A Review of the Dynamics Progress of Bubble Collapse within Droplet and Droplet Splash

Abstract

The dynamics of a cavitation bubble within a droplet is one of the hot research topics at present. The present paper summarizes the research progress of bubble collapse within droplets and associated droplet splash. Firstly, three typical structures of bubble collapse are introduced, together with the collapsing shock waves and the distribution of collapsing forces. Secondly, several typical forms of droplet splash are shown in terms of splash speed, splash direction, and dynamic mechanisms. Finally, the unsolved problems in the field of cavitation bubbles within droplets are proposed with perspectives.

1. Introduction

Cavitation bubbles within droplets are one of the hot research issues in the field of cavitation and bubble dynamics [1,2]. Figure 1 shows the fields involved in cavitation bubbles within droplets. During the fuel atomization [3,4,5,6], fuel droplets in internal combustion engines could be atomized into fine droplets by cavitation. Due to the decrease in the pressure inside the droplet, which is less than the local saturated pressure, a bubble could be formed inside the droplet. The degree of droplet cavitation determines the combustion efficiency of fuel and thus affects the operating conditions of internal combustion engines [7,8]. In the process of inkjet printing [9,10,11,12,13], a laser beam can produce a bubble in a high viscosity small ink droplet, thereby improving inkjet technology. In the process of military underwater explosions [2,2,14,15,16,17], the bubble migration process, shock wave, and liquid–surface splash process caused by bubble collapse are closely related to the research of cavitation bubbles. The bubble collapse jet generally points to a rigid wall, and the velocity is usually about 100 m/s. In the research of micropumps [18,19,20,21], the process of bubble collapse can transport the bubble vapor cloud to the bottom of the pipeline and achieve a millisecond response speed without fluid mechanics [22]. In the medical field [23,24,25,26,27,28,29,30], acoustic droplet vaporization (ADV) is a physical process in which the pressure wave of ultrasound induces a phase transition, resulting in superheated liquid nanodroplets forming bubbles that provide ultrasound imaging contrast and other functions. It is widely used in many medical treatment [31,32], such as in the generation of cavitation bubbles within droplets in blood vessels and drug transport, imaging, embolization therapy, and treatment delivery. In the development of needle-free injection technology [33,34,35,36,37], the vaccine needle-free injection process is achieved by effectively utilizing the collapse of a shock wave within the droplet to form a jet. In the field of aerosols in the atmospheric environment, the jets and cavitation bubbles within droplets generated by the collapse of oceanic bubbles are one of the main sources of aerosols in seawater. The reasons for their formation are as follows. First, when a bubble rises to the water surface and bursts, and the bubble is entrained in the water column by previously broken waves, the bubble collapses and the surface film ruptures, and membrane droplets will be generated. When the surface shear stress is high enough, the water droplets will be torn from the surface wave [38,39,40,41,42,43].

In many fields, cavitation bubbles within droplets have broad application prospects. The essence of cavitation bubbles within droplets is that the process of bubble collapse leads to a severe multiphase flow phenomenon, accompanied by rich fluid phenomena. The main features include the formation of microjets [45,46,47], bubble migration [48,49], and the formation of microbubbles [45,50] during the collapse process of the cavitation bubble. During the splash process of the droplet, splashes and jets of different modes [51,52,53] and directions [54,55] are formed. A shock wave is generated during the collapse of the bubble within a droplet and causes a change in the pressure field in the droplet. When the pressure wave propagates to the free surface of the droplet, it will cause the surface of the droplet to oscillate and excite the droplet to splash. During the non-spherical collapse of the bubble under the free surface, a high-speed jet will be generated while moving away from the free surface. The current research on the dynamics of bubbles within droplets is still in the early stage. Many unsolved problems limit its wide applications. For example, the theoretical model suitable for the dynamics of cavitation bubble collapse within droplets still suffers many limitations. Based on these, the topic of cavitation bubbles within droplets is being given increasing attention by researchers [56].

