre of the droplet, with $R_b$ and $R_d$ denoting the radii of the bubble and droplet, respectively. The surfaces of the bubble and droplet are represented by $S_b$ and $S_d$, respectively. (b) The bubble is not positioned centrally within the droplet, and the eccentricity between the bubble and the droplet centroids is denoted by $e$.

The bubble and droplet radii are denoted by $R_b$ and $R_d$, respectively. The densities of fluid 1 and fluid 2 are denoted by $\rho _1$ and $\rho _2$, respectively. The flow dynamics in both fluids is governed by the Laplace equation

(2.1)\begin{equation} \nabla^2{\varphi_i}=0\quad (i = 1, 2), \end{equation}

where $\varphi$ represents the velocity potential, and the subscript ‘$i$’ represents the fluid type. The velocity and velocity potential in the entire flow field can be expressed in unified forms, irrespective of the fluid type, as follows:

(2.2a,b)\begin{equation} \boldsymbol{\nabla}{\varphi_i}=\frac{\dot{R_b} R_b^2}{r^2},\quad \varphi_i=-\frac{\dot{R_b}R_b^2}{r}\quad (i = 1, 2), \end{equation}

where the dot presents the time derivative, and $r$ denotes the radial position in the flow field from the centre of the bubble.

Considering the effect of surface tension, we can express the relationship between the gas pressure inside the bubble ($P_g$) and the liquid pressure on the bubble surface ($P_{b1}$) as follows:

(2.3)\begin{equation} P_g=P_{b1}+\frac{2\sigma_1}{R_b},\end{equation}

where $\sigma _1$ is the surface tension coefficient for the bubble surface. Similarly, the relation between the pressures just inside and outside the droplet surface, denoted by $P_{d1}$ and $P_{d2}$ respectively, can be expressed as

(2.4)\begin{equation} P_{d1}=P_{d2}+\frac{2\sigma_2}{R_d},\end{equation}

where $\sigma _2$ presents the surface tension coefficient of the fluid–fluid interface.

The unsteady Bernoulli equation is applicable to both fluids. As a result, we can express the dynamic boundary conditions on the bubble surface and the fluid–fluid interface as follows:

(2.5)$$\begin{gather} {\left.\frac{{\partial\varphi_1}}{{\partial}t}\right|_{r=R_b}} +\frac{1}{2}{\dot{R}_b^2}+ \frac{P_{b1}}{\rho_1}={\left.\frac{{\partial\varphi_1}}{{\partial}t}\right|_{r=R_d}}+ \frac{1}{2}\left(\frac{\dot{R_b}R_b^2}{R_d^2}\right)^2+ \frac{P_{d1}}{\rho_1} \quad {\rm on}\ S_{b}, \end{gather}$$

(2.6)$$\begin{gather}{\left.\frac{{\partial\varphi_2}}{{\partial}t}\right|_{r=R_d}} +\frac{1}{2}\left({\frac{\dot{R_b}R_b^2}{R_d^2}}\right)^2+\frac{P_{d2}}{\rho_2}=\frac{P_\infty}{\rho_2} \quad {\rm on}\ S_{d}, \end{gather}$$

where $P_\infty$ is the hydrostatic pressure at infinity.

The bubble dynamic equation can be obtained from (2.2a,b)–(2.6), written as

(2.7)\begin{align} R_b\ddot{R_b} + \frac{3}{2} \dot{R}_b^2+(1-\alpha)\left[-2\theta \dot{R}_b^2-\theta R_b\ddot{R_b}+\frac{1}{2}\theta^4 \dot{R}_b^2\right] = \frac{P_g-P_\infty}{\rho_1}-\frac{2\sigma_1}{\rho_1R_b}-\frac{2\theta\sigma_2}{\rho_1R_b}, \end{align}

where $\theta =R_b/R_d$ is the bubble-to-droplet size ratio that varies with time and $\alpha =\rho _2/\rho _1$ is the density ratio between the two fluids. This equation is an extension form of Rayleigh–Plesset (RP) equation and one of its derivatives. When $\alpha =0$, (2.7) reduces to the form obtained by Obreschkow et al. (Reference Obreschkow, Kobel, Dorsaz, De Bosset, Nicollier and Farhat2006), which describes the bubble dynamics inside a droplet that is levitated in air (without inertia effects). As $\theta$ approaches 0, the RP equation is recovered. Our approach can be extended to derive equations governing the dynamics of bubbles within more fluid layers under spherically symmetric conditions.

To model and match the experimental data, we employ the adiabatic approximation to compute the gas pressure (Klaseboer & Khoo Reference Klaseboer and Khoo2004b; Zeng, Gonzalez-Avila & Ohl Reference Zeng, Gonzalez-Avila and Ohl2020; Han et al. Reference Han, Zhang, Tan and Li2022; Zhang et al. Reference Zhang, Li, Cui, Li and Liu2023b), given by

(2.8)\begin{equation} P_g=P_{0}\left(\frac{V_{0}}{V}\right)^{\lambda},\end{equation}

where $V$ denotes the bubble volume, $P_0$ the initial bubble pressure, $\lambda = 1.4$ the ratio of the specific heats and the subscript ‘0’ represents initial quantities. This simplicity can be justified by the large associated Péclet number ${\sim }O(10^3)$ of the bubbles in our experiments.

The viscosity is neglected here but can be incorporated through the conditions on the normal stresses at the bubble wall and the droplet surface. To estimate the viscous effect, we define the Reynolds number for bubble dynamics as $Re=\rho _1 R_{b,max}U/{\mu _1}$, where the characteristic velocity is taken as $U=\sqrt {P_\infty /\rho _1}$, $\mu _1$ is the viscosity of fluid 1, and $R_{b, max}$ is the maximum radius of the bubble. For submillimetre-sized laser-induced bubbles generated in a viscous oil droplet, the Reynolds number can be estimated as $Re\sim 10^2$. The deviation of the bubble oscillation period between the results obtained from (2.7) and the equation with viscous terms (Raman et al. Reference Raman, Rosselló, Reese and Ohl2022b) is within 2 %. For bubbles generated in a W/O droplet, the variation of the bubble oscillation period due to viscous effects is within 0.2 %. In applications such as ultrasonic emulsification, where the maximum acoustic bubble radius typically falls within the range of 30 to $150\ \mathrm {\mu }$m (Wu et al. Reference Wu, Eskin, Priyadarshi, Subroto, Tzanakis and Zhai2021; Yamamoto et al. Reference Yamamoto, Matsutaka and Komarov2021; Udepurkar et al. Reference Udepurkar, Clasen and Kuhn2023), it is expected that viscosity would have a more pronounced influence. However, the experimental data with adequate spatio-temporal resolution for such small bubbles is currently lacking. As a result, our investigation is confined to millimetre-sized bubbles, and the examination of viscous effects on micron-sized bubbles is beyond the scope of this study. Interested readers can refer to the relevant literature (Popinet & Zaleski Reference Popinet and Zaleski2002; Minsier, De Wilde & Proost Reference Minsier, De Wilde and Proost2009; Zeng et al. Reference Zeng, Gonzalez-Avila, Voorde and Ohl2018; Kannan et al. Reference Kannan, Balusamy, Karri and Sahu2020; Wang et al. Reference Wang, Liu, Corbett and Smith2022) for more discussion.

