Collapsing and Splashing Dynamics of Single Laser-Induced Cavitation Bubbles within Droplets

Abstract

In the present paper, the cavitation bubble dynamics model for a single bubble oscillating within a droplet is improved based on the classical Rayleigh–Plesset bubble dynamics equation and the effects of liquid surface tension and viscosity are both considered. In the aspect of the experiment, the collapsing dynamic process of a bubble within a droplet is carried out by building a high-speed photography experimental platform. In addition, the numerical solution of the dynamic equation for the collapse time of the bubble within the droplet is also carried out. The findings are given as follows: (1) The bubble dynamic equation considering liquid surface tension and viscosity of bubble within droplet is proposed. (2) The surface of liquid droplets induced by the bubble motion could be divided into three modes: no splashing, scattered splashing, composite splash consisting of scattered and flaky splash. (3) The bubble interface during the first collapsing stage could be divided into three types: spherical, conical, and fungiform. (4) The numerical solution shows an accurate prediction of the bubble collapse time within the droplet especially under the condition of medium radius ratio.

1. Introduction

In the current study, the atomization performance of liquid fuel is one of the representative issues in the field of droplet cavitation [1]. The combustion process of liquid fuel in fluid machinery mainly includes fuel jet splitting, droplet breakage, atomization and evaporation, and fuel air mixing [2]. The process of fuel entering the low-pressure region from the high-pressure region of the internal combustion engine is accompanied by the primary and secondary cavitation processes inside the liquid droplet. It can be seen that the degree of fuel atomization is closely related to the process of cavitation bubble within droplet growth [3,4] and collapse [5,6]. In general, with intense cavitation bubble collapse within droplet, the droplet can be dispersed into smaller droplets, resulting in improved fuel combustion efficiency.

Presently, many researchers focused on the bubble dynamics in the infinite fluid [7,8,9]. The interaction between the bubble and the free surface in the confined-area fluids is enhanced, resulting in changes in the dynamics and the time of bubble collapse of the free surface. Droplet cavitation is one of the typical bubble dynamics problems in confined fluid domains. Lv et al. [4] deduced a new analytical form of a discrete equation that could be employed to describe the perturbation growth rate of cavitation bubble growth instability within fuel droplets. It was found that the perturbation growth rate is affected by dimensionless parameters, such as the Reynolds number. Obreschkow et al. [10] proposed a bubble dynamics equation suitable for the cavitation bubble within the droplet. The collapse time under the solution of the classical Rayleigh–Plesset equation and the model above were compared. Finally, they presented a curve of the bubble collapse modification coefficient as a function of the radius ratio of the cavitation bubble within the droplet. However, the effect of droplet surface tension and viscosity on the collapse time has not been considered in their research.

The Weber number represents the ratio of the inertial force to the surface tension force of the liquid. Generally, the size of the cavitation bubble within the droplet is of the micron scale. Under this condition, the Weber number is small, and the droplet surface tension has a significant impact on the bubble collapse and droplet splash dynamics. In addition, different viscosity of droplets [11,12,13] can affect the stress on the wall of the bubble; thereby, the growth and collapse process of the bubble is greatly affected [14] with the modifications of the collapse time. It can be seen that from a mechanical perspective, it is necessary to further improve the Rayleigh–Plesset equation by considering the effects of surface tension and viscosity of droplets, making it more general for the dynamics of cavitation bubble within the droplet.

Experimental research is one of the important ways to explore cavitation bubble dynamics [15]. For this, researchers [16,17,18,19] often use a high-speed camera to capture the bubble dynamics process and generate bubbles through laser-induced energy [20]. In the study of the dynamics of cavitation bubbles within droplets [21,22], there are three main types of splashes [23], including rapid atomization, flaky splashes, and droplet breakage. In addition, Liang [24] and Avila [23] both proposed that the relative size of the cavitation bubble within the droplet would affect the droplet splash morphology. However, the correlationship between bubble collapse dynamics and droplet splash dynamics has not been reported in the literature. Kobel et al. [25] prepared a nearly spherical droplet structure through a support pipe under microgravity conditions, and a bubble within the droplet was prepared. Unfortunately, in this study, the cavitation bubble within the droplet was prepared by an electric spark, which would have an impact on the droplet structure. Liang et al. [24] prepared bubbles in a vacuum pipe by changing the internal pressure of the droplets. However, in this manner, the repeatability of the same droplet cavitation structure is weak. Avila et al. [23] used a laser to prepare a bubble within the suspension droplet, avoiding the interference induced by the process of an electric-spark-induced cavitation bubble and explored the dynamic process of the cavitation bubble inside the droplet. However, the bubble collapse process of suspended droplets can be influenced by the sound field, and the bubble collapse process of the droplet is disturbed. Based on the above, it is an effective method to prepare cavitation bubbles within the droplet by preparing droplets in a pipe and combining them with a laser-induced cavitation bubble.

