
==== Front
Ultrason Sonochem
Ultrason Sonochem
Ultrasonics Sonochemistry
1350-4177
1873-2828
Elsevier

S1350-4177(24)00298-0
10.1016/j.ultsonch.2024.107050
107050
Original Research Article
Theoretical study on the movements of bubbles
Zhang Lingling a
Chen Weizhong wzchen@nju.edu.cn
b⁎
a School of Electronic and Information Engineering, Changshu Institute of Technology, Changshu 215506, China
b The Key Laboratory of Modern Acoustics, Ministry of Education, Institution of Acoustics, Nanjing University, Nanjing 210093, China
⁎ Corresponding author. wzchen@nju.edu.cn
30 8 2024
11 2024
30 8 2024
110 1070504 7 2024
12 8 2024
26 8 2024
© 2024 The Authors. Published by Elsevier B.V.
2024

https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The radial and translational motions of multiple interacting spherical bubbles are obtained using classical Newton mechanics. It is seen that bubbles not only move in straight line, but also in circular motion. The tracks of the bubbles show that the interactions among them include attractive, repulsive and dynamic equilibrium. There are three types of straight line corresponding to attraction, coexistence of attraction and repulsion and dynamic equilibrium, and two types of circular movement corresponding to attraction and dynamic equilibrium. The results can provide an explanation for cavitation chain and profile in cavitation field.

Keywords

Bubble movement
Interaction
Bubble dynamics
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pmc1 Introduction

Cavitation cloud can form different structures in acoustic field. It has been observed experimentally that bubbles can form stable clusters, namely ”bubble grapes” in a weak acoustic field [1], [2] and arrange in branch-like patterns that are referred to as acoustic streamers in a strong acoustic field [3], [4], [5]. Akhatov et al. conducted linear stability analysis and numerical simulation on the formation of acoustic streams [6]. Woodrow and Pinton analyzed the structure, which results from the same underlying bifurcation to wake instability [7]. Desjouy et al. observed cavitation bubbles located in the vicinity of this ring may exhibit a spiraling behavior around the pressure nodal line [8]. Other cavitation structures have also been observed in experiments such as bows, smokers and webs [9], [10], [11], [12], [13].

Various cavitation structures have indicated that the interaction between bubbles is not simply attraction and repulsion[14], [15], [16], [17], [18]. In recent years, many scholars are keen on the study of two bubbles[19], [20], [21], [22], [23], [24] or multiple bubbles[25], [26], [27], [28], [29], [30] to explore the interactions among bubbles in acoustic field. Ougz et al. found that in certain parameter ranges the force between the bubbles has a sign opposite to what would be expected on the basis of the linear theory of Bjerknes forces that if two bubbles are driven below or above their natural frequencies, they attract each other, whereas if the driving frequency is in between the two natural frequencies, the bubbles repel each other [19]. Doinikov theoretically concluded that two interacting bubbles can form a bound pair rather than collide and coalesce in acoustic field with pressure amplitude exceeding 1 bar [20]. The studies assumed that two bubbles move along their center line. In multi-bubble dynamics, Cui et al. mainly explored the impact of the bubble–bubble interaction on bubble pulsation in a vortical flow field, and indicated that bubble–bubble interaction influences the amplitudes and periods of bubble oscillations severely, but has small effects on bubble trajectories [25]. Bubbles move under the action of the primary Bjerknes forces in vortical flow field. Doinikov explored the precession of microbubbles about the pressure node plane under the primary Bjerknes force [26]. Chahine and Duraiswami discussed dynamical interactions in a multi-bubble cloud and indicated the influence of the mutual interaction on the dynamics of multiple bubble cloud [27].

In the above literatures, the influence of primary Bjerknes force on bubble movement is mainly highlighted, while the influence of secondary Bjerknes force on bubble movement is relatively small. Especially, the non straight line motion caused by the secondary Bjerknes force is studied relatively little in multi-bubble system. The characteristic of the formation of cavitation cloud is the complex motions of bubbles, and the movement is localized. Obviously, the formation of cavitation cloud cannot be explained by straight line motion, so it is essential to study the non straight line motions of bubbles. The present work is an investigation on the translations of multi-bubble on plane mainly subjected to the secondary Bjerknes force. Next, the coupled equations of radial and translational motions are derived using classical Newton mechanics. Then, the straight and circular motions of multiple bubbles are discussed in cavitation field. And there are conclusions in the last section.