Next, we derive a modified Rayleigh collapse time that describes the duration from the bubble's maximum radius $R_{b,max}$ to the point where the bubble is completely filled up under a constant pressure difference $\Delta P = P_\infty - P_g$ (Rayleigh Reference Rayleigh1917). Surface tension is neglected here for the sake of clarity. The energy conservation equation can be solved for $\dot {R_b}$ and is expressed as

(2.9)\begin{equation} \dot{R_b}=-\sqrt{\frac{2\Delta P(R^{\prime-3}-1)}{3\rho_1 \left[1+\dfrac{(\alpha-1)}{\sqrt[3]{\xi^{-3}R^{\prime-3}+1}} \right] }},\end{equation}

where $R^{\prime }=R_b/R_{b, max}$ and $\xi =R_{b, max}/R_{d, min}$. In this context, we find it convenient to use a fixed value of the bubble-to-droplet size ratio $\xi$ for a given case.

Integrating this equation from $R_b=R_{b, max}$ to $R_b=0$ yields an analytical expression for the bubble collapse time

(2.10)\begin{equation} T_c=\eta(\alpha, \xi) \cdot R_{b, max}\sqrt{\frac{\rho_1}{\Delta P}}, \end{equation}

with

(2.11)\begin{equation} \eta(\alpha, \xi)=\sqrt{\frac{1}{6}}\int_{0}^{1} \sqrt{1+\frac{\alpha-1}{(\xi^{-3}s^{-1}+1)^{1/3}}}\cdot \frac{{\rm d}s}{s^{1/6}(1-s)^{1/2}}, \end{equation}

where $s$ substitutes $R^{\prime 3}$. Note that the modified Rayleigh factor $\eta$ is influenced by both density ratio $\alpha$ and size ratio $\xi$. If $\alpha =1$ or $\xi \to 0$, (2.11) reduces to the classic Rayleigh factor $\eta \approx 0.9147$. More quantitative discussion about the influence of $\alpha$ and $\xi$ on bubble collapse time will be given in § 4.2.

Finally, if the bubble oscillates with a low amplitude, the bubble's natural frequency can be obtained in a similar manner as Minnaert (Reference Minnaert1933), given by

(2.12)\begin{equation} f(\alpha, \theta_e)=\frac{1}{2{\rm \pi} R_e}\sqrt{\frac{3\lambda \left(P_{\infty}+\dfrac{2\sigma_1}{R_e}+\dfrac{2\sigma_2\theta_e}{R_e} \right)- \dfrac{2\sigma_1}{R_e}-\dfrac{2\sigma_2\theta_e^4}{R_e} }{\rho_1 [1+(\alpha-1)\theta_e] }}, \end{equation}

where $R_e$ represents the equilibrium radius of the bubble and $\theta _e=R_e/R_d$. This formula would be useful for the community of acoustic bubbles.

2.2. Boundary integral method

When the bubble is not centred in the droplet, as shown in figure 2(b), non-spherical oscillations and jetting behaviours of the bubble can be expected. Therefore, we utilize a well-verified boundary integral (BI) method (Li et al. Reference Li, van der Meer, Zhang, Prosperetti and Lohse2020a; Yi et al. Reference Yi, Li, Jiang, Lohse, Sun and Mathai2021; Han et al. Reference Han, Zhang, Tan and Li2022; Li, Zhang & Han Reference Li, Zhang and Han2023) to investigate the dependence of the non-spherical bubble dynamics inside a droplet on the governing parameters. Here, we provide a brief overview of the BI method. We use a cylindrical coordinate system $(r,\theta, z)$, with the origin $O$ located at the centre of the bubble and the positive $z$-axis pointing from the initial droplet centre towards the bubble centre. Through the application of Green's second theorem, the Laplace equation (2.1) can be transformed into a BI equation, expressed as follows:

(2.13)\begin{equation} c(\boldsymbol{{r}})\varphi_i(\boldsymbol{{r}})=\iint_S \left(\frac{\partial \varphi_i(\boldsymbol{{q}})}{\partial\it n} \frac{1}{|\boldsymbol{{r}}-\boldsymbol{{q}}|}-\varphi_i (\boldsymbol{{q}})\frac{\partial}{\partial n} \left(\frac{1}{|\boldsymbol{{r}}-\boldsymbol{{q}}|}\right)\right){\rm d} S_i(\boldsymbol{{q}}),\quad (i=1,2), \end{equation}

where $c$ is the solid angle, $\boldsymbol{{r}}$ and $\boldsymbol{{q}}$ stand for the control and source points, respectively, $\partial /\partial n$ denotes the normal derivative and $S$ refers to the droplet surface and the bubble surface when $i = 1$ (flow domain 1), while refers to the droplet interface only when $i = 2$ (flow domain 2).

The dynamic boundary conditions on the bubble surface and the droplet interface are given by

(2.14)$$\begin{gather} \frac{{\rm D}{\varphi_1}}{{\rm D}t}=\frac{P_{\infty}}{\rho_1}- \frac{P_{g}}{\rho_1}+\frac{1}{2}{|{\nabla \varphi_1}|^2+ \frac{\sigma_1 \kappa}{\rho_1}}\quad {\rm on}\ S_{b}, \end{gather}$$

(2.15)$$\begin{gather}\frac{{\rm D}{(\varphi_1-\alpha\varphi_2)}}{{\rm D}t}= \frac{1}{2}{|{\nabla \varphi_1}|^2}+\frac{\alpha}{2}{|{\nabla \varphi_2}|^2}-\alpha\boldsymbol{\nabla}\varphi_1 \boldsymbol{\cdot}\boldsymbol{\nabla}\varphi_2-\frac{\sigma_2 \kappa}{\rho_1} \quad {\rm on}\ S_{d}, \end{gather}$$

where $\kappa$ denotes the curvature.

The kinematic boundary condition on both $S_b$ and $S_d$ is expressed as

(2.16)\begin{equation} \frac{{\rm D}{\boldsymbol{{r}}}}{{\rm D}t}=\boldsymbol{\nabla}\varphi_1.\end{equation}

The BI equations (2.13) are solved for the velocities on bubble and droplet surfaces. Then, we update the velocity potential using (2.14)–(2.15) and the position of the surfaces using (2.16). All simulations are conducted in a non-dimensional form, where the maximum radius of the bubble $R_{b, max}$, hydrostatic pressure $P_\infty$ and density of the droplet $\rho _1$ serve as the three fundamental quantities. Four non-dimensional variables of the system are given below

(2.17a–d)\begin{equation} \alpha=\frac{\rho_2}{\rho_1},\quad \frac{1}{\xi}=\frac{R_{d, min}}{R_{b, max}},\quad \gamma=\frac{R_{d, min}-e}{R_{b, max}},\quad \varepsilon=\frac{P_0}{P_\infty}, \end{equation}

where the density ratio $\alpha$ and size ratio $\xi$ have the same definition as in § 2.1, $\gamma$ is the standoff parameter that measures the non-dimensional distance from the bubble centre to the nearest droplet surface ($e$ is the eccentricity), $\varepsilon$ is the strength parameter that describes the initial bubble pressure. As for the toroidal bubble dynamics, we use a vortex ring model (Wang et al. Reference Wang, Yeo, Khoo and Lam1996; Curtiss et al. Reference Curtiss, Leppinen, Wang and Blake2013; Zhang, Li & Cui Reference Zhang, Li and Cui2015; Han et al. Reference Han, Zhang, Yang, Han and Li2023) coupled with the BI method to continue the simulation after jet impact.