The study of the dynamic process of a cavitation bubble within a droplet mainly includes two parts. For the one hand, the bubble collapse process is investigated, such as bubble wall motion, bubble centroid migration, and micro jet [26,27,28]. For the other hand, the droplet splashing process is accompanied by processes such as droplet jet [29], droplet explosion [30], and liquid film evolution [31,32]. Thorodsen et al. [33] studied the droplet splash process and revealed the relationship between hemispherical droplet splash and bubble collapse. It was found that changing the size of the cavitation bubble and changing the laser position would have a significant impact on droplet splash morphology. Zhang et al. [34] conducted a quantitative study on the splash caused by bubble collapse, including physical quantities such as the length and height of the splash, droplet tip jet, bubble centroid displacement, and bubble collapse time. Based on this, it can be found that the process of bubble collapse [35,36] and droplet splash are the primary focus of cavitation bubbles within droplet.

In summary, the dynamic equation of the cavitation bubble within a droplet is different from the classical Rayleigh–Plesset bubble dynamics equation. The dynamics of a cavitation bubble within droplet collapse belongs to one of the restricted domain fluid problems involving both bubble collapse and droplet splash. Compared with the bubble collapse process in infinite domain [37], cavitation bubble collapse process in the droplet is more complex [38,39]. Based on this, in the present paper, a bubble dynamic equation suitable for the cavitation bubble within a droplet is proposed, and a numerical calculation method is used to predict the collapse time of the cavitation bubble within the droplet. In experimental research, high-speed photography combining a laser-induced cavitation bubble are employed for revealing the physical process of cavitation bubble collapse and droplet splash.

The structure of this article is given as follows. In Section 2, the dynamic equation of a single oscillating bubble within a droplet involving surface tension and viscosity is proposed. In Section 3, the experimental equipment and parameter settings of cavitation bubble within droplet high-speed photography are introduced. In Section 4, three cases of cavitation bubbles collapse within droplets are discussed and analyzed, respectively. In Section 5, three cases with different radius ratios are quantitatively compared. In Section 6, the numerical solutions of bubble collapse time are verified. In Section 7, the main conclusions of this paper are summarized.

2. Physical Model

2.1. Theory Model

In this section, the equations of radial motion and the physical model of the bubble wall within a droplet are presented. This section focuses on the equations of motion for the bubble wall within a liquid droplet. The bubble is assumed to be of radial oscillations. The ideal gas law is employed as Equation (1):

$$PV=n{R}_{\mathrm{g}}{T}_V$$

(1)

where P is the ideal gas pressure inside the bubble; V is the instantaneous volume of the bubble; n is the quantity of gas; R_g is the general gas constants; T_V is the temperature inside the bubble. For an adiabatic process of an ideal gas, the pressure and volume follow the Equation (2):

$$P{V}^{\kappa }=\mathrm{const}$$

(2)

where κ is the polytropic exponent. The following Equation (3) can be derived, where the subscript “0” indicates the initial value.

$$P_b=\frac{P_{b0}{V}_{b0}^{\kappa }}{V_b^{\kappa }}=P_{b0}{(\frac{R_{b0}}{R_b})}^{3\kappa }$$

(3)

where P₀ is the environmental pressure in equilibrium; P_b is the gas side pressure at the surface of bubble; R_b is the instantaneous radius of a single bubble within a droplet; the R_b0 is the radius of the bubble within a droplet at equilibrium.