2 Model of pulsations and translations

There are N spherical bubbles in incompressible fluid with liquid density ρ and sound velocity c. The geometry of the system is shown in Fig. 1. The radius Ri and location pi (i=1,2,…,N), of course, obey their dynamic equations including pulsation equations and translation equations. pi and pj (j=1,2,…,N) are the position vectors of the i-th and j-th bubbles with respect to the global Cartesian coordinates O-xy, respectively. A coordinate system Oi-riθi at the center of the i-th bubble Oi is established, and the position of point Q on this coordinate system is observed as (ri,θi). Usually, the ultrasonic field is described by the velocity potential ϕ. In incompressible liquid, the velocity potential satisfies the Laplacian equation ▵ϕ=0. The velocity potential of the multi-bubble system is nonspherical and approximately solvable, unlike spherical and exactly solvable for the single bubble system. Because bubble distribution is irregular and the system is asymmetric, the spherical harmonic function is used to express the velocity potential ϕ, then(1) ϕ=∑n=0∞∑m=-nnAnm(i)(t)ri-n-1+∑i=1,i≠jNBnm(ij)(t)rinYnm(θi,εi),

here Ynm(θi,εi) is the spherical harmonic function and Anm(i) and Bnm(ij) the time-related coefficients, and they are determined in Doinikov’s work [26] and degenerate into a two-dimensional case.Fig. 1 Geometry of the system.

The liquid pressure is obtained through the generalized Bernoulli equation(2) pL-p∞(t)ρ=-∂∂t-Ui·▽ϕ-12|∇ϕ|2,

where Ui is the translation speed of the i-th bubble center, and the pressure of far field is written as(3) p∞(t)=p0-pasinωt,

with p0 being the hydrostatic pressure, ω the circular frequency and pa the amplitude of the driving pressure. Substituting the velocity potential into Eq. (2), the liquid pressure at the i-th bubble wall pLWi can be obtained.

There are also some additional pressures on the interface between the liquid and gas, such as the pressures due to surface tension and viscosity and the pressure in gas inside bubble. Because the bubbles are assumed to be spherical balls, the interfacial pressures are spherically symmetric and can be expressed usually as(4) pIi=2σRi+4ηR˙iRi,i=1,2,…,N,

(5) pGi=p0+2σRi0Ri0Ri3γ,i=1,2,…,N,

where Ri and Ri0 are the time-varying radius and the ambient radius of the i-th bubble, respectively, the overdot denotes the time derivative, γ is the polytropic exponent of the gas within the bubbles, η is the liquid viscosity, and σ is the surface tension of the liquid.

The bubble wall is usually assumed to be geometrical surface with zero-mass, so that the mechanical equilibrium requires(6) ∫Si(pGi-pIi-pLWi)erdSi=0.

According to the Eq. (6), the dynamical equations of radial pulsations of N coupled spherical cavitation bubbles in the two-dimensional case have been derived, and in order to make more adequate for large forcing amplitudes, the result can be written as(7) 1-R˙icRiR¨i+32-R˙i2cR˙i2-Piρ1+R˙ic-RiρcdPidt=p˙i24-∑j=1,j≠iNRj2R¨j+2RjR˙j2Dij+Rj22Dij3(pi-pj)·(Rjp¨j+R˙jp˙i+5R˙jp˙j)-Rj34Dij3[p˙j·(p˙i+2p˙j)+3Dij2[p˙j·(pj-pi)][(pi-pj)·(p˙i+2p˙j)]],

where c is the speed of sound in liquid, Dij=|pj-pi|. The acoustic pressure Pi is taken in the form(8) Pi=p0+2σRi0Ri0Ri3γ-2σRi-4ηR˙iRi-p0+pasinωt.

Considering the liquid viscosity, the net force Fni and the viscous force Fηi have(9) Fni+Fηi=midUidt,i=1,2,…,N,

where Fni=-∫SipLndSi, and n is the outward unit normal to the bubble surface. mi=ρb4πRi3/3 is the mass of the gas bubble, and ρb is the gas density. Due to ρb/ρ≪1, the resultant force is about 0. According to the Eq. (9), the dynamical equations of translations of N coupled spherical cavitation bubbles have been derived(10) Rip¨i3+R˙ip˙i=∑j=1,j≠iNpi-pjDij3ddtRiRj2R˙j-Rj22Dij3[RiRjp¨j+(5RiR˙j+RjR˙i)p˙j-2RiR˙jp˙i]+3Rj2(pi-pj)2Dij5[(pi-pj)·[RiRjp¨j+(5RiR˙j+RjR˙i)p˙j-2RiR˙jp˙i]]+Fηi2πρRi2,

where the viscous drag Fηi is taken in Doinikov’s work [26] and an additional term appears on the right side of the Eq. (10) compared to the work.