2.3. Experimental set-up

As sketched in figure 3, experiments were conducted to investigate the dynamics of cavitation bubbles within a water droplet (W/O system) in a tank measuring $100\ {\rm mm}\times 100\ {\rm mm}\times 100\ {\rm mm}$ at room temperature (${\sim }25\,^{\circ }$C) and atmospheric pressure ($\sim$97.1 kPa). The tank was initially filled with sunflower oil (50.6 centistokes) to a depth of 90 mm, with a density of 914 kg m$^{-3}$. To achieve precise control over the volume and position of millimetre-sized liquid droplets in the oil bulk, we installed an injection syringe fitted with an ultrafine flat-tipped needle (inner diameter 0.08 mm, outer diameter 0.2 mm) on a three-axis mobile platform (precision 0.02 mm). This set-up allowed us to generate a millimetre-scale droplet hanging at the needle opening. The associated Bond number, defined as $Bo=(\rho _1-\rho _2)gR_{d, min}^2/\sigma _2$, can be estimated to be $O(10^{-2})$. The surface tension coefficient of the water–oil interface $\sigma _2$ is about 0.029 N m$^{-1}$. The difference between the horizontal radius $R_r$ and the vertical radius $R_z$ of the droplet is within 4 %. Therefore, we can assume the droplet is initially spherical. In experiments involving deionized water as the host fluid and sunflower oil as the droplet (O/W system), the needle opening was positioned upward, and the droplet was supported by the needle. While one may concern about the needle's influence on the transient bubble–droplet interaction, the experimental results demonstrated that the impact of the needle on bubble oscillation and jetting behaviours was minimal due to large difference in size.

Figure 3. Experimental set-up for cavitation bubble dynamics inside a droplet that surrounded by a different host fluid. A submillimetre-scale cavitation bubble is generated by focusing a pulsed laser inside a droplet that is hanging or held by a thin flat-tipped needle.

The generation of submillimetre-scale cavitation bubbles was achieved using a frequency-doubled Nd:YAG laser (Nimma-900, pulse duration 8 ns, wavelength 532 nm, pulse energy $16\sim 32$ mJ). A microscope objective lens (M Plan Apo L 10$\times$, numerical aperture ${\rm NA} = 0.28$) focused the parallel pulsed laser beam inside the droplet. A 30 mm diameter hole was created in the sidewall of the tank, with a 0.25 mm thick sapphire glass embedded to minimize the effects of refraction, thereby enhancing laser focusing. The energy at the focal point exceeded the breakdown threshold of the liquid medium, resulting in transient ‘avalanche’ ionization, forming a high-temperature and high-pressure plasma cavity. Subsequently, it rapidly expanded into a cavitation bubble with a maximum diameter of approximately 1 mm.

For uniform background illumination, a continuous LED light source (300 W) filtered through matt glass was utilized. To capture both the transient bubble behaviours and the droplet evolutions, a high-speed camera (Phantom V2012) equipped with a macro lens (LAOWA, 100 mm, F2.8) was triggered simultaneously with the laser. The camera recorded the phenomena at a resolution of $256\times 128$ pixels, with 340 000 frames per second and an exposure time of $1\ \mathrm {\mu }$s. To measure the bubble pulsation period and micro-jet velocity more precisely, we only look at the area near the equator of the bubble-droplet system, enabling us to record at a frame rate of up to 656 000. Due to the droplet's optical-lens-like effect (Gonzalez Avila & Ohl Reference Gonzalez Avila and Ohl2016), a direct measurement of the bubble size is questionable. Instead, we calculate the bubble radius based on the changes in droplet volume. We notice that for the cases where bubbles are located at the centre of O/W and W/O droplets, the actual sizes of the bubble are 0.94–0.97 and 1.13–1.21 times those observed through the high-speed images, respectively.

3.1. Bubble initiation in an oil-in-water droplet

Figure 4 illustrates three representative experiments in which bubbles were initiated within O/W droplets ($\alpha =1.093$). The bubble-to-droplet size ratios of the three experiments are approximately the same ($\xi \approx 0.6$), and the bubble dynamic behaviour exhibits significant variations with variations of the standoff parameter (or the eccentricity). In the first experiment shown in (a), the cavitation bubble originates at the centre of the oil droplet, and frame 1 shows the expansion of the bubble. Frames 2, 7 and 9 capture the moments when the bubble reaches its maximum size during the first, second and third oscillation cycles, respectively. Furthermore, frames 4 and 8 depict the instances when the bubble reaches its minimum volume at the end of the first and second cycles, respectively. Owing to the initial spherical symmetry condition, both the bubble and droplet surfaces maintain nearly spherical shapes throughout the entire process.

Figure 4. Three representative experiments in which bubbles were initiated within O/W droplets ($\alpha =1.093$). (a) The bubble was initiated at the centre of the droplet ($e = 0$, $R_{d, min } = 0.99$ mm, $R_{b, max } = 0.56$ mm, $\xi = 0.57$, $\gamma = 1.77$), maintaining a spherical shape throughout the oscillation process. (b) The bubble was initiated off centre of the droplet ($e = 0.39$ mm, $R_{d, min } = 0.93$ mm, $R_{b, max } = 0.56$ mm, $\xi = 0.61$, $\gamma = 0.96$), and during the rebound stage, a liquid jet forms. (c) The bubble was initiated off centre of the droplet ($e = 0.63$ mm, $R_{d, min } = 0.95$ mm, $R_{b, max } = 0.56$ mm, $\xi = 0.60$, $\gamma = 0.57$). After reaching its minimum volume, the bubble produces a rapid ‘needle-like’ jet, piercing through the droplet surface. During the third cycle, the main body of the bubble escapes the droplet and enters the water bulk. Non-dimensional times are indicated in the lower-left corners of each frame. The time scale for non-dimensionalization ($R_{b, max}\sqrt {\rho _1/P_\infty }$) in these cases is $54.23\ \mathrm {\mu }$s. The horizontal width of each frame is 4.2 mm. For reader's convenience, black, red and blue boxes represent the first, second and third cycles of bubble oscillation, respectively.

In the second experiment shown in figure 4(b), the cavitation bubble nucleates off centre and in proximity to the left interface of the droplet ($\gamma = 0.96$). Throughout the first oscillation cycle (frames 1–4), the bubble maintains a nearly spherical shape. Notably, a high-speed liquid jet forms around the end of the first cycle of the bubble (frame 4). However, this jet fails to penetrate the droplet surface (frames 5–6). Subsequently, during the rebound stage of the third cycle, we observe the second jet that forms around the end of the second cycle finally penetrates the droplet surface (frame 9). This effectively transports oil droplets into the surrounding water bulk. The black residual produced after the jet penetration in frame 9 might be a mixture of bubble gas and oil, with an approximate diameter of $50\ \mathrm {\mu }$m (with an uncertainty of $25\ \mathrm {\mu }$m), which is approximately 0.03 times the droplet size.

In the third experiment shown in figure 4(c), the cavitation bubble is initiated closer to the left surface of the droplet ($\gamma = 0.57$). The left side of the bubble surface appears slightly flattened due to the interaction with the left surface of the droplet (frames 1–2), resulting from the higher inertia of water. Subsequently, after reaching its minimum volume, the bubble generates a rapid ‘needle-like jet, which pierces through the droplet surface (frames 5–6). At the same time, the bubble migrates towards the left side. During the third cycle, the main body of the bubble separates from the droplet and enters the water bulk (frame 9). As expected, this scenario promotes stronger fluid mixing compared with the second experiment.