Considering the effect of the viscous term and surface tension of the droplet and the bubble, Equations (4) and (5) could be given.

$$p_b=p_{in}-\frac{2\sigma }{R_b}-\frac{4{\mu }_L}{R_b}\frac{d{R}_b}{dt}$$

(4)

$$P_{\mathrm{in}}=\left[P_0+\frac{2\sigma }{R_{b0}}\left(1+\frac{R_{b0}}{R_{d0}}\right)\right]{\left(\frac{R_{b0}}{R_b}\right)}^{3\kappa }$$

(5)

where P_in is the gas side pressure at the surface; σ is surface tension coefficient; μ_L is the liquid viscosity.

The combined radial force at the surface of the droplet is expressed as Equation (6). The gas side pressure at the surface between the droplet and the external air is expressed as Equation (7), where compressibility in the liquid environment is not taken into account.

$$p_d=p_{out}+\frac{2\sigma }{R_d}+\frac{4{\mu }_G}{R_d}\frac{d{R}_d}{dt}$$

(6)

$$P_{out}=P_0$$

(7)

where P_d is the liquid side pressure at the surface; P_out is the gas side pressure at the surface between the droplet and the external gas; R_d is the instantaneous radius of the droplet containing a bubble.

Combined the above equations, the kinetic equation for the bubble within a droplet can be expressed as Equation (8).

$$\begin{array}{l}\left(R_b-\frac{R_b{}^2}{R_d}\right)\frac{d^2{R}_b}{d{t}^2}+\left(\frac{4{\mu }_L}{{\rho }_L{R}_b}+\frac{4{\mu }_G{R}_b{}^2}{{\rho }_L{R}_d{}^3}\right)\frac{d{R}_b}{dt}+\left(\frac{3}{2}-\frac{2R_b}{R_d}+\frac{R_b{}^4}{2R_d{}^4}\right){\left(\frac{d{R}_b}{dt}\right)}^2\\ =\frac{1}{{\rho }_L}\left\{\left[P_0+\frac{2\sigma }{R_{b0}}\left(1+\frac{R_{b0}}{R_{d0}}\right)\right]{\left(\frac{R_{b0}}{R_b}\right)}^{3\kappa }-\frac{2\sigma }{R_b}\left(1+\frac{R_b}{R_d}\right)-P_0\right\}\end{array}$$

(8)

where ρ_L is the density of the liquid; R_d0 is the instantaneous radius of the droplet containing a bubble at equilibrium.

2.2. Numerical Calculation Settings

Due to the complexity and nonlinear characteristics of bubble dynamics, the numerical solution of differential equations is essential. Based on the fourth-order Runge–Kutta numerical solution method, the numerical simulation analysis of the bubble model in a droplet is carried out in this paper. The constants used in the numerical calculation process are as follows: κ = 1.4, ρ_L = 1000 kg/m³, μ_G = 1.8 × 10⁻⁵ Pa·s, σ = 0.0728 N/m, P₀ = 101,325 Pa, P_v = 2335 Pa, P_a = 0, μ_L = 10⁻³. In addition, in case 1: λ = 0.581, R_b0 = 1.549 × 10⁻³ m. In case 2: λ = 0.828, R_b0 = 2.391 × 10⁻³ m. In case 3: λ = 0.926, R_b0 = 2.853 × 10⁻³ m. Here, λ is the ratio of bubble radius to droplet radius at the maximum bubble volume.

3. Experimental System

3.1. Experimental Equipment

Figure 1 shows the high-speed photography experimental platform of laser-induced bubbles within the droplet inverted. Among them, the upper subplot is the schematic diagram, and the lower one is the real setting. The main components include a high-speed camera (Qianyan Wolf X113); the camera speed is set to be 40,000 fps together with time delay generator (DG535), the laser generator (Penny-100A-SC), the three-dimensional electric translation platform (LSTOUCH-04), the microinjection pump (LD-P2020II), the focus lens (LMH-10X-532), and the computer.

The experimental system consists of four subsystems. In the micro-injection system, a millimeter droplet are prepared in a transparent tube through a micro-injection pump prepares. In the laser-induced bubble system, different size bubbles within the droplet are prepared by adjusting the laser voltage quantitatively. In the high-speed camera system, the process of bubble collapse is captured, and the data are transferred to the computer through the high-speed camera. The delay generation system, the capture process of the laser generator, and the high-speed camera are coordinated to ensure that the bubble dynamic process of droplet bubble collapse is captured effectively.