The research progress of cavitation bubbles within droplets could be briefly introduced as follows. Avila et al. [52] used laser-induced suspended droplets and captured the process of bubble collapse within droplets through high-speed photography. The different situations were observed on the surfaces of droplets, including atomization, film formation, and droplet fragmentation. The propagation process of the shock wave inside the droplet was researched. Obreschkow and Mohamed [45,50,57] found that the collapse of a cavitation bubble within a droplet can generate two droplet jets. In addition, the surface of the droplet will reflect the shock wave generated by the collapse of the bubble, thereby inducing the generation of a large number of microbubbles. Subsequently, the evolution process of the shock wave of bubble within a droplet was verified through numerical simulation. Lv et al. [7] established a mathematical model focusing on the effect of viscosity. Zeng et al. [58] conducted numerical simulation (Open FOAM) research on the splash processes of suspended droplets. It showed that the radial acceleration of the droplet splash was caused by the oscillation of an internal bubble. Thoroddsen et al. [59] investigated the bubble initial position within the droplet and its effects on the form of the droplet splash.

This paper mainly introduces the dynamic processes of bubble collapse and induced droplet splash and summarizes the important research achievements in recent years. The main purposes of the following sections are given as follows: in Section 2, the forms of bubble collapse within droplets is introduced, including its typical forms and characteristics, and a mechanism explanation of bubble motion; in Section 3, the research on collapse shock waves is introduced; in Section 3.3, the distribution of forces during the collapse process is introduced; in Section 4, the typical droplet splash patterns are introduced together with the influencing dimensionless parameters, radius ratios, and eccentricity; in Section 5, the splash velocity is introduced with splash velocity, flow field analysis, and splash mechanism; in Section 6, the splash direction is introduced with splash direction control; in Section 7, the conclusion of this paper is given with perspectives.

2. Typical Forms of Cavitation Bubble Collapse

2.1. Microjets

Due to the influence of gravity on the droplet structure, researchers have conducted droplet cavitation experiments in spacecrafts [45,57]. Figure 2 shows the microjets formed during the collapse process of a cavitation bubble within a droplet [45]. Under the condition of microgravity, it shows a nearly spherical drop structure, and the cavitation bubble was induced by electric spark. Figure 2a shows the maximum volume state of a cavitation bubble within a droplet at an eccentric position. Figure 2b shows the state of a bubble forming a microjet within a droplet with microjet velocity greater than 50 m/s. The jet on the right side of Figure 2c shows the formation of a double jets structure on the droplet surface caused by bubble collapse, with a jet velocity of 6 m/s. It should be noted that the structure on the left side of the droplet is a droplet splash structure. In addition, this research indicates that the process of jets formation during cavitation bubble collapse within droplets is consistent with the process of bubble collapse under the free surface. There are two kinds of micro-jets formed during the collapse of a droplet cavitation, the micro-jet formed by the cavitation inside the droplet and the micro-jet formed on the surface of the droplet caused by the shock wave induced by the bubble collapse.

Figure 3 shows the jet structure during the collapse process of a cavitation bubble at a liquid surface [60]. During the process of bubble collapse, the bubble first collapses near the free surface, then the microjet passes through the center of the bubble, and finally evolves into a mushroom-shaped vapor cloud. At the same time, in the opposite direction to the bubble microjet, a pinnacle splash on the liquid surface forms. The vapor cloud mass is caused by the non-spherical evolution in the process of bubble collapse, and the vapor bubble moves away from the free surface. Additionally, the splash at the top is due to the increase of the pressure between the droplet and the bubble wall, resulting in the rupture of the free surface.

Figure 4 shows the motion process of the bubble wall and the formation mechanism of the vortex ring structure [61]. Among them, Figure 4a shows that the bubble wall begins to approach the free surface at the initial moment. Figure 4b shows the reverse direction of the pinnacle jet and the microjet formed by the bubble collapse on the free surface. The reflected flow passes through the bubble, which also leads to the second collapse starting from the lower side of the bubble. Figure 4c shows the expansion of the bubble volume after a single collapse process. Figure 4d shows the vortex ring structure generated during the later stage of the bubble collapse, caused by the pressure gradient at the edge of the bubble wall. And, a high-pressure stagnant flow ring is formed.