3.2. Bubble initiation in a water-in-oil droplet

In this section, we present three representative experiments in which bubbles were initiated within W/O droplets ($\alpha =0.915$). In the first case, shown in figure 5(a), the cavitation bubble originates at the centre of the water droplet and oscillates in a spherical shape. However, the rebound bubble exhibits slightly unstable features (frame 4) compared with that in figure 4(a). This discrepancy arises due to the large viscosity of the oil medium, which acts as a stabilizing factor (Prosperetti Reference Prosperetti1977; Zeng et al. Reference Zeng, Gonzalez-Avila, Voorde and Ohl2018) for bubble oscillations and is 50 times greater than that of water. A similar phenomena can be found in Kannan et al. (Reference Kannan, Balusamy, Karri and Sahu2020), in which the dependence of the bubble dynamics on viscosity of the flow field (single phase) was experimentally studied. Back to figure 5(a), the non-dimensional oscillation period of the bubble decreases in comparison with that in figure 4(a). Further discussion on the influence of droplet confinement on the collapse time of the bubble will be given in § 4.2.

Figure 5. Three representative experiments in which bubbles were initiated in W/O droplets ($\alpha =0.915$). (a) the bubble was generated at the droplet's centre ($e = 0$, $R_{d, min} = 1.14$ mm, $R_{b, max } = 0.78$ mm, $\xi = 0.69$, $\gamma = 1.46$). (b) The bubble was initiated off centre from the droplet's centre ($e = 0.47$ mm, $R_{d, min } = 1.12$ mm, $R_{b, max } = 0.77$ mm, $\xi = 0.69$, $\gamma = 0.84$), leading to the formation of a weak jet directed towards the droplet's centre. (c) The bubble was initiated off centre from the droplet ($e = 0.80$ mm, $R_{d, min } = 1.14$ mm, $R_{b, max } = 0.77$ mm, $\xi = 0.68$, $\gamma = 0.44$), resulting in a tiny water-hump-like protrusion on the left side of the droplet. The residual bubble vortex continues moving to the right, causing the pinch-off of the droplet. Non-dimensional times are indicated in the lower-left corners of each frame. The time scales for non-dimensionalization ($R_{b, max}\sqrt {\rho _1/P_\infty }$) are 78.99, 77.97 and $77.97\ \mathrm {\mu }$s, respectively. The horizontal width of each frame is 4.6 mm. For reader's convenience, black, red and blue boxes represent the first, second and third cycles of bubble oscillation, respectively, while green boxes represent the evolution of the droplet after multiple bubble cycles.

In the second case, as depicted in figure 5(b), the bubble is generated off centre within the droplet. During the rebound phase (frame 4), a weak jet emerges, directed towards the centre of the droplet. Subsequently, the vortex ring bubble undergoes a migratory path from the left side to the right side of the droplet (frames 5–8). The bubble pushes the right side of the droplet surface, inducing a noticeable bulge. Nonetheless, no pinch-off of the droplet can be observed in this case. Under the influence of surface tension, the droplet progressively regains its spherical shape (not shown here).

In the third case, illustrated in figure 5(c), the bubble forms in closer proximity to the droplet surface, resulting in a more pronounced bubble–droplet interaction. Consequently, a stronger jet forms and the bubble obtains a faster migration velocity, resulting in a breakup of the protrusion on the right side of the droplet, as evidenced in frame 8. This phenomenon could potentially constitute a secondary mechanism for fluid mixing within the current W/O system. Additionally, we notice a tiny water-hump-like protrusion on the left side of the droplet as the bubble migrates away (frames 6–7). This phenomenon is much more pronounced when the water–oil interface is initially flat (Han et al. Reference Han, Zhang, Tan and Li2022). Even for a much smaller standoff parameter case in the present bubble–droplet system, the protrusion on the left side of the droplet hardly develops to a large size and consequently no pinch-off can be found (refer to Appendix A). The significant difference between the present system from Han et al. (Reference Han, Zhang, Tan and Li2022) is the curvature of the water–oil interface. The dependence of the bubble–droplet interaction on the curvature parameter $\xi$ will be presented in § 5.2.

In our current experimental set-up, we have noticed that the phenomenon of jet piercing through the bubble surface is typically most prominent during the rebound phase. This observation closely resembles what has previously been characterized as the ‘weak-jet’ phenomenon, as documented in Supponen et al. (Reference Supponen, Obreschkow, Tinguely, Kobel, Dorsaz and Farhat2016). By applying the concept of the anisotropy parameter $\zeta =0.195 \gamma ^{-2}(\rho _{1}-\rho _{2})(\rho _{1}+\rho _{2})^{-1}$, as introduced by Supponen et al. (Reference Supponen, Obreschkow, Tinguely, Kobel, Dorsaz and Farhat2016), for a rough estimation of jet characteristics, we calculated $\zeta$ values of 0.012 and 0.045 for the second and third experiments in figure 5, respectively. While these $\zeta$ values place them within the ‘intermediate-jet’ category, our observations more closely align with the ‘weak-jet’ phenomenon. This discrepancy can be primarily attributed to the presence of non-flat boundaries within our experimental system, whereas the anisotropy parameter was originally derived under the assumption of flat boundaries.

4.1. Comparison of experiments with theoretical results

In this section, we discuss the dynamics of a spherical bubble located at the centre of a spherical droplet. First, we try to reproduce the spherical bubble dynamics in the experiments using the theoretical model given in § 2.1. In order to facilitate a meaningful comparison with the experimental results, it is imperative to define appropriate initial conditions for the calculations, namely $R_0$ and $\varepsilon$. We use a simplified method (Zeng et al. Reference Zeng, Gonzalez-Avila, Voorde and Ohl2018; Saade et al. Reference Saade, Jalaal, Prosperetti and Lohse2021; Zeng et al. Reference Zeng, An and Ohl2022) to initialize the bubble, setting it as an initially stationary high-pressure gas bubble that corresponds to the moment when the bubble becomes visible in the experiment. We need to adjust only the strength parameter $\varepsilon$ to match the experimental data, while the initial bubble radius is calculated using the principle of energy conservation (Klaseboer & Khoo Reference Klaseboer and Khoo2004b; Han et al. Reference Han, Zhang, Tan and Li2022). A satisfactory result can be obtained if $\varepsilon$ is set as 2000. More detailed discussion on the sensitivity of the results on $\varepsilon$ can be found in Appendix B.

Figure 6 displays the time evolutions of bubble and droplet radii for the two experiments shown in figures 4(a) and 5(a). As can be seen, our theoretical model accurately predicts the bubble and droplet dynamics. Note that the non-dimensional bubble oscillation period deviates a lot between the two experiments. This is mainly attributed to the difference in the density ratio $\alpha$. We will discuss the influences of the density ratio $\alpha$ and size ratio $\xi$ on the bubble collapse time in § 4.2. Although the numerical results obtained from the BI method are not shown here (for the sake of image clarity), we want to emphasize that the difference between the results obtained from BI simulations and the theoretical model is indistinguishable.