The experimental process is given as follows. Firstly, water is injected into the inverted, transparent pipe through the micro-injection pump, and the millimeter size droplet is formed. Secondly, the laser generator is adjusted for the preheating and the operation. The three-dimensional translation table is quantitatively controlled to ensure that the laser generator can make the bubble at the center of the liquid droplet. By adjusting the voltage of the laser generator, the maximum radius of the bubble is quantitatively modified. Thirdly, the high-speed camera is employed to adjust the position of the bubble, ensuring that the bubble is located in the center of the frame. Fourthly, the time delay generator is adjusted for the process.

3.2. Experimental Parameters and Process

Figure 2 shows the content of the parameters related to the experiment. R_bx represents the radius of the short axis of the bubble; R_by represents the radius of the long axis of the bubble. R_dx represents the radius of the short axis of the droplet; R_dy represents the radius of the long axis of the droplet. Point O in the figure represents the initial position of the bubble, which is also the position of the induced cavitation in the liquid droplet by the laser. The X axis and Y axis are the principal axes of the reference coordinate system in the process of droplet bubble collapse. In the experiment, the droplet is not completely spherical due to the effects of the pipeline and the droplet. After reasonable equivalence, the droplet and the bubble are regarded as ellipsoids. The dimensionless parameter radius ratio λ is defined as follows:

$$\lambda ={\left(\frac{R_{bx}^2{R}_{by}^{}}{R_{dx}^2{R}_{dy}^{}}\right)}^{\frac{1}{3}}=\frac{R_{b\max }}{R_{d\max }}$$

(9)

4. Experimental Results and Analysis

4.1. Case 1: No Splash

Figure 3 shows high-speed photographs of the process of cavitation bubble collapse within the droplet under the condition of a small radius ratio (λ = 0.581), defined as case 1. In case 1, the main characteristics of the bubble collapse and the droplet splash process are summarized as follows. During the first stage of the bubble collapse, the bubble wall is nearly spherical with the center of mass moving vertically upward, and the droplet surface being no splash. Among them, frames (1)–(8) show the first stage of the bubble collapse, in which the droplet shape hardly changes with the growth of the bubble. Frames (9)–(13) show the second stage of the bubble collapse, in which the centroid of the bubble moves obviously. Frames (14)–(17) show the third stage of the bubble collapse, in which the area of the vapor cloud gradually decreases. Frames (18)–(20) show the fourth stage of the bubble collapse, in which the area of the vapor cloud is further reduced and finally disappears. Under this condition, the droplet shape has little effect on the bubble collapse.

Figure 4 shows the outline of the process of cavitation bubble collapse within droplets under the condition of a small radius ratio (λ = 0.581). Different curves are used to show the distribution of the bubble wall and bubble centroid in the droplet at four moments. In addition, the black curve represents the corresponding droplet profile when the bubble is R_max. The outlines of the bubble wall at different times were obtained. In the following, no longer repeated introduction. When t = 50 μs, the bubble is squeezed by the liquid drop; it is close to an ellipsoidal shape. Additionally, the bubble keeps contracting in an ellipsoidal shape. When t = 150 μs, the cavitation tends to be spherical. In 50~100 μs, the displacement of the bubble wall is very small, and the bubble wall outline almost coincides. In 100~150 μs, the bubble wall displacement increases significantly, reflecting the gradual increase of the instantaneous velocity of bubble wall shrinkage. Points of different shapes represent the distribution of centroids at different times. In 50~150 μs, the centroids of the bubble slowly move upward.

Figure 5 shows the image of a cavitation bubble within a droplet in the X axial direction under the condition of a small radius ratio (λ = 0.581). In order to better show the interaction of droplet and the bubble in the collapse process, the high-speed photographic image is cut at equal intervals with the X axis as the symmetry line. And, the region containing the left and right end points of the droplets and bubbles are taken, and finally, the image is synthesized. The X axis is the actual length of the droplet bubble and the Y axis is the time. As shown in Figure 5, during the stages of bubble growth and the first collapse, the droplet boundary only fluctuated slightly. During the second collapse stage of the bubble, the droplet boundary returns to its original position. It can be seen that when the droplet bubble radius is relatively small, the influence of the bubble collapse process on the droplet boundary is very limited.