Based on this, it can be seen that due to the influence of a spherical free surface, the microjet process of the cavitation bubble within a droplet [46] is more complex than in the infinite fluid domain [62,63]. There are not only microjet processes, the evolution of a collapsed vapor cloud [64], and secondary cavitation jets in a cavitation bubble within a droplet, but also splash jets on the surface of the droplet.

2.2. Bubble Migration

The process of bubble migration is an important feature of the bubble collapse process [65]. Figure 5 contains high-speed photographs of the bubble migration process in the process of bubble collapse in a droplet under microgravity conditions [57]. Figure 5a–c show the first collapse process of the bubble, and Figure 5d–f show the rebound process of the bubble. It is obvious that the bubble undergoes significant migration during the rebound process and moves in the opposite direction away from its initial position.

If the local spherical surface near the initial position of the bubble is approximated to a free surface, its movement trend is consistent with the interaction between the free surface and the bubble. Figure 6 shows the theoretical verification process of bubble migration under a free surface based on the Kelvin impulse theory. It can effectively deal with the migration and deformation law of cavitation bubble’s centroid. In the Kelvin theory, it is assumed that the fluid domain is an incompressible, inviscid, and irrotational ideal fluid, and the bubble will undergo non-spherical deformation and migration along the axial symmetry direction during the volume oscillation. When the bubble evolves non-spherically near the free surface, a high-speed jet is often generated, and the bubble moves away from the free surface. Among them, the numbers 1~6 in the Figure 6a show the process of bubble growth under the free surface; the numbers 6~10 in the Figure 6b show the process of bubble collapse under the free liquid surface [48,66]. The reasons of cavitation migration could be explained as follows. Firstly, combined with the Kelvin impulse theorem, the migration away from the free surface occurs during the evolution of the cavity. Secondly, the fluid domain inside the droplet is affected by the surface tension of the free surface of the droplet. Thirdly, during the process of bubble collapse, the vapor mass moves downward.

2.3. Microbubble Cloud

The generation of a large number of microbubbles during the process of bubble collapse is a typical phenomenon as given by Obreschkow et al. [45,50,57]. It should be emphasized that this phenomenon is less significant under gravity. The reasons for this include that the ambient pressure under microgravity is low, and the spherical droplet structure could be remained.

Figure 7 shows the formation of microbubbles during the process of bubble collapse [45]. This type of submillimeter-sized microbubble could last for about 50 s during the process of bubble collapse. Figure 7a shows the initial moment of cavitation, and there are no microbubbles generated. Figure 7b shows the moment of the first collapse generating the microbubbles. Figure 7c shows the later stage of bubble collapse, with the disappearance of the microbubbles.

3. Shock Wave of Bubble Collapse

The shock wave formed by bubble collapse is often accompanied by an intense energy dissipation [67,68]. The collapse of the bubble transforms the potential energy stored in the bubble into the kinetic energy of the surrounding flow. There is mutual interations between the shock wave and the bubble collapse process [69,70], inducing the formation of microjets from the bubble [71] and erosion [72,73,74]. Hence, this section mainly introduces the impact of shock waves on the collapse process of cavitation bubbles within droplets, the evolution process of a shock wave inside the droplet [52,75], and the stress analysis.

3.1. Form of Shock Wave

The collapse process of a cavitation bubble within a droplet is usually accompanied by an intense shock wave [76]. Due to the uniquely restricted environment within a droplet, during the bubble collapse, a shock wave will continuously reflect and focus inside the droplet, further affecting the bubble collapse and droplet splash. Figure 8 shows the microbubbles that form through shock waves under different eccentric positions [50]. The subfigure on the left side is the experimental diagram of microbubbles with small eccentricity. The lower subfigure on the left is the experimental figure of microbubbles with large eccentricity. The right subfigure is the numerical simulations of energy density under the corresponding conditions. The red region represents high energy density; the blue region represents low energy density. The experimental and numerical results are consistent, verifying that the energy distribution of a shock wave during the collapse of an eccentric bubble is altered by the reflection of a liquid surface. When the eccentricity is small, it is the unilateral energy is primary. When the eccentricity is large, the bilateral energy is of great importance. It can be seen that the distribution of microbubbles during the collapse process of a bubble can measure the energy distribution of the bubble collapse. On the other hand, it also reflects that the structure of a cavitation bubble within a droplet can also could impact the propagation of the shock wave.