Figure 6. Time evolutions of bubble and droplet radii obtained from our experiments (denoted by the circles and rectangles with error bar) and the theoretical model (denoted by the solid lines). In the first experiment, the bubble was initiated at the centre of an O/W droplet, with the following parameters: $R_{d, min } = 0.99$ mm, $R_{b, max } = 0.56$ mm, $\xi = 0.57$, $\gamma = 1.77$. In the second experiment, the bubble was initiated at the centre of a W/O droplet, with the following parameters: $R_{d, min } = 1.14$ mm, $R_{b, max } = 0.78$ mm, $\xi = 0.69$, $\gamma = 1.46$. The time scales for non-dimensionalization ($R_{b, max}\sqrt {\rho _1/P_\infty }$) of the two experiments are 54.23 and $78.99\ \mathrm {\mu }$s, respectively. The non-dimensional initial gas pressure and radius of the bubble in computations are set as $\varepsilon =2000$ and $R_0=0.0591$.

4.2. Modified Rayleigh collapse time

Rayleigh (Reference Rayleigh1917) first derived the analytical expression for the collapse time of a vacuum bubble in an infinite medium, namely, $T_c=\eta R_{b, max}\sqrt {{\rho _1}/{\Delta P}}$, where the Rayleigh factor $\eta$ is approximately 0.9147. Obreschkow et al. (Reference Obreschkow, Kobel, Dorsaz, De Bosset, Nicollier and Farhat2006) revealed a remarkable reduction in the collapse time for a bubble collapsing inside a droplet surrounded by air. In this study, we derive a modified Rayleigh collapse time for bubbles within a droplet, which is characterized by the density ratio $\alpha$ and bubble-to-droplet size ratio $\xi$.

To start, we compare the experimental results with the theoretical predictions. In our experimental trials, we can adjust the bubble-to-droplet size ratio in the range of $0.4 < \xi < 2$. The Rayleigh collapse time in our experiments, denoted as $T_c$, is defined as half of the first oscillation period of the bubble. The Rayleigh factor $\eta$ is calculated using the formula $T_c/( R_{b, max} \sqrt {{\rho _1}/{P_{\infty }}})$. For the purpose of comparison, we analyse the experimental data alongside both the analytical estimation and the results obtained from the extended RP (ERP) equation. These computations encompass scenarios in which bubbles are initiated within both W/O and O/W droplets, and the results are depicted in figure 7. Similar to the results illustrated in figure 6, the strength parameter is consistently set at $\varepsilon =2000$ for calculations utilizing ERP. In cases where bubble initiation occurs within W/O droplets ($\alpha =0.915$), it is evident that the Rayleigh factor $\eta$ exhibits a decreasing trend with increasing $\xi$. It is important to note that the analytical estimation provides a lower limit due to the assumption that the bubble's interior is a vacuum. Notably, the outcomes obtained through the ERP align well with the experimental data. Conversely, for bubbles initiated within O/W droplets ($\alpha =1.093$), there is a noticeable increase in the Rayleigh factor $\eta$ as $\xi$ increases.

Figure 7. Variation of the Rayleigh factor $\eta$ over the size ratio $\xi$. The upper and lower parts of the figure represent two different situations of bubble generation within O/W droplets ($\alpha = 1.093$) and W/O droplets ($\alpha = 0.915$), respectively. The orange circles represent the experimental data, the blue solid lines represent the results obtained from ERP equation ($\varepsilon = 2000$) and the green solid lines denote the analytical solution (calculated from (2.11)).

Next, we evaluate the modified Rayleigh factor over an extended parameter space, specifically $10^{-4}\le \alpha \le 10$ and $10^{-3}\le \xi \le 10^2$. As shown in figure 8(a), the contour represents the value of $\eta$ in the $\xi$–$\alpha$ space. This plot can be interpreted in several ways. First, when $\alpha =1$, the classic Rayleigh collapse factor is obtained, as indicated by the upper horizontal line $\eta =0.9147$. Above this line ($\alpha >1$), $\eta$ increases with $\alpha$, particularly for larger values of $\xi$. This is because the heavier outer fluid (fluid 2) is more difficult to accelerate, resulting in an increase in the bubble collapse time. Notably, the influence of $\alpha$ becomes increasingly pronounced as $\xi$ rises, corresponding to a reduction in droplet size. Second, for very small size ratios $\xi <10^{-2}$, $\eta$ is almost independent of $\alpha$, implying that the outer fluid phase has little effect on the bubble if the droplet is much larger than the bubble. Third, the Rayleigh factor decreases with decreasing $\alpha$ and increasing $\xi$, corresponding to a reduction in the mass of fluid that needs to be accelerated by the bubble.

Figure 8. Variation of the Rayleigh factor $\eta$ in the $\xi$–$\alpha$ space. In (a), the white dashed lines denote isolines of $\eta = 0.1$, 0.2, 0.4, 0.6, 0.8, 0.9 and 0.9147 (the classic Rayleigh factor), respectively. In (b), the case where $\alpha =0$ corresponds to the result obtained by Obreschkow et al. (Reference Obreschkow, Kobel, Dorsaz, De Bosset, Nicollier and Farhat2006). They considered a constant-pressure boundary condition on the droplet surface but neglected the influence of air flow.

Figure 8(b) depicts the variation of $\eta$ with respect to $\xi$ for different $\alpha$. When $\alpha <1$, we observe that $\eta$ decreases slowly in two different parameter regimes, i.e. $\xi \lesssim 10^{-1}$ and $\xi \gtrsim 10$, while it decreases rapidly in the range of $10^{-1}\lesssim \xi \lesssim 10$. Thus, if there is a large size difference between the bubble and droplet, the bubble dynamics is mainly dominated by the inertia of one fluid. However, when the bubble and droplet have comparable size, the Rayleigh factor is more sensitive to the variation of $\xi$. Additionally, the criterion for neglecting the effect of the outer phase on bubble dynamics is worth discussing. The case of $\alpha =0$ (the lowest blue line) corresponds to the result given by Obreschkow et al. (Reference Obreschkow, Kobel, Dorsaz, De Bosset, Nicollier and Farhat2006). For comparison, let us take the case of $\alpha =10^{-3}$ (which corresponds to air and is denoted by the yellow line). The results for the $\alpha =0$ and $\alpha =10^{-3}$ cases are almost identical when $\xi \lesssim 1$, and deviations can be observed only when $\xi >10$. This suggests that the effect of air can be neglected for small bubbles in a large droplet.

5.1. Comparison between experimental and BI simulation results

In figure 9, we present a comparison between experimental observations and BI simulations regarding bubble and droplet profiles. Each frame is divided into two halves: the left side displays the experimental observations, while the right side shows the simulation results. The bubble and droplet profiles are depicted using solid red and blue lines, respectively. In figure 9(a), a bubble is generated inside an O/W droplet, characterized by the following parameters: $R_r=0.96$ mm, $R_z = 0.95$ mm, $R_{b, max}=0.53$ mm, $e = 0.61$ mm, $\gamma =0.64$ and $\alpha =1.093$. In frame 1, as the bubble grows to its maximum size in an almost spherical shape, no interfacial instability is observed on the droplet surface. During the subsequent collapse stages (frames 2–3), the bubble retains its nearly spherical shape due to the minimal density difference between the two liquids. Simultaneously, the droplet gradually regains its spherical form. In the final collapse phase, between frames 3 and 4, the bubble generates a thin liquid jet that propels towards the upper interface of the droplet. By frame 5, the jet tip ascends and ultimately impacts the droplet interface, transporting oil into the surrounding water bulk. It is worth noting that the times of the BI simulation for frames 4–5 in (a) are 1.93 and 1.99, respectively, while the times for frames 1–3 align with those of the experimental observations. The slight deviation between the simulation and experiment during the rebounding stage of the bubble can be attributed to the omission of energy loss in the current BI model. Nonetheless, our numerical model effectively captures the fundamental physics underlying the interaction between the bubble and droplet, particularly the critical process of the jet impacting the droplet surface. Further quantitative comparisons of jet velocity and maximum impact velocity on the droplet surface will be presented in § 6.1.