4.2. Case 2: Scattering Splash

Figure 6 shows high-speed photographs of the process of cavitation bubble collapse within the droplet under the condition of middle radius ratio (λ = 0.828), defined as case 2. In case 2, the main characteristics of the bubble collapse and the droplet splash process are demonstrated. During the first stage of the bubble collapse, the wall of the bubble remained basically spherical with the center of the bubble shifted significantly. And, the droplet surface showes a scattering splash. Among them, frames (1)–(10) show the first stage of the bubble collapse, in which the bubble wall shrinkage basically remains spherical, and there is no splash on the droplet surface. Frames (11)–(17) show the second stage of the bubble collapse, in which the center of the bubble moves up significantly, and a scattering splash appears at the droplet surface. Frames (18)–(22) show the third stage of the bubble collapse, in which the vapor cloud basically disappears, and the droplet scattering splash is further strengthened. Frames (23)–(40) show the development stage of droplet splashing, in which the scattered splashing water column gradually evolves into granular. With the oscillation of the droplet, the obvious deformation occurs at the junction of the droplet and the pipe. In this condition, the droplet shape is significantly affected by the bubble collapse, resulting in the formation of scattered droplet splash.

Figure 7 shows the outline of the process of cavitation bubble collapse within the droplet under the condition of the middle radius ratio (λ = 0.828). Different curves are used to show the distribution of the bubble wall and bubble centroid in the droplet at four moments, respectively. In addition, the black curve represents the corresponding droplet outline when the bubble is R_max. When t = 100 μs, the wall of the bubble is basically spherical, and the radius is R_max. The volume deformation of the droplet is caused by the expansion of the bubble volume. When t = 175 μs, the bubble is basically spherical, and the wall of the bubble close to the surface of the droplet shrinks significantly, while the wall of the bubble close to the pipe shrinks weakly. When t = 200 μs, the wall of the bubble shrinks unevenly and begins to deform gradually. When t = 225 μs, the bubble wall appears conical. In 100~175 μs, in the stage of bubble collapse, the contraction of the bubble wall is basically spherical. In 200~225 μs, during the stage of bubble collapse, the bubble wall gradually deforms and evolves from an irregular shape to a conical shape. In addition, compared with the case 1, the displacement of the center of the bubble in the case 2 is more obvious.

Figure 8 shows the image of a cavitation bubble within the droplet in the X axial direction under the condition of middle radius ratio (λ = 0.828). Because the effect of bubble collapse on droplet splash is significantly increased in the middle radius ratio, the time range is changed to 0~2000 μs. As shown in Figure 8, during the stages of bubble growth and the first collapse, a relatively obvious fluctuation occurred in the droplet surface. After the second stage of bubble collapse, the droplet boundary returns to its original position. On the other hand, the bubble rebound process causes the oscillation of the droplet surface, and the splash at the droplet boundary begins. It can be clearly seen that the number and the intensity of splash on the right side of the droplet are higher than those on the left side. This phenomenon may be due to the influence of the surface tension of the droplet surface during the stage of the bubble collapse, resulting in the loss of symmetry of the bubble oscillation and, thus, the loss of symmetry of the droplet splash. It can be seen that with the increase of the bubble radius ratio of the droplet, the influence of the bubble collapse dynamics on the droplet splash process is significantly enhanced. From the second stage of the bubble collapse, the droplet splash basically presents a scattering trend.

