Compressible effect on the mutual interaction of two cavitation bubbles in radial oscillations and translational motion

Abstract

In this study, we considered the compressible effect on the mutual interaction of two cavitation bubbles by correcting the sound field emitted by one bubble in the radial equations of the other bubble to first order in the Mach number of the flow, and the effect is represented by the incident wave acting on bubbles. The results illustrates that the incident wave can enhance the resonance response at the redistributed resonance frequency, which leads to an increase in radial acceleration and the secondary Bjerknes force, and rapid approach of bubbles. Furthermore, the influence of incident wave on the interaction of bubbles driven at lower frequencies is more significant, due to resonance enhancement caused by the proximity of natural frequencies and frequency multiplications of the external sound field. Our findings reveal that the compressible effect is not only critical to interaction in radial oscillations, but also in translational motion.

1. Introduction

Cavitation bubbles undergo radial oscillations and translational motion in an acoustic field [1], [2], controlled by acoustic and hydrodynamics forces, leading to the mutual interaction between bubbles discovered by Bjerknes [3]. The sound field emitted by bubbles leads to mutual interaction in the radial oscillations [4], and its gradient is the acoustic force known as the secondary Bjerknes force [5], [6], [7], affecting the translational motion of bubbles and their distance.

Based on the linear theory and the assumption of incompressible viscous flow, the secondary Bjerknes force is repulsive if the driving frequency is between the natural frequencies of the two bubbles, otherwise the force is attractive [8], [5], which illustrates that the radial oscillations have an impact on the secondary Bjerknes force. Oguz and Prosperetti [9] theoretically analyzed the nonlinear interaction of two bubbles based on the assumption of inviscid and incompressible fluid, and found the sign in the secondary Bjerknes force opposite to that in the linear theory, but the distance between two bubbles is much higher than the characteristic size of bubbles. Under different initial distance between two bubbles, Mettin et al. [4] investigated the relation between radial oscillations of two interacting cavitation bubbles and secondary Bjerknes force by equations coupled by bubble pressure emission terms, and the results reveal that the magnitude and sign of the secondary Bjerknes force depends on the mutual interaction in radial oscillations of two bubbles, but no analysis was conducted on the translational motion. The secondary Bjerknes can affect the variation of the distance between two bubbles, so it is necessary to investigate the influence of the radial oscillations of two bubbles on the translational motion. Under the assumption of incompressible fluid, Ida [10] studied the characteristic frequency of two bubbles with a coupled-oscillator model, which was considered to affect the mutual interaction in the radial oscillations, but its influence on the secondary Bjerknes force and translational motion was not discussed. Considering the fluid compressibility, Doinikov [11] corrected the equations for radial oscillations derived from the Lagrangian formalism based on the Keller-Miksis’ approach [12], and constructed the translational equations making direct calculation of the translational motion in a strong acoustic field possible. As for the radial equation of a bubble, the acoustic radiation induced by radial oscillations of the corresponding bubble is contained, and the compressible effect on the sound field emitted by the other bubble is neglected, so the terms related to the other bubble are valid to zero order in the Mach number of the flow according to the expressions for the pressure field constructed by Lezzi and Prosperetti [13]. Recently, the secondary Bjerknes force was obtained based on the nonspherical perturbations, and the influence of the bubble deformation on the secondary Bjerknes force was studied [14]. Furthermore, the equation coupling radial oscillations, translational motion and deformation was proposed based on the velocity potential and perturbation theory, and the influence of bubble motion on the secondary Bjerknes force was obtained [15]. However, in the above studies on the deformation of interacting bubbles, the sound field emitted by one bubble acting on the other bubble is also at zero order.

It implies that the mutual interaction in radial oscillations is critical to translational motion, so it is necessary to consider the compressible effect on the interaction by correcting the sound field emitted by bubbles to first order in the equation for radial oscillations. Based on the corrected radial equations and spectral analysis, the compressible effect on the mutual interaction of two bubbles in radial oscillations and translational motion will be studied in this paper. The results will be of interest for understanding and modeling bubble dynamics in an acoustic field, and further promote technological applications such as pollutant degradation, polymer synthesis, cleaning, nanoparticle production, food science, and medical applications [16].