Figure 9. Comparison between experimental observations (left-hand half of each frame) and BI simulations (right-hand half of each frame) for a jetting bubble inside a droplet. (a) An O/W droplet with the following parameters: $R_r=0.96\ {\rm mm},\ R_z=0.95\ {\rm mm}$, $e = 0.61$ mm, $R_{b, max} = 0.53$ mm and $\alpha =1.093$. The width of each frame is 1.10 mm, and the time scale for non-dimensionalization is $51.33\ \mathrm {\mu }$s. (b) A W/O droplet with the following parameters: $R_r=1.25\ {\rm mm},\ R_z=1.20\ {\rm mm}$, $e = 0.92$ mm, $R_{b, max} = 0.75$ mm and $\alpha =0.915$. The width of each frame is 1.50 mm, and the time scale for non-dimensionalization is $75.95\ \mathrm {\mu }$s. Non-dimensional times are indicated in the lower right corner. The times of the BI simulation for frames 4–5 in (a) are 1.93 and 1.99, respectively, while the times for other frames match those of the experiment. Due to light refraction, the actual sizes of the bubbles in these two experiments are 0.97 and 1.15 times the sizes observed in the high-speed images, respectively. For the convenience of comparison, all the experimental images are rotated $90^\circ$ clockwise.

In figure 9(b), we examine the generation of a bubble in close proximity to the interface of a W/O droplet, characterized by the following parameters: $R_r=1.25$ mm, $R_z = 1.20$ mm, $R_{b, max}=0.75$ mm, $e = 0.92$ mm, $\gamma =0.37$ and $\alpha =0.915$. As the bubble grows to its maximum size, a significant curvature protrusion emerges on the upper surface of the water droplet (frame 1). This phenomenon arises because the fluid's inertia around the upper surface of the bubble is smaller compared with other directions. Notably, we observe a thin water film separating the bubble from the oil, which poses a numerical challenge. To ensure precise simulation, the element size on both surfaces must be smaller than the liquid layer's thickness. To tackle this issue, we employed 500 elements on both the bubble and droplet surfaces, ensuring accuracy and stability in the simulation. Throughout most of the collapse stage (frames 2–3), the bubble maintains an approximately spherical shape. Subsequently, a jet forms toward the centre of the droplet at approximately the moment when the bubble reaches its minimum volume (frame 4). Thereafter, the toroidal bubble rapidly rebounds and migrates downwards (frame 5). In this case, due to light refraction, the actual size of the bubble is approximately 1.15 times the size observed in the high-speed images. As evident, our BI model also reproduces the primary features of this experiment.

5.2. Dependence of bubble dynamics on the droplet curvature $\xi$

Previous research had examined the interaction between a cavitation bubble and a planar fluid–fluid interface (Chahine & Bovis Reference Chahine and Bovis1980; Klaseboer & Khoo Reference Klaseboer and Khoo2004a; Orthaber et al. Reference Orthaber, Zevnik, Petkovšek and Dular2020; Han et al. Reference Han, Zhang, Tan and Li2022). For the present bubble–droplet system, the curvature of the droplet surface is an important parameter. The impact of water–air interface curvature on bubble dynamics has been emphasized (Obreschkow et al. Reference Obreschkow, Kobel, Dorsaz, De Bosset, Nicollier and Farhat2006), and therefore we aim to examine the relationship between bubble dynamics and the bubble-to-droplet size ratio $\xi$ (denotes half of the droplet curvature) via a series of BI simulations. The simulation results discussed below encompass a broad parameter range: $10^{-3}\le \xi \le 1.5$, with $\gamma =0.6$ being the fixed standoff parameter. We will further explore the effect of $\gamma$ in § 5.3.

Figure 10(a) shows the overall interaction between the bubble and droplet for a typical value of $\xi =0.3$ and two different density ratios, $\alpha =0.8$ (a) and $\alpha =1.2$ (b). When the density of the host fluid is smaller than the droplet density ($\alpha =0.8$), the bubble is repealed by the nearby droplet surface, resulting in a high-speed liquid jet directed away from the interface during the collapse phase. Conversely, when the density of the host fluid is larger than the droplet ($\alpha =1.2$), the jet moves towards the droplet surface. Notably, the latter scenario, with a larger $\alpha$, contributes more to fluid mixing from the perspective of jet penetration. To evaluate the asymmetry of the collapse, we introduce the concept of Kelvin impulse (expressed in non-dimensional form, with the density term omitted here)

(5.1)\begin{equation} \boldsymbol{{I}}=\int_{S_{b}} \varphi_1 \boldsymbol{{n}}\,{\rm d} S. \end{equation}

Figure 10. Quantitative study on non-spherical bubble dynamics inside a droplet for different $\alpha$ and $\xi$. (a) Bubble collapse patterns inside a droplet with $\alpha =0.8$ (on the left half) and $\alpha =1.2$ (on the right half). The standoff parameter is fixed at $\gamma =0.6$ and the size ratio is $\xi =0.3$. (b) Variations of the Kelvin impulse of the bubble $|I_z|$ at the jet impact moment vs $\xi$. Note that $|I_z|$ or $I_z/A$ is maximized at $\xi =0.3$, independent of $\alpha$. The inset shows a normalized $I_z$ by the Atwood number $A=(\alpha -1)/(\alpha +1)$.

Since we limited ourselves to the axisymmetric configuration, only the vertical component of the Kelvin impulse $I_z$ is considered in this context. Figure 10(b) shows the variation of $|I_z|$ over $\xi$ for different $\alpha$. As $\alpha$ approaches 1, which indicates a reduction in the asymmetry of the bubble collapse, the value of $|I_z|$ diminishes. Previous studies (Blake, Leppinen & Wang Reference Blake, Leppinen and Wang2015; Supponen et al. Reference Supponen, Obreschkow, Tinguely, Kobel, Dorsaz and Farhat2016; Han et al. Reference Han, Zhang, Tan and Li2022) have shown that the non-dimensional Kelvin impulse of a bubble near a flat fluid–fluid interface scales with the Atwood number $A=(\alpha -1)/(\alpha +1)$. This can be expressed as follows:

(5.2)\begin{equation} I_z\propto \frac{\alpha-1}{\alpha+1}. \end{equation}

As shown in the inset of figure 10(b), we are surprised that the data nearly collapse together when we normalized $I_z$ with the Atwood number. This suggests that the scaling of (5.2) still holds well for the current curved fluid–fluid interface in a large parameter space.

Next, we examine how bubble dynamics is affected by $\xi$. As $\xi$ increases from a very small value (where the fluid–fluid interface is nearly flat), $|I_z|$ initially increases and then decreases. Interestingly, $|I_z|$ or $I_z/A$ is maximized at $\xi \approx 0.3$, independent of $\alpha$. Compared with a flat fluid–fluid interface, a curved interface is overall closer to the bubble, resulting in stronger bubble–droplet interaction. Therefore, the value of $|I_z|$ increases with increasing $\xi$ at first. When $\xi$ is very small, the droplet is relatively large, and the lower surface of the droplet is far from the bubble, while the upper surface of the droplet is primarily responsible for the asymmetrical motion of the bubble. However, when $\xi$ exceeds 0.3, the droplet size gradually becomes comparable to the bubble size, and the lower surface of the droplet also influences bubble dynamics. Consequently, $|I_z|$ decreases with $\xi$ when $\xi$ is greater than 0.3. This critical value of $\xi \approx 0.3$ applies to a wide range of parameters: $0.5 \leq \alpha \leq 1.6$ and $\gamma \leq 1.1$.