4.3. Case 3: Composite Splash

Figure 9 shows high-speed photographs of the process of cavitation bubble collapse within the droplet under the condition of a large radius ratio (λ = 0.926), and it is defined as case 3. In case 3, the main characteristics of the bubble collapse and droplet splash process are given. During the first stage of the bubble collapse, the wall of the bubble finally contracts into a mushroom shape. In the later stage of the bubble collapse, not only scattered splash appears on the droplet surface, but also an obvious flaky splash appears at the connection between the droplet and the pipe. In this paper, this kind of splash is defined as a “composite splash”. Among them, frames (1)–(11) show the first stage of the bubble collapse, in which the bubble wall shrinks from spherical to mushroom-shaped, and the droplet surface splashes significantly. Frames (12)–(20) show the second stage of the bubble collapse. During this process, the droplet surface oscillates violently, scattering splash increases, and flaky splash occurs at the same time. Frames (21)–(33) show the third stage of the bubble collapse. In this process, the vapor cloud moves into the pipe, and the compound splash effect of the liquid droplet is further strengthened. Frames (34)–(40) show the development stage of droplet splash, in which the scattered splash water column gradually evolves into granular, and the flaky splash disappears. Under the oscillation of the droplet, the droplet becomes flat. Compared with case 2, the effect of bubble collapse on droplet morphology is further enhanced under this condition, leading to the formation of a “composite splash”.

Figure 10 shows an outline of the process of cavitation bubble collapse within a droplet under the condition of a large radius ratio (λ = 0.926). Different curves are used to show the distribution of the bubble wall and bubble centroid in the droplet at four moments. In addition, the black curve represents the corresponding droplet outline when the bubble is R_max. When t = 100 μs, the wall of the bubble is basically ellipsoid, and the radius is R_max. The volume expansion of the bubble at this moment directly results in the droplet splash. When t = 200 μs, the bubble wall shrinks unevenly and begins to deform gradually. When t = 250 μs, the empty bubble wall finally takes on a fungiform shape. In t = 125~225 μs, the bubble wall shrinkage develops from spherical to irregular shape during bubble collapse. In t = 200~225 μs, the bubble wall gradually deforms during bubble collapse, from irregular shape to mushroom shape. In addition, compared with case 2, the displacement of the centroid of the bubble in case 3 is further increased.

Figure 11 shows the image of a cavitation bubble within the droplet in the X axial direction under the condition of a large radius ratio (λ = 0.926). Under this condition, the time is within 0~2000 µs in the figure. As shown in Figure 11, in the stages of bubble growth and the first collapse, the droplet surface oscillates obviously, and the droplet splash also occurs. After, the bubble rebound process causes the oscillation of the droplet surface, and the droplet boundary begins to splash violently. In case 3, the droplet boundary oscillates significantly with time, the number of droplet splashes increases, and the degree of splashes increases significantly, compared with case 1 and case 2.

5. Analysis of Three Droplet Splash Cases

5.1. Comparison of Three Cases under Different Radius Ratios

Figure 12 shows the variations of three cases of droplet splash under different radius ratios. The X axis in the figure is the radius ratio, and the Y axis is the equivalent radius of the maximum volume of the bubble. The red dot indicates case 1, which is no splash; The blue square represents case 2, which is a scattering splash; The orange triangle represents case 3, which is the “composite splash” formed by scattered splash and flaky splash. As shown in Figure 12, when λ ≤ 0.6, the bubble collapse process has little effect on the droplet splash. With the increase of λ, the influence of bubble collapse on droplet splash increases, resulting in a scattered splash. When λ ≥ 0.85, the bubble collapse further strengthens the droplet splash, leading to the “composite splash”.

Table 1. The summary of the characteristics of three cases of droplet splash.

Case	Phenomenon	λ	Rbmax (mm)	Collapse Time (μs)
1	The upward movement of the center of the bubble is not obvious, and the droplets have no splash.	0.463~0.581	1.221~1.549	50~100
2	The center of the bubble moves upward obviously, and there are scattering splashes on both sides of the droplet.	0.581~0.834	1.549~2.205	100~130
3	The bubble oscillates violently, the center of the bubble moves significantly, and the “composite splash” is formed on the surface of the droplet.	0.834~0.953	2.205~2.935	130~170

shows the summary of three splash cases of the droplet, and three splash cases are classified. Among them, the phenomenon, range of radius ratio λ, the maximum equivalent radius of bubble R_bmax, the collapse time of the bubble, and four main contents are displayed. As can be seen from the data in the table, the phenomenon of droplet splashing and bubble collapse is closely related with the radius ratio. For case 1 and case 3, the radius ratio gradually increases and the maximum equivalent radius of the bubble also gradually increases. Finally, it should be emphasized that the interval of the collapse time is gradually increasing. It can be seen that the radius ratio λ is a significant parameter influencing the dynamic properties of droplet cavitation bubble collapse within the droplet, which can be regarded as an effective control parameter.