2. Mathematical model for two interacting bubbles

Considering two cavitation bubbles undergoing radial oscillations and translational motion in an acoustic field, which are controlled by both acoustic and hydrodynamic forces, and the forces are mainly along the y axis. Suppose that the distance between the two bubbles is larger than their radius so that they remain spherical at all time. As shown in Fig. 1, the relation of the velocity potential with the boundary conditions at the bubble surfaces is [17]:

$$\mathrm{\nabla }\phi =a_i^{\prime }+y_i^{\prime }\cos {\theta }_i\text{at}r=a_i(t)i=1,2,$$ (1)

where φ is the velocity potential, $\mathrm{\nabla }\phi$ is the gradient of velocity potential, a_i(t) and y_i(t) are the time-dependent radius and position of the center of the ith bubble, respectively, and the prime detonates the time derivative. It is generally assumed that the velocity potential satisfies the Laplace equation $\mathrm{\Delta }\phi =0$[11], to simplify the expression of velocity potential φ and construct the Lagrangian function of the system L=T−U. The kinetic energy T is determined by the velocity of the host liquid, and the potential energy U of the liquid is represented as

$$U=-\frac{4}{3}\pi p_1{a}_1^3-\frac{4}{3}\pi p_2{a}_2^3-y_1{F}_{ey1}-y_2{F}_{ey2}$$ (2)

where p_i is the pressure on the surface of the ith bubble, and F_eyi is the external force acting on the ith bubble.

Fig. 1.

Geometry of the two interacting bubbles.

With the Lagrangian equations

$$\frac{d}{\mathit{dt}}\left(\frac{\partial L}{\partial q_i^{\prime }}\right)-\frac{\partial L}{\partial q_i}=0$$ (3)

where the generalized coordinates $q_i$ and velocities $q_i^{\prime }$ are taken as $a_i$, $y_i$, $a_i^{\prime }$ and $y_i^{\prime }$, and the equations for radial oscillations and translational motion are obtained. The radial equations are similar to the Rayleigh-Plesset equations, where the compressibility of the flow is neglected. For large forcing amplitudes, the velocity of radial oscillation increases and the wave effect is obvious, so the Keller-Miksis model [12] is used to replace the terms representing ith bubble motion in the ith bubble radial equations [11], expressed as follows:

$$\begin{array}{c}\left(1-\frac{a_i^{\prime }}{c}\right)a_i{a}_i^{″}+\left(\frac{3}{2}-\frac{a_i^{\prime }}{2c}\right){\left(a_i^{\prime }\right)}^2-\frac{1}{{\rho }_l}\left(1+\frac{a_i^{\prime }}{c}\right)p_i-\frac{a_i}{{\rho }_{l}c}\frac{d{p}_i}{\mathit{dt}}\\ =\frac{{\left(y_i^{\prime }\right)}^2}{4}-\frac{a_j^2{a}_j^{″}+2a_j{\left(a_j^{\prime }\right)}^2}{D}+\frac{a_j^2\left(y_i^{\prime }{a}_j+a_j{y}_j^{″}+5a_j^{\prime }{y}_j^{\prime }\right)}{2D^2}-\frac{a_j^3{y}_j^{\prime }\left(y_i^{\prime }+2x{y}_j^{\prime }\right)}{2D^3}\\ i=3-j,i=1,2\end{array}$$ (4)

where c is the sound velocity, D is the distance between two bubbles, ρ_l is the density of the liquid. However, the compressible flow related to the jth bubble radial oscillations is not considered in the ith bubble radial equations. As discussed in the multiple-bubble interaction models [18], the flow field should contain contributions from both two bubbles, but where the compressible effect is also not considered in the flow field.

Doinikov [11] considered adiabatic compression of an ideal gas within the bubble and the influence of acoustic pressure field $p_{\mathit{ex}}\left(t\right)=p_a\sin \left(\omega t\right)$, and obtained the formula of pressure p_i on the ith bubble surface:

$$p_i=\left(p+\frac{2\sigma }{a_{i0}}\right){\left(\frac{a_{i0}}{a_i}\right)}^{3\gamma }-\frac{2\sigma }{a_i}-\frac{4\mu a_i^{\prime }}{a_i}-p_0-p_a\sin \omega t$$ (5)

where a_i0 is the equilibrium radius and p₀ is the hydrostatic pressure. However, the incident wave acting on the ith bubble caused by the compressible effect related to the jth bubble radial oscillations is neglected. According to the Bernoulli equation, the pressure p in the fluid is related to velocity potential φ of the fluid:

$$\frac{\partial \phi }{\partial t}+\frac{1}{2}{\left|\mathrm{\nabla }\phi \right|}^2+gy+\frac{p}{{\rho }_l}=0$$ (6)

where g is the gravitational acceleration. And combined with the Laplace equation, which yields

$$\frac{\partial \left({\phi }_1+{\phi }_2\right)}{\partial t}+\frac{1}{2}{\left|\mathrm{\nabla }\left({\phi }_1+{\phi }_2\right)\right|}^2+gy+\frac{p}{\rho }=0$$ (7)

where φ₁ and φ₂ is the potential of two bubbles. The equation above indicates that the pressure p_i is related to the motion of two bubbles. Correspondingly, the terms on the left-side of Eq. (4) not containing pressure p_i and sound velocity c represent the influence of the ith bubble motion on the pressure p_i, and the terms on the right-side of the Eq. (4) represent the effect of the jth bubble motion on the pressure p_i. The relation of the two bubbles motion with the pressure p_i on the ith bubble surface in the Eq. (4) is consistent with the perturbation series constructed by Lezzi and Prosperetti [13]:

$$h=h_0=\frac{p_i}{\rho }=a_i^2{a}_i^{″}+\frac{3}{2}{a}_i{\left(a_i^{\prime }\right)}^2+\frac{{\left(a_j^2{a}_j^{\prime }\right)}^{\prime }}{D}$$ (8)

where the term of the order of D⁻⁴ is omitted, h is the enthalpy in the liquid, h₀ is its zero-order approximation and the expression for the pressure p_i is valid to zero order in the Mach number. This zero-order approximation focuses on the inertia effect and neglects the compressibility of flow, which is corresponding to the relation of kinetic energy with the potential energy in the Lagrangian system L=T−U. In addition, those terms containing sound velocity c in the Eq. (4) represent the compressible effect related to the ith bubble, but the terms representing the compressible flow induced by the jth bubble radial oscillations is missing in the Eq. (4). Referring to the perturbation solution of the pressure p to order one, we find the pressure p_i on the surface of the ith bubble:

$$h=h_0+\epsilon h_1=\frac{p_i}{{\rho }_l}=a_i^2{a}_i^{″}+\frac{3}{2}{\left(a_i^{\prime }\right)}^2-\frac{{\left(a_i^2{a}_i^{\prime }\right)}^{″}}{c}+\frac{{\left(a_j^2{a}_j^{\prime }\right)}^{\prime }}{D}-\frac{{\left(a_j^2{a}_j^{\prime }\right)}^{″}}{c}$$ (9)

where ε is the perturbation parameter, and h₁ is of the order of the ε⁻¹. The one-order sound field emitted by the jth bubble is represented by the fourth and fifth term in the Eq. (9), and the fifth term on the right-side of the Eq. (9) was considered as the incident wave acting on the ith bubble, which represents the compressible effect on the mutual interaction. By moving the fifth term in right-side of the Eq. (9) to the left side, the formula of pressure p_i containing the incident wave is obtained:

$$p_i=\left(p_0+\frac{2\sigma }{a_{i0}}\right){\left(\frac{a_{i0}}{a_i}\right)}^{3\gamma }-\frac{2\sigma }{a_i}-\frac{4\mu a_i^{\prime }}{a_i}-p_0-p_a\sin \omega t+\frac{\rho {\left(a_j^2{a}_j^{\prime }\right)}^{″}}{c}$$ (10)

Upon substitution of pressure p_i in the Eq. (4) into the rewritten formula (Eq. (10)), we find the radial equations considering the compressible effect on the interaction. The fourth term and the fifth term on the right-side of the Eq. (9) play a similar role in the radial oscillations to the secondary Bjerknes force in the translational motion, because which is defined as the force between two bubbles due to the ‘‘secondary’’ sound fields emitted by other bubbles. To avoid the appearance of term containing $a^{‴}$, since the error in the Eq. (10) is with one-order Mach accuracy, it is sufficient to approximate the term ${\left(a^2{a}^{\prime }\right)}^{″}$ to order one. The approximation of term ${\left(a_i^2{a}_i^{\prime }\right)}^{″}$ was obtained from the dimensionless Rayleigh-Plesset equation [13], which are the terms containing sound velocity c, and the approximation of term ${\left(a_j^2{a}_j^{\prime }\right)}^{″}$ has the same form, except that subscript changes from i to j.