3 Translations of multiple bubbles

Take two bubbles, three bubbles with the arrangement of equilateral triangle and four bubbles with the arrangement of square as examples, the average distance among bubbles is defined D‾ as 1N∑j>i,i≠jNDijand the results are obtained using the MATLAB based on the Eqs. (7), (10). The driving frequency in water is 20 kHz. The other parameters are set to p0=1 bar, ρ = 998 kg/m3,c=1500 m/s, σ=0.0725 N/m, γ=1.4and η=0.001 Pa.s.

3.1 Straight line motion

3.1.1 Two bubbles

The ambient radii of bubbles in acoustic field are not of the same size, and there is a size distribution. Here the ambient radii are below 10 μm and the initial distance of bubbles is 300 μm. Figure 2 shows the tracks of two bubbles and the average distance D‾. The arrows with solid line indicate the direction of bubble translation, the black curve corresponds to the track of the bubble 1, and the red curve corresponds to that of the bubble 2 in Fig. 2, Fig. 2, Fig. 2(e). It is seen that the interactions among them are manifested as attraction in Fig. 2(a), repulsion in Fig. 2(c), and dynamic equilibrium in Fig. 2(e) that is consistent with the Doinikov’s work [20]. The corresponding average distance D‾ in Fig. 2(b) decreases, increases in Fig. 2(d) and remains unchanged in Fig. 2(f).Fig. 2 The tracks of two bubbles and the average distance. (a)/(c)/(e) is the tracks and (b)/(d)/(f) is the average distance D‾. (a)-(b). R10=5μm,R20=5.6μm and pa=1.15 bar; (c)-(d). R10=1.8μm,R20=4μm and pa=1.32 bar; (e)-(f). R10=4.2μm,R20=5.2μm and pa=1.2 bar.

3.1.2 Multiple bubbles

Fig. 3, Fig. 4 show the trajectories of multiple bubbles and the corresponding average distance D‾, the blue curve represents the trajectory of the bubble 3, and the green curve represents that of the bubble 4. The red dots are the initial positions of each bubble. The bubbles move along their center line exhibiting attraction when the ambient radii of the bubbles are the same in Fig. 3, Fig. 4(a), and the average distance D‾ is getting smaller and smaller in Fig. 3, Fig. 4(b), which indicates that bubbles with in-phase pulsation are mutually attracted. In Fig. 3, Fig. 4(c), the bubbles move away from each other at first, and then two (three) of the three (four) bubbles are getting closer to each other, while one of the bubbles always stays far away from the other bubbles, which exhibiting the coexistence of attraction and repulsion. It can also be seen that the average distance D‾ increases first and then slowly decreases in Fig. 3, Fig. 4(d). The distance D12 in Fig. 3(d) and D14(D12,D24) in Fig. 4(d) both increase slowly and then decrease, and the distance D13 (D23) in Fig. 3(d) and D23 (D13,D34) in Fig. 4(d) both increase. When the phase difference between bubbles is large enough, the bubbles can repel each other [31]. There is another situation besides attraction and repulsion, where the bubbles move at a dynamical equilibrium separation distance neither attracting nor repelling in Fig. 3, Fig. 4(e), and the average distance remains unchanged in Fig. 3, Fig. 4(f), which is the same as Fig. 2, Fig. 2(f). The result can provide an explanation for cavitation streamers. In addition, the trajectory of bubble movement is also related to its initial position. For instance, the three bubbles shown in Fig. 3(e) are manifested as attraction for initial distance Dij0 = 90 μm.Fig. 3 The trajectories of three bubbles, the average distance and the distance of the centers of bubble i and bubble j. (a)/(c)/(e) is trajectories and (b)/(d)/(f) is the average distance D‾ and the distance Dij. (a)-(b). R10=R20=R30=4μm and pa=1.1bar; (c)-(d). R10=R20=2μm,R30=4μmand pa=1.3bar; (e)-(f). R10=6.7μm,R20=5.2μm,R30=4.2μmand pa=1.17bar.

Fig. 4 The trajectories of four bubbles, the average distance and the distance between the centers of bubble i and bubble j. (a)/(c)/(e) is trajectories and (b)/(d)/(f) is the average distance D‾ and the distance Dij. (a)-(b). R10=R20=R30=R40=4μm and pa=1.1 bar; (c)-(d). R10=R20=R40=2μm,R30=4μm and pa=1.3 bar; (e)-(f) R10=6.12μm,R20=3.695μm,R30=4.09μm and R40=5μm and pa=1.161 bar.