3.2. Evolution of Shock Wave

Figure 9 shows the evolution process of a shock wave inside cavitation bubbles within droplets [52]. Figure 9a shows a high-speed photography time-domain diagram of a cavitation bubble within 1.45 µs. The long axis of the droplet radius is 1.7 mm and the short axis is 1.1 mm. Here, the cavitation bubble is placed at an eccentric position. Figure 9b shows the numerical simulations corresponding to the shock wave evolution of bubble collapse within the droplet. The COMSOL 5.0 solver was employed for the numerical solution. Among them, the shock wave reflects at the free surface of the droplet, and then being focused. Figure 9c shows the top view corresponding to the image in Figure 9b, where red lines represent compressive pressure and blue lines represent tensile pressure. Furthermore, the blue low-pressure area represents the predicted area of secondary cavitation occurrence. In Figure 9d, the negative pressure region is circular and located at the equatorial position of an ellipsoidal droplet. It is also verified that the secondary cavitation of the droplet occurs in the area where the shock wave is reflected by the free surface. Based on this, it can be seen that the surface structure of the droplet and the relative position of the bubble have a significant impact on the reflection and focusing of the shock wave.

3.3. Force in The Early Stage of Collapse

3.3.1. Types of Force

Based on the classical Rayleigh–Plesset equation, the dynamic equation of the cavitation bubblecan be expressed as Equation (1) [77]:

$$p-p_{\infty }=\frac{2\sigma }{R}+{\rho }_{1}R\frac{d^{2}R}{d{t}^2}+\frac{3}{2}{\rho }_1{\left(\frac{dR}{dt}\right)}^2+\frac{4{\mu }_1}{R}\frac{dR}{dt}$$

(1)

In Equation (1), $P$ is the pressure inside the bubble and $P_{\infty }$ is the environmental pressure, $\sigma$ is the surface tension coefficient, ${\rho }_1$ is the liquid density, ${\mu }_1$ is the liquid viscosity, and $R$ is the bubble radius. The first item on the right side of Equation (1) represents the surface tension term, the second and the third items represent the inertial force term, and the fourth item represents the viscous force term. It can be seen that the dynamic process of droplet bubble formation is mainly influenced by surface tension, inertial force, and viscous force [77].

3.3.2. Distribution of Force

Figure 10 shows the distribution of force over time during the evolution of the cavitation bubble within a droplet [77]. Among them, Figure 10a shows the effect of surface tension on the evolution of a cavitation bubble within a droplet. I–III represent three stages respectively. At the initial stage (I), it is mainly affected by surface tension. And, the influence of the surface tension gradually decreases in the second stage (II). At the third stage, the influence of the surface tension is weakened.

Figure 10b shows the effect of an inertial force on the evolution of the cavitation bubble within the droplet. The promoting effect of inertial force on the evolution of the bubble at the first stage can be basically ignored. At the second stage (II), the inertial force shows a slow growth trend and gradually increases its impact on the evolution of the bubble. At the third stage (III), the inertial force sharply increases, and the evolution rate of the bubble is significantly affected by the inertial force.

Figure 10c shows the effect of viscous force on the evolution of the cavitation bubble within a droplet. At the first stage, the inhibitory effect of viscous forces on the bubble evolution slowly increases. At the second stage (II), the effect of viscous force on the bubble shows a fluctuating pattern. At the third stage, the effect of viscous force increases sharply firstly and then decreases sharply. It is worth emphasizing that the growth stage of the bubble wall determines the size of the potential energy of the bubble wall, which is affected by the movement of the bubble wall in the subsequent bubble collapsing stage and the rebounding process. The shock wave induced by bubble collapse is also related with the bubble wall motion.