The dynamic behavior of bubble is related to radial oscillations, as well as the translational motion. The translational equations are derived from the Lagrangian equations, which is influenced by radial oscillations and determined by external acoustic force and hydrodynamics force [18]:

$$\begin{array}{c}\frac{a_i{y}_i^{″}}{3}+a_i^{\prime }{y}_i^{\prime }+{\left(-1\right)}^{i+1}\frac{1}{D^2}\frac{d}{\mathit{dt}}\left(a_i{a}_j^2{a}_j^{\prime }\right)-\frac{a_j^2\left(a_i{a}_j{y}_j+a_j{a}_i^{\prime }{y}_j^{″}+5a_i{a}_j^{\prime }{y}_j^{\prime }\right)}{D^3}=\frac{F_{\mathit{eyi}}}{2\pi {\rho }_l{a}_i^2}\\ i=3-j,i=1,2\end{array}$$ (11)

since the translational velocity is small compared with the sound velocity and the wave effect is not significant. The terms containing $a_j$, $a_j^{\prime }$ and $a_j^{″}$ represents the influence of jth bubble radial oscillations on the translational motion of the ith bubble, and are the effect of secondary Bjerknes force on the translational motion, considered as the internal force in the Lagrangian system. Due to the gradient of the incident wave being zero, the Eq. (11) remains the same after the correction in the Eq. (10). Therefore, the incident wave has an indirect impact on the translational motion by directly affecting the mutual interaction in the radial oscillations. Furthermore, the Eq. (11) presents that the external force F_eyi affects the translational motion, which includes viscous drag F_d, buoyancy force F_buoy and primary Bjerknes force F_p.

If the translational Reynolds numbers $R{e}_t\gg 1$ or the oscillation Reynolds numbers $R{e}_r\ll 1$, the viscous drag F_d [19] is written as:

$$F_d=-12\pi \mu a{y}^{\prime }-\frac{2}{3}\pi \rho \frac{d}{\mathit{dt}}\left(a^3{y}^{\prime }\right)$$ (12)

Whereas if translational Reynolds numbers $R{e}_t\ll 1$ or the oscillation Reynolds numbers $R{e}_r\ll 1$, one obtains

$$\begin{array}{c}{F}_d=-4\pi \mu a{y}^{\prime }-8\pi \mu {\int }_0^t\exp \left(\frac{9\mu }{\rho }{\int }_{\tau }^{t}a{\left(t^{\prime }\right)}^{-2}d{t}^{\prime }\right)\\ \times \text{erfc}\left[\sqrt{\frac{9\mu }{\rho }{\int }_{\tau }^{t}a{\left(t^{\prime }\right)}^{-2}}\right]\frac{d\left[a(\tau )y^{\prime }(\tau )\right]}{d\tau }d\tau -\frac{2}{3}\pi \rho \frac{d}{\mathit{dt}}\left(a^3{y}^{\prime }\right)\end{array}$$ (13)

where ρ is the density of water.

The buoyancy force F_buoy is given by

$$F_{\mathit{bouy}}=\frac{4}{3}\pi g{a}^3\left({\rho }_l-{\rho }_g\right)$$ (14)

where ρ_g is the density of the gas within the bubble.

The primary Bjerknes force F_p [20] is expressed as:

$$F_p=\frac{4}{3}\pi k{p}_a{a}^3\sin \left(\mathit{kd}\right)\sin \left(\omega t\right)$$ (15)

where k is the wave number, d is the distance between the center of the bubble and the nearest pressure antinode in the y-direction.

3. Results and discussion

A bubble observation system similar to that introduced by Jiao [21], [22] and Ashokkumar [23] was applied in the capturing the cavitation bubbles. All devices, degassing process and image processing method are the same to our previous report [24]. The experimentally observed translational movements of two bubbles in an acoustic field and the corresponding numerical simulation results are presented in the Fig. 3. The numerical simulation results are calculated by two models: the first model considers the compressible effect on the mutual interaction and contains the first-order sound field emitted by bubbles, including Eq. (4) and Eqs. (10)−(15), and the second neglects the compressible effect, including Eqs. (4)−(5) and Eqs. (11)−(15). (In the following, the model introduced first is called “the first-order model”, and the other is called “the zero-order model”.) The amplitude of acoustic pressure p_a is 30 kPa, and the frequency is 20 kHz, 40khz, 60 kHz and 80 kHz respectively. The sound velocity c is 1480 m/s, density of water is 998 kg/m³, hydrostatic pressure p₀ is 101.3 kPa, adiabatic exponent γ is 1.4, surface tension σ is 0.0725 N/m and viscosity of the liquid μ is 0.001 kg/(m·s). It should be noted that, the multiple experiments were conducted and produced similar results at 20 kHz, 40khz, 60 kHz and 80 kHz respectively.