3.2 Circular motion

The motions of bubbles are complex and variable in cavitation field. Bubbles not only move in straight line, such as attracting, repelling and equidistant translation, but also in circular motion. Taking three and four bubbles as an example, the trajectories and the corresponding average distance D‾ are shown in Fig. 5, Fig. 6. The hollow circle represents time t1, the hollow triangle represents time t2, and t1 occurs earlier. It can be seen that two of the bubbles collide with each other in the process of circling in Fig. 5, Fig. 6(a), and the average distance D‾ is getting smaller and smaller in Fig. 5, Fig. 6(b). The bubbles can maintain stable dynamic equilibrium while circling counterclockwise in Fig. 5(c) and clockwise in Fig. 6(c), neither attracting nor repelling, and the average distance D‾ fluctuates within a very small range in Fig. 5, Fig. 6(d), which is consistent with straight line motion.Fig. 5 The trajectories of three bubbles and the average distance. (a)/(c) is trajectories and (b)/(d) is the average distance D‾. (a)-(b). R10=5.8μm,R20=4.8μm,R30=4.1μm and pa=1.16 bar. (c)-(d). R10=5.926μm,R20=4.98μm,R30=4.08μm and pa=1.1655 bar.

Fig. 6 The trajectories of four bubbles and the average distance. (a)/(c) is trajectories and (b)/(d) is the average distance D‾. (a)-(b). R10=4.2μm,R20=5.92μm,R30=3.75μm,R40=5μm and pa = 1.16 bar; (c)-(d). R10=4.2μm,R20=6μm,R30=3.7μm,R40=5μm and pa = 1.161 bar.

In the two-bubble system, bubbles only move in a straight line by the secondary Bjerknes force. Different from the two-bubble system, bubbles can exhibit circular motion in three-bubble or four-bubble system. Bubbles with different initial radii have phase differences, which determines the polarity of the secondary Bjerknes force. The secondary Bjerknes forces are not a pair of action-reaction forces. In the linear case, the secondary Bjerknes forces acting on two bubbles are equal in magnitude and opposite in direction, similar to action and reaction. However, in the case of nonlinearity, the secondary Bjerknes forces acting on two bubbles are not necessarily equal in magnitude and opposite in direction. The synergistic effect of other bubbles leads to circular motion of the bubble.The study of circular motion can provide an explanation for the formation of cavitation cloud, and the non straight line motion is an important factor for its formation.

4 Dependancy on the ambient radius

Taking three bubbles as an example, the correlation between bubble translation and the ambient radius is discussed. Figure 7 shows the distribution of the translations of three bubbles in ambient radius space. The red area indicates that the average distance D‾ is decreasing, that is, ΔD‾<0, the blue area represents that the D‾ is a constant,and the yellow area shows ΔD‾>0. The numbers 1 and −1 only represent greater than 0 and less than 0, without any meaning. It can be seen that both red and blue areas exist, with a high proportion of red area but a lack of yellow area at pressure amplitude 1.17 bar in Fig. 7(a). The red area corresponds to the bubble attraction, such as Fig. 3, Fig. 5(a). The D‾ of the blue area is a constant, which can be a constant straight line motion or constant circular motion, corresponding to Fig. 3, Fig. 5(c). In Fig. 7(b), there are yellow and red areas at pressure amplitude 1.3 bar, indicating that the repulsion between bubbles exists in yellow area, corresponding to Fig. 3(c), and the red area still indicates bubble attraction.Fig. 7 The distribution of the translations of three bubbles. (a) pa = 1.17 bar and R10=6.2μm and (b) pa = 1.3 bar and R10=4μm.

5 Conclusion

In this paper, the radial and translational motions of multiple interacting spherical bubbles are obtained to simulate the translations of bubbles in the framework of classical Newton mechanics. Using the model, the translations of multiple bubbles have been made. The bubbles not only move in straight line but also in circular motion. The tracks of the bubbles show that the interactions among them include attractive, repulsive and dynamic equilibrium. In the case of straight line motion, there are three type including attraction, coexistence of attraction and repulsion and dynamic equilibrium. In circular motion, the bubbles can also maintain dynamic equilibrium, neither attracting nor repelling. In addition, the bubbles also attract in the process of circling. The translations of bubbles in the ambient radius space are also discussed. The results can provide an explanation for cavitation chain and profile in the ultrasonic cavitation field.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (No. 12074185).
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