5.2. Splash Displacement Comparison

Figure 13 shows the change rule with the time of the displacement difference d* between the left and right end points at the X axis of the droplet splash of different radius ratios. When λ = 0.581, it belongs to case 1, and the displacement difference basically remains unchanged; when λ = 0.828, it belongs to case 2, about t = 200 μs, the displacement difference increases significantly; when λ = 0.926, it belongs to case 3, the displacement difference increases sharply from the initial moment. It can be seen that the greater the radius ratio, the more significant impact on droplet splash. With the increase of the radius ratio, the droplet splash becomes intensive.

6. Theoretical Verification of Bubble Collapse Time

Figure 14 shows the comparisons between theoretical and experimental values of bubble collapse time under three cases. The variations of cavitation bubble within the droplet collapse versus time under three radius ratios is shown, respectively. From the scatter diagram, it can be observed that the larger the slope of the curve, the shorter the bubble collapse time.

The Y axis represents the bubble collapse time, and the X axis represents the instantaneous radius of the bubble. The dotted line represents the experimental results of three cases, and the solid line represents the numerical solutions of the bubble dynamics equations of three cases. From the comparisons of experimental values and numerical solutions, the theoretical results and experimental results in case 2 have the highest prediction accuracy, with an error rate of only 0.27%. The prediction error between theoretical results and experimental results in case 3 is the largest, with an error rate of 40.31%. In case 1, the prediction error rate of theoretical results and experimental results is 16.11%. It can be seen that with the change in the radius ratio of the bubble within the droplet, there is a significant difference in the prediction accuracy between the theoretical and the experimental results of case 1. The theoretical bubble collapse time in this paper has a better prediction for the cases of the middle values of the radius ratio of the bubble within droplet. The main reasons for different prediction accuracy could be explained as follows. Case 2 is closer to the ideal structure of a cavitation bubble within the droplet, so the prediction accuracy is the best. In case 1, the volume of the bubble is smaller than that of the droplet, and the surface tension of the droplet is not evenly distributed during the experiment, which leads to the uneven force of the bubble, and the collapse time is shortened. In case 3, the bubble volume is close to the droplet volume, and the mutual attraction between the bubble wall and the droplet surface is enhanced during the experiment, which induces the oscillation process of the bubble and the collapse time are prolonged. In addition, the large error under the condition of a large radius ratio may be the increase of the influence of the pipe on the collapse process under this condition.

7. Conclusions

In the present paper, the bubble dynamics model is improved considering the effects of liquid surface tension and viscosity. The dynamic process of cavitation bubble collapse within the droplet was carried out by a high-speed photography experimental platform. In addition, the numerical calculation of the theoretical collapse time of the bubble dynamic equation within the droplet was carried out to verify the experimental results. Several conclusions are obtained as follows:

• By adding the surface tension and viscosity terms, the bubble dynamics equation is further improved, and the cavitation bubble dynamics within the droplet equation model is proposed.
• With the increase of radius ratio, the influence of the bubble collapse process on the droplet is gradually enhanced. When there is a small radius ratio, the bubble collapse process has little effect on the droplet surface; when there is a medium radius ratio, the bubble collapse process causes a scattering splash on the droplet surface; when there is a large radius ratio, the bubble collapse process causes a “composite splash” on the surface of liquid droplets, including both scattered splash and flaky splash.
• There is an interaction between bubble collapse and droplet splash. Basically, with the increase of radius ratio, the center of the bubble moves more significantly compared with the shape before the first collapse of the bubble. Under the small radius ratio, the bubble contracted into a sphere. Under the middle radius ratio, the bubble contracted into a cone. Additionally, under the large radius ratio, the bubble contracted into a mushroom shape.
• Through numerical calculation, the time of bubble collapse within the droplet can be accurately predicted under the condition of a medium radius ratio.

In future work, we will further explore the bubble collapse and droplet splash dynamics under different droplet structures. In addition, the theory of droplet bubble collapse time is further improved.