Fig. 3.

Time-averaged secondary Bjerknes force acting on bubble 1 at (a) 80 kHz, (b) 60 kHz, (c) 40 kHz and (d) 20 kHz. The time-averaged secondary Bjerknes force calculated from the first-order model (red lines). The time-averaged secondary Bjerknes force calculated from the zero-order model (blue lines).

The Fig. 2 presents the experimentally observed and calculated distance between two bubbles along the y axis. The bubbles are with similar size (provided in details later), driven under the same acoustic pressure (p_a = 30 kPa) but different frequencies (20 kHz, 40 kHz, 60 kHz and 80 kHz). Under different frequencies, at the beginning Bubble 1 is near the pressure antinode and Bubble 2 is 300 μm far away from the pressure antinode. All the initial condition of the experiments is the same as those in the numerical simulation. In the Fig. 2(a) and (b), initially, the radii of bubble 2 away from the pressure antinode is 15 μm, while the radii of bubble 1 near the pressure antinode is 17 μm. Driven by the acoustic and hydrodynamic forces, bubble 2 approaches the pressure antinode and thus the distance between two bubbles gradually decreases. Although the deviation is not significant, the results calculated from the first-order model are closer to the experimental results than those calculated from the zero-order model. The displacement of two bubbles driven at 60 kHz are shown in the Fig. 2(c) and (d), with radii of 17 μm and 15.5 μm. Although the variance between calculation results from two models are still not significant, those from the first-order model are also closer to the experimental results. The displacement of two bubbles driven at 40 kHz (with radii of 17 μm and 16.5 μm) and 20 kHz (with radii of 17 μm and 17.7 μm) is presented in the Fig. 2(e), (f), (g) and (h), respectively. And the calculated results from the first-order model are consistent with the experimental results, but there is an evident difference between those calculated from the zero-order model and the experimental results. It is found that the compressible effect is not significant at 60 kHz and 80 kHz, but is important to the bubbles driven at 20 kHz and 40 kHz.

Fig. 2.

Distance between two bubbles at (a) 80 kHz, (c) 60 kHz, (e) 40 kHz and (g) 20 kHz. Captured images of two bubbles at (b) 80 kHz, (d) 60 kHz, (f) 40 kHz and (h) 20 kHz. Experimentally observed distance between two bubbles (marked by circles). Calculated distance from the first-order model (solid lines). Calculated distance from the zero-order model (dashed lines).

To proceed with the compressible effect on mutual interaction in the translational motion, we calculate the time-averaged secondary Bjerknes force F_Bi(t)

$$F_{\mathit{Bi}}\left(t\right)=\frac{{\int }_0^t{\left(-1\right)}^i\frac{1}{D^2}\frac{d}{\mathit{dt}}\left(a_i{a}_j^2{a}_j^{\prime }\right)+\frac{a_j^2\left(a_i{a}_j{y}_j+a_j{a}_i^{\prime }{y}_j^{″}+5a_i{a}_j^{\prime }{y}_j^{\prime }\right)}{D^3}dt}{t}$$ (16)

which is shown in the Fig. 3. The Fig. 3(a) and (b) demonstrates the time-averaged secondary Bjerknes force calculated from the first-order model is slightly higher at 80 kHz and 60 kHz, so the calculated distance between two bubbles from the corresponding model decreases faster, as shown in the Fig. 2(a) and (c). At 40 kHz and 20 kHz presented in the Fig. 3(c) and (d), the time-averaged secondary Bjerknes force calculated from the first-order model is significantly higher, especially as the distance between bubbles further decreases over time. Comparing the formula of pressure p_i applied in two models, Eq. (5) and Eq. (10), it was considered that the incident wave increases the pressure p_i on the surface of the ith bubble, and leads to the increase in the radial acceleration based on the relation of pressure and radial acceleration in the Eq. (4). The growth in the radial acceleration contribute to the amplitude of radial velocity and radius, and according to the Eq. (11) and Eq. (16), we consider that the incident wave causes the enhancement of the time-averaged secondary Bjerknes force, further leading to the distance between two bubbles decreases more rapidly. It illustrates that the fluid compressibility can indirectly affect the secondary Bjerknes force and approach of two bubbles, so which is essential to mutual interaction in radial and translational motion.