5.3. Bubble dynamics in the $\alpha$–$\gamma$ parameter space

In the current system, the interaction between bubbles and droplets is primarily governed by three key parameters: $\xi$, $\alpha$ and $\gamma$. While we discussed the influence of $\xi$ in the previous section, our attention now shifts to exploring the dynamics of bubbles within the $\alpha$–$\gamma$ parameter space. In this section, our focus narrows to scenarios with small standoff parameters, where $\gamma \leq 1$. This choice is motivated by the fact that for large standoff parameters ($\gamma \gtrsim 0.9$), as will be discussed in § 6.1, the jet cannot effectively penetrate water–oil interface. We will delve further into the critical standoff parameter required for successful jet penetration of the droplet surface.

Figure 11 illustrates the relationship between bubble jet volume, jet velocity and kinetic energy concerning $\gamma$ and $\alpha$ while maintaining $\xi$ at a typical value of 0.6. These parameters provide valuable insights into the mass transport capabilities of the bubble. Figure 11(a) depicts the variation of jet volume with respect to $\gamma$, accompanied by a schematic representation of the jet volume in the upper inset. Notably, increasing $\alpha$ results in a proportional increase in jet volume and inertia. This indicates that a higher $\alpha$ value can yield a larger jet. Moving on to figure 11(b), we examine the behaviour of jet velocity ($V_{jet}$), defined as the velocity of the jet tip just prior to impact. Here, we observe that $V_{jet}$ decreases as $\alpha$ increases but increases with $\gamma$. This behaviour is attributed to the weakening of jet acceleration as the jet's volume and mass expand.

Figure 11. Dependence of bubble dynamics on $\alpha$ and $\gamma$ at a fixed $\xi =0.6$. (a) Bubble jet velocity, (b) jet volume and (c) kinetic energy of the jet. The upper inset of (a) shows a sketch of the bubble jet. All the panels share the same legend as (b).

Following Pearson, Blake & Otto (Reference Pearson, Blake and Otto2004), Li, Prosperetti & van der Meer (Reference Li, Prosperetti and van der Meer2020b) and Han et al. (Reference Han, Zhang, Tan and Li2022), we introduce the kinetic energy associated with the liquid jet

(5.3)\begin{equation} E_{jet}=\frac{1}{2}\rho_1 \int_{S_{jet}} \varphi_1\frac{\partial \varphi_1}{\partial n}{\rm d} S, \end{equation}

where $S_{jet}$ represents a closed surface delineating the confinement of the liquid jet within the bubble. This surface is depicted by the red solid line in the upper inset of (a). In figure 11(c), we explore the dependence of $E_{jet}$ on the parameters $\gamma$ and $\alpha$. Notably, due to its incorporation of mass (or liquid volume) and velocity components, the kinetic energy, $E_{jet}$, does not exhibit a straightforward monotonic relationship with $\gamma$. However, it attains its maximum value at an optimal standoff parameter, denoted as $\gamma _o$. It is noted that $\gamma _o$ increases with the parameter $\alpha$.

After the jet impacts the bubble, it transforms into a toroidal shape. To model this phenomenon, we introduce a vortex ring within the bubble. Additional details regarding the numerical model are available in Wang et al. (Reference Wang, Yeo, Khoo and Lam1996) and Zhang et al. (Reference Zhang, Li and Cui2015). Figure 12 shows the evolution of the toroidal bubble and the corresponding droplet deformation under the jet impact with $\gamma =0.6$, $\xi =0.6$ and $\alpha =1.1$. The contour denotes the magnitude of the flow velocity. As shown in frame 1 ($t=1.912$), the velocity within the bubble jet is much higher than in other positions of the flow field. Subsequently, the liquid jet goes upward and pushes the droplet surface outward rapidly (frames 2–5). The jet cannot impact the droplet surface directly as the liquid layer between the bubble top and the droplet surface acts like a cushion. The bubble rebounds at this stage, with both the width and jet velocity decreasing. The velocity at the top of the droplet surface (denoted by $V_{d}$) initially increases and reaches a peak velocity as the jet tip approaches the droplet surface (frame 4). After the jet starts to enter the surrounding liquid phase, $V_{d}$ rapidly decreases (frame 5). In the numerical simulation, the liquid jet can penetrate deeply into the surrounding phase until the thickness of the liquid layer between the bubble and droplet becomes less than the grid size. It appears that the liquid jet has the potential to transport the droplet phase into the outer phase, which may result in the production of very fine droplets. Our BI model struggles to capture the fragmentation of the liquid jet that follows. More discussion on the jet penetration of the droplet surface from the experimental aspect will be provided in § 6.1.

Figure 12. Evolution of a toroidal bubble and the jet impact on the droplet surface for $\xi =0.6$, $\gamma =0.6$ and $\alpha =1.1$. The contour denotes the magnitude of the flow velocity. The panels in the figure feature a unified legend on the left side. Horizontal and vertical axes are $0\le r\le 0.5$, $-0.2\le z \le 1$. The dimensionless times are marked at the upper left corners.

6.1. Jet penetration

As discussed in § 3.1, a very thin liquid jet forms around the end of the bubble collapse phase, culminating in the jet impacting the droplet surface. This impact creates a sharp protrusion on the droplet surface, which may explain how fine droplets are produced in the O/W system. In our experiments, we notice that the liquid jet can penetrate the droplet interface at a relatively small standoff parameter. If we only consider the phenomena during the first collapse and the subsequent rebound process of the bubble, the upper bound of the standoff parameter (denoted by $\gamma _c$) for jet penetration of the droplet surface is about $0.88\pm 0.03$, which is summarized from more than 60 experiments, as shown in figure 13. Interestingly, this critical value is very close to the configuration with an initially flat fluid–fluid interface and a similar density ratio (Han et al. Reference Han, Zhang, Tan and Li2022). This implies that the jet penetration condition is insensitive to the curvature of the fluid–fluid interface in the parameter range of the present experiment ($0.58 < \xi < 1.05$). Additionally, if the standoff parameter ranges from 0.85 to 1.05, we may observe the jet penetration of the droplet surface when the bubble forms a second jet around the end of the second cycle, as denoted by the yellow rectangles in figure 13. When $\gamma \gtrsim 1.07$, we can hardly observe the jet penetration of the droplet surface during the whole bubble lifetime, which implies that no fluid mixing takes place.

Figure 13. Variation in jet impact velocity with respect to the standoff parameter $\gamma$. The red circles and yellow rectangles signify the penetration of the droplet surface by the first and second bubble jets, respectively. Meanwhile, the green triangles indicate scenarios where the bubble jet cannot penetrate the droplet surface. The critical standoff parameter $\gamma _{c}$, which divides the first two regimes of jet behaviours, lies within the range of 0.85 to 0.91. If $\gamma \gtrsim 1.07$, the jet cannot penetrate the droplet surface during the whole lifetime. The bubble-to-droplet size ratios of these experiments range from 0.58 to 1.05. The parameters in BI simulations are set as $\varepsilon = 2000$, $R_0=0.0591$, $\alpha =1.093$ and $\xi =0.6$.