4. Typical Forms of Droplet Splash

The form of the droplet splash is an important characterization of droplet splash dynamics [53,78]. The setup of droplet splash experiments, the induction of splash forms [46,79], the quantitative analysis of splash, and the final mechanism analysis are all important components of the research on the dynamics of droplet splash. On this basis, this section introduces a high-speed photography experimental platform for the splash dynamics of a bubble within a suspended droplet and four typical splash cases. It summarizes the quantitative research methodology for splash dynamics, explores the changes in splash velocity and splash direction of the droplet [80,81], and finally provides a highly recognized explanation mechanism of the splash dynamics [66,82].

4.1. Experimental System

Figure 11 shows the experimental setup of a cavitation bubble within a suspension droplet and the definition of its eccentricity. In Figure 11a, the core components of the experimental setup system are the transducer with a frequency of 27.4 kHz and the aluminum acoustic reflector. The droplet suspension is maintained by controlling the sound radiation pressure. Afterwards, a bubble is induced by a laser in the suspension droplet. Figure 11b shows the relevant definitions of the dimensionless parameters of the radius ratio and the eccentricity of the droplet. The radius ratio dimensionless parameter is related to laser energy, and the energy scale $E$ is the work done by a droplet to form a bubble with a volume of $V_d$ under environmental pressure with $E=\kappa E_1/V_d$. Among them, $\kappa$ is the percentage of laser energy and $E_1$ is the laser energy. The dimensionless parameter of eccentricity is related to the initial position of the cavitation bubble, where $\phi$ is the ratio of the short axis radius $R_x$ to the long axis radius $R_y$. ${\epsilon }_y$ and ${\epsilon }_x$ are the short axis eccentricity and the long axis eccentricity, respectively.

4.2. Typical Splash Forms

Figure 12 shows four typical splash forms of the droplet during the collapse process of a bubble within a suspended droplet. By changing the ratio radius, different splash forms of the droplet were achieved. Among them, Figure 12a shows the atomization case, in which the droplet is completely shattered into tiny droplets during the bubble collapse process, presenting a mist-like appearance. Figure 12b shows the unstable liquid film form, in which the droplet initially appears as a grid-like splash, then evolves into a liquid film, and finally disperses into small droplets. Figure 12c shows a stable liquid film form, in which the droplet gradually exhibits a hemispherical liquid film shape, and ultimately maintains the liquid film structure. Figure 12d shows the coarse crushing form, in which multiple scattered splashes are generated in the short axis direction of the droplet during the bubble collapse process, and then the ellipsoidal droplet is stretched into a long strip shape.

5. Splash Velocity

5.1. Definition of Splash Structure

Figure 13 shows the definition of splash structure [83]. It shows the stable coronal splash of the droplet, including the main peak of the splash, the liquid film of splash, and other structures. Among them, it mainly includes the height of the main peak of the splash H_spike, the height of the liquid film of the splash H_spray, the splash width W_base, and the width of the liquid film of the splash W_spray. The structured definition of droplet splash is the basis for quantitative research. Currently, the research on the main peak of droplet splash has attracted most of attention.

5.2. Comparison of Splash Velocity

The jet velocity of the main peak is an important factor in droplet splash [84,85,86]. Figure 14 shows the distribution of the vertex velocity of droplet splash [59]. It can be seen that the velocity of the main peak of the droplet splash is between 1100 and 1400 m/s [87]. Here, the main peak of the splash will be rapidly decomposed into small spray within time duration about 4 μs. The speed of the splash will drop sharply to the speed of sound (in the air). It can be seen that the splash form changes sharply with the evolution stage of bubble collapse, and its dynamic characteristics also undergo significant change.

5.3. Splash Flow Field

The analysis of the flow field of a droplet splash is an important basis for revealing the dynamic operation mechanism of the splash. Researchers mainly explain the internal mechanism of droplet splash by researching the distribution of velocity and pressure in the flow field of the droplet splash [88,89,90,91]. Figure 15 shows the distribution of the flow field of droplet splash [58]. At the bubble expansion stage ($t=20\mathsf{\mu }\mathrm{s}$), the surface of the droplet has no splash. At the later stage ($t=50\mathsf{\mu }\mathrm{s}$), the non-uniform distribution of velocity and pressure along the surface of the droplet leads to droplet fluctuations. At the stage of bubble collapse ($t=70\mathsf{\mu }\mathrm{s}$), the liquid flows towards the center of the droplet. At the bubble collapse stage ($t=110\mathsf{\mu }\mathrm{s}$), the disturbance amplitude of the surface of the droplet increases. Afterwards, pressure pulses are generated by bubble expansion ($t=126\mathsf{\mu }\mathrm{s}$). This accelerates the movement of the fluid surface, causing the trough to transform into a peak, and the jet is formed at the position of the previous trough. The velocity of the jet is increased by the pressure pulse, resulting in more jets being generated after the second bubble collapsing process ($t=240\mathsf{\mu }\mathrm{s}$).