From the calculated acceleration of bubble 1 shown in the Fig. 4, it can be seen that the amplitude of the acceleration calculated from the first-order model is higher, as we analyzed previously. The increase at 80 kHz and 60 kHz is not obvious, as presented in Fig. 4(a) and (b), while the enhancement at 40 kHz and 20 kHz is significant. Furthermore, at 80 kHz and 60 kHz the difference between calculated acceleration of two models is relatively obvious in the early stage compared to that in the later stage, and decreases over the time due to the viscous force, which causes the energy dissipation and weaken the influence of incident wave. On the contrary, at 40 kHz and 20 kHz the difference increases over the time, shown in the Fig. 4(c) and (d), indicating that the influence of the incident wave is dominant. Therefore, at 40 kHz and 20khz the dominant incident wave causes the enhancement of the amplitude of radial acceleration, and further results in the increase in the secondary Bjerknes force. It demonstrates that the compressible effect on the mutual interaction of two bubbles varies under different frequencies.

Fig. 4.

Calculated radial acceleration of bubble 1 at (a) 80 kHz, (b) 60 kHz, (c) 40 kHz and (d) 20 kHz. The radial acceleration of bubble 1 calculated from the first-order model (red lines). The radial acceleration of bubble 1 calculated from the zero-order model (blue lines).

The Lomb-Scargle periodogram (LSP) is a method for detecting and characterizing the periodic signals in unevenly-sampled data, and it can extract the weak periodic signals and weaken the false signals generated by the non-uniformity of the temporal sequences [25], [26]. To further investigate the difference in the compressible effect on the interaction between bubbles driven at different frequencies, we used LSP to extract the radial oscillations frequency, presented in the Fig. 5, where the vertical grid lines represent the frequency of external sound field and its frequency multiplications. Except in Fig. 5(e) and (g), most of the power spectrum peak concentrates at the vertical grid lines, reflecting the dominance of external sound field, and with a small portion concentrate at the natural frequency f, which is related to the equilibrium radius a₀ [4]. The interaction of two bubbles depends on the natural frequency [10], and at the frequency the consistency of power spectrum peak of two bubbles represents the resonance strength, which affects the secondary Bjerknes force. The Fig. 5(a) and (b) present that the natural frequencies of bubble 1 and bubble 2 are 197.9 kHz and 225.4 kHz, respectively. The product of the natural frequencies and corresponding equilibrium radius is 3.38 m/s and 3.36 m/s, consistent but higher than 3 m/s [4], which might be related to the compressible effect and the viscous drag. The resonance response at two natural frequencies in the Fig. 5(a) is more obvious that in the Fig. 5(b), due to effect of incident wave, so the amplitude of radial acceleration and secondary Bjerknes force is higher. And the resonance response of two bubbles is also obvious at 278.5 kHz, which is corresponding to the subtranstion frequencies (STFs) [10].

$$f_{\mathit{iSFTs}}^2=\frac{f_{j0}^2}{1-a_{j0}/D}$$ (17)