Due to the limited spatio-temporal resolution of high-speed recordings, we can hardly trace the jet head within the bubble interior. Hence, the jet impact velocity can be roughly evaluated from two adjacent frames before and after the jet impact moment. These measured velocities should be considered as lower bounds of the actual values. Nevertheless, the maximum value of jet velocity reaches more than 300 m s$^{-1}$. We also added the results obtained from BI simulations in figure 13. As anticipated, the numerical results exceed the experimental data. We notice that the jet impact velocity increases with $\gamma$ in BI simulations, as already discussed and explained in § 5.3. However, in the experiments, $V_{jet}$ has a tendency of increase in the range of $\gamma <0.9$ and decrease in the range of $\gamma >0.9$. This phenomenon can be explained as follows. When $\gamma >0.9$, the width and kinetic energy of the bubble jet decreases rapidly with $\gamma$. Although the jet possesses a very high velocity prior to the moment of impact, it would experience rapid deceleration after the impact. Take the $\gamma =0.6$ case for example, we find from the BI simulation that the jet velocity increases from 100 to 360 m s$^{-1}$ within $0.87\ \mathrm {\mu }$s, and then decreases rapidly after the jet impact. However, the time interval is more than $1.5\ \mathrm {\mu }$s in our experiments. It is a very challenging work for us to measure the jet velocities accurately at present. Nevertheless, our BI simulations can provide a reference for the high-speed liquid jet formation.

As inferred from previous discussion, the jet cannot impact the droplet surface directly as the liquid layer between the bubble and the droplet surface acts like a cushion. Therefore, we measure the maximum velocity at the impact position of the droplet surface (denoted by $V_{d}$), which may be a more direct way to quantify the jet penetration process that is responsible for fluid mixing. Figure 14 shows the dependence of $V_{d}$ on $\gamma$ and the numerical results are also added for comparison. The maximum value of $V_{d}$ reaches more than 100 m s$^{-1}$. Although the data have a certain degree of discreteness, the plot reveals an overall decrease of $V_{d}$ over $\gamma$. Remarkably, our BI simulation reproduces the upper bound of the experimental data quite well when $\gamma \lesssim 0.8$. We note that the value of $V_d$ at $\gamma =0.9$ measured in the experiments is overestimated by our BI model. This is because the jet tip reaches the droplet surface when the bubble rebounds to the maximum size in this case, thus the impact velocity on the droplet is much influenced by the energy loss of the rebounding bubble, which is not considered in the present BI model. Finally, our experiments demonstrate that the minimum value of $V_{d}$ for jet penetration is about 31 m s$^{-1}$. In the work by Yamamoto et al. (Reference Yamamoto, Matsutaka and Komarov2021), a condition for droplet fragmentation was proposed, which relies on a balance between interfacial and kinetic energies, along with a comparison between dynamic and Laplace pressures. Their study concluded that the minimum velocity of the liquid jet required to fragment a micro-sized droplet is approximately 40 m s$^{-1}$, which is close to our experimental data considering that our length scale is approximately 15 times larger.

Figure 14. Dependence of the maximum velocity at the impact position of the droplet surface (denoted by $V_{d}$) on $\gamma$. The red circles signify the penetration of the droplet surface by the first bubble jet that forms around the end of the first cycle. The green triangles indicate scenarios where the first bubble jet cannot penetrate the droplet surface. The minimum value of $V_{d}$ for jet penetration is approximately 31 m s$^{-1}$. The bubble-to-droplet size ratios of these experiments range from 0.58 to 1.05. The numerical results (denoted by the black cross) are also added for comparison, which align well with the upper bound of the experimental data. The parameters in BI simulations are set as $\varepsilon = 2000$, $R_0=0.0591$, $\alpha =1.093$ and $\xi =0.6$.

6.2. Droplet pinch-off

In addition to the jet penetration in the O/W system, another significant mechanism for fluid mixing in the W/O system involves the pinch-off of water droplets. This process occurs on a larger time and length scale compared with jet penetration. Refer to figure 5(c) for a representative experiment. As the vortex ring bubble travels over a considerable distance, it induces a global motion in the water droplet. In the context of droplet pinch-off, it is imperative for inertia to overcome surface tension in the presence of viscous dissipation. Consequently, we introduce the local Reynolds and Weber numbers as follows:

(6.1a,b)\begin{equation} Re=\frac{\rho_2 R_p V_p}{\mu_2},\quad We=\frac{\rho_1 V_p^2 R_p}{\sigma_2}, \end{equation}

where $R_p$ represents the radius of the protrusion driven by the bubble and $V_p$ signifies the maximum velocity of the protrusion tip. Here, $R_p$ is approximately equivalent to the radius of the satellite droplet following pinch-off. More than 100 experiments were conducted to establish the droplet breakup criterion. In the phase diagram depicted in figure 15, we identify the critical Weber number denoted as $We_c$, which demarcates two distinct regimes: (i) the creation of a satellite droplet and (ii) the absence of droplet pinch-off. Notably, the critical value $We_c$ is approximately 16, with a corresponding Reynolds number of around 7. This suggests that both surface tension and viscosity play significant roles in this process. Unfortunately, this phenomenon falls beyond the applicability of our BI model. To comprehend this process, we provide an explanation based on the balance between interfacial and kinetic energies. Firstly, the increase in interfacial energy during the droplet pinch-off can be expressed as

(6.2)\begin{equation} \Delta E_i= 4{\rm \pi} R_p^2 n \sigma_2, \end{equation}

where $4{\rm \pi} R_p^2 n$ represents the maximum increment in the interface area during the process, and $n$ is a constant, assuming a value of 1 when the satellite droplet is much smaller than the main droplet. The kinetic energy associated with the migrating remnant bubble can be roughly estimated as

(6.3)\begin{equation} E_k=\tfrac{1}{2} \rho_1 ({\rm \pi} R_p^2 l) V_p^2, \end{equation}

where $l$ denotes the length of the protrusion jet at the moment when the droplet surface reaches its maximum velocity. Since the vortex ring (bubble) typically carries a portion of liquid that assumes an ellipsoidal shape, with the ratio of semi-minor to semi-major axes typically falling between 1 and 2 (Didden Reference Didden1979; Dabiri & Gharib Reference Dabiri and Gharib2004; Sullivan et al. Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008), we have the relationship $R_p < l < 2R_p$.

Figure 15. Water-in-oil droplet behaviours driven by a migrating vortex ring bubble on a large time scale. The Reynolds and Weber numbers are defined by (6.1a,b). We identify the critical Weber number denoted as $We_c\approx 16$, which demarcates two distinct regimes: (i) the creation of a satellite droplet and (ii) the absence of droplet pinch-off.

Given that the Reynolds number is less than 10, viscosity leads to considerable energy dissipation. Therefore, the kinetic energy $E_k$ must exceed $\Delta E_i$, resulting in the inequality

(6.4)\begin{equation} We=\frac{\rho_1 V_p^2 R_p}{\sigma_2}>\frac{8nR_p}{l}. \end{equation}

We determined the value of $n$ through our experiments precisely at the point of droplet pinch-off, yielding a range between 1.3 and 1.8. Consequently, the right-hand side of (6.4) spans from 5.2 to 14.4, aligning remarkably well with our experimental observations ($We_c\approx 16$).