5.4. Splash Mechanism

5.4.1. Singularity Theory of Interface Depression

Currently, the theory of interface depression singularity is one of the important mechanisms [92]. Figure 16 shows the theoretical schematic of the interface depression singularity [92]. The red line represents the initial stage of splash, the blue line represents the middle process of splash, and the green line represents the further strengthening of splash. The three red dots in the figure represent the singularity of the free surface, and the two arrow directions represent the directions that may cause a splash of the droplet during the surface depression and collapse process. The curvature change of the free surface is the main cause of the splash due to the radial outward movement of the liquid surface at the depression.

5.4.2. Taylor Instability Mechanism

Another explanation for droplet splash is the Taylor instability mechanism. Figure 17 shows a schematic of the Taylor instability mechanism of bubble collapse within droplets [93]. Firstly, the surface of a droplet is disturbed by the shock wave of bubble collapse. When the bubble oscillates, the surface of a droplet is accelerated, and the disturbance amplitude of the surface is increased. When the bubble wall and droplet splash are in contact with the disturbed surface, interfacial instability occurs [94]. Among them, $n$ is the number of wave peaks on the droplet surface, $R_d$ is the droplet radius, $\eta$ is the disturbance amplitude of the surface, $P_d$ is the pressure on the droplet surface, $R_b$ is the bubble radius, and $P_R$ is the pressure at the bubble boundary.

6. Splash Direction

6.1. Splash Direction Control

The direction control technology of the droplet splash is being developed [54]. Figure 18 shows the variation of the droplet jet direction versus the position and the time of bubble initiation [55]. Figure 18a is variation of Ejection angle with displacement. It can be observed that there is a linear relationship between the initial injection direction of the liquid column and the position of the bubble initiation. A liquid column injection within the range of −45° to 45° could be achieved through varying x (distance between droplet center and initial position of bubbles) between −900 μm and 900 μm. Due to the radial symmetry of the droplet, the liquid column can be discharged in any direction within a 90° cone. Figure 18b is variation of Ejection angle with time of different displacement. It can be found that under the three displacement relationships, the ejection angle gradually decreases with time. It shows that the splash will gradually stabilize over time.

6.2. Splash Direction Mechanism

The control of droplet splash could be accomplished through the interface-focusing mechanism [95,96]. Figure 19 is a schematic of the shock wave reflection path for the cavitation bubble within a droplet [55]. It can be seen that, after the first rebound of the waveform on the spherical droplet interface, the shock wave is focused and reflected to a certain position, which is different with the initial position of the bubble. This provides strong theoretical support for the controllable operation of droplet splash.

7. Conclusions and Perspecitves

7.1. Conclusions

(1) The movement of the bubble wall during the collapsing process of a cavitation bubble within a droplet is significantly influenced by the droplet surface, leading to three typical structures (microjets, bubble migration, and the microbubble group respectively).

(2) The bubble collapsing shock wave could be formed during the evolution of microbubble groups within the droplet. And, the reflection and focusing of the shock wave inside the droplet lead to the complex droplet structure.

(3) The forms, velocity, and direction of droplet splash are closely related to the evolution process of the bubble collapse and the shock wave. The quantitative control of droplet splash speed and the direction can be achieved by controlling the bubble collapse.

7.2. Perspective

(1) The theoretical aspects of the dynamics of cavitation bubbles within droplets are still not clear enough.

(2) The implementation of a controllable droplet structure under a gravity environment is important.

(3) The quantitative control and implementation of droplet splash are not yet unified.