where $f_{\mathit{iSFTs}}$ is the subtransition frequency of the ith bubble and $f_{j0}$ is the natural frequency of the jth bubble. The calculated subtransition frequency of bubble 1 is 236.5 kHz, but the corresponding resonance response is not obvious as the frequency is close to a frequency multiplication of external sound field (240 kHz). The subtransition frequencies are also related to mutual interaction of bubbles, so at which the consistency of power spectrum peak of two bubbles represents the same significance as the natural frequency. Both the external sound field and sound field emitted by bubbles contribute to the resonance response near the 240 kHz, and the power spectrum peak at 240 kHz is higher than that others except for that at the frequency of external sound field. The same condition also occurs for the bubbles driven at 60 kHz (Fig. 5(c) and (d)), which is not be elaborated further. As can be seen in Fig. 5(e), the power spectrum peak at 198.4 kHz is higher than others except for that at the frequency of the external sound field. This frequency (198.4 kHz) is between a frequency multiplication of external sound field (200 kHz), the natural frequency of bubble 1 (calculated before, 197.9 kHz), and bubble 2 (the calculated results is 204.0 kHz). It reveals that the resonance response at natural frequencies and at a frequency multiplication of the external sound field are superimposed at 198.4 kHz, which is between the natural frequencies and frequency multiplication. Different from the resonance response of bubbles driven at 80 kHz and 60 kHz, at 40 kHz, except for the resonance response at the frequency of the external sound field, all other the resonance frequencies deviate the frequency multiplications of the external sound field. It indicates that when not at the frequency of the external sound field, the incident wave dominates instead of other external force. However, the Fig. 5(f) shows that the second highest power spectrum peak (at 200 kHz) is 2.7 dB/Hz lower than the second highest one in the Fig. 5(e) (at 198.4 kHz). It was concluded that when radius of two bubbles is relatively close (bubbles with radius of 17 μm and 16.5 μm), the incident wave representing compressible effect can lead to the redistribution of resonance frequencies, as discussed previously, which makes the amplitude at natural frequencies (197.9 kHz and 204.0 kHz) and the amplitude at a nearby frequency multiplication (200 kHz) of the external sound field superimposed at the redistributed resonance frequency (198.4 kHz). Comparing Fig. 5(e), (f), (g) and (h), it can be found that the influence of incident wave on the resonance response of bubbles driven at 20 kHz is more significant. Therefore, when the driving frequency is lower the interval between the frequency multiplications of external sound field is shorter, making frequency multiplications of the external sound field closer to natural frequencies, and ultimately leading to a more pronounced superposition of resonance responses. As the analysis of resonance response at 40 kHz, the incident wave also causes the redistribution of resonance frequences, the higher power spectrum peak ranges from 157.8 kHz to 227.1 kHz in the Fig. 5(g), while those ranges from 80 kHz to 100 kHz in the Fig. 5(h). However, the higher power spectrum peak ranges from 80 kHz to 100 kHz, deviating significantly from the natural frequencies, can’t cause the obvious growth in the radial acceleration and secondary Bjerknes force, as presented in the Fig. 4(d) and Fig. 3(d). It was considered that an increase in the resonance amplitude at the frequency close to the natural frequencies can contribute to the obvious improvement of radial acceleration, secondary Bjerknes, and rapid approach of two bubbles.

Fig. 5.

Spectral analysis of radial oscillations of bubbles at (a-b) 80 kHz, (c-d) 60 kHz, (e-f) 40 kHz and (g-h) 20 kHz. Spectral analysis of calculated radial oscillations of bubble 1 (red lines). Spectral analysis of calculated radial oscillations of bubble 2 (blue lines). Spectral analysis of radial oscillations calculated from the first-order model (a, c, e, g). Spectral analysis of radial oscillations calculated from the zero-order model (b, d, f, h).

4. Conclusion

In this study, we considered the compressible effect on the mutual interaction of two bubbles by correcting the sound field emitted by one bubble in the radial equations of the other bubble to first order in the Mach number of the flow, and the effect is represented by the incident wave acting on bubbles. Compared to the zero-order model, the calculated translational motion from the first-order model is in better agreement with the experimentally observed results. The incident wave results in a redistribution of resonance frequencies, and makes the amplitude at natural frequencies and the amplitude at a nearby frequency multiplication superimposed at the redistributed resonance frequency, leading to enhanced resonance response. The enhanced resonance response leads to an increase in radial acceleration and secondary Bjerknes force, ultimately resulting in the rapid approach of bubbles. When the driving frequency is lower, the frequency multiplication of the external sound field is closer to the natural frequencies, which explains why at lower frequencies the compressible effect on the mutual interaction is more obvious. Our findings reveal that the compressible effect is not only critical to interaction in radial oscillations, but also in translational motion. It should be noted that this research belongs to the field of acoustic cavitation, so the conclusion can provide reference for the study of the mutual interaction between cavitation bubbles (∼μm) driven by ultrasound field (with frequency ∼ 20–80 kHz).

CRediT authorship contribution statement

Hancheng Wang: Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. Junjie Jiao: Writing – review & editing, Methodology, Investigation, Conceptualization. Xuchao Pan: Validation, Software, Data curation. Yanjie Qi: Writing – original draft, Methodology, Investigation, Data curation. Feng Shan: Methodology, Investigation, Conceptualization. Zhong Fang: Writing – review & editing, Validation, Data curation. Chuanting Wang: Writing – review & editing, Data curation, Conceptualization. Yong He: Writing – review & editing, Methodology, Investigation, Conceptualization.

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.