Interaction between double nonspherical bubbles in compressible liquid under the coupling effect of ultrasound and electrostatic ﬁ eld

– A dynamic model for a double-bubble system in compressible liquid under the coupling effect of ultrasound and electrostatic ﬁ eld was developed here. In this study, we mainly discussed the effect of the interaction on the investigated bubble using the numerical solutions to the theoretic model. The variable parameters are the distance between bubble centers and the initial radius of the adjacent bubble. In addition, we applied approximate equations to analyse variations of the internal gas pressure and temperature of a bubble. We found that, the oscillation amplitude of a bubble with an adjacent bubble signi ﬁ cantly reduces, compared to that of an isolated bubble.


Introduction
As an efficient, environmentally-friendly and economical technology, the combination of an electric field with ultrasound has been widely used in laboratories and industrial fields in recent years.A study by Jung et al. [1] revealed that this combined treatment can be used to disintegrate waste activated sludge.After 60 min treatment time, the average particle size of the sludge decreased significantly more, compared to ultrasonic treatment alone.In addition, Yang et al. [2] used the ultrasound-synergized electric field extraction (UEE) to extract flavonoids from hemerocallis citrina baroni (a citron daylily).They found that the yield of flavonoids was 10% higher using UEE than conventional ultrasound extraction, and it was 18% higher than water bath extraction.A work by Yang et al. [3] reveals the mechanism how electric stress strengthens ultrasonic cavitation.However, many details of interactions between nonspherical bubbles are not known.In fact, when cavitation occurs in the liquid, it can generate a high number of cavitation bubbles, instead of a single one.The interactions between these adjacent bubbles affect the oscillation of the individual bubbles significantly, which should be considered in the mathematic model.To describe the effect of an adjacent bubble on the cavitation of an investigated bubble, we need to develop a dynamic equation-set including the interaction between double bubbles.
Many researchers helped advance the theoretical model for two interacting bubbles.Li et al. [4], Pityuk et al. [5], and Wang et al. [6] provided three different mathematical models for two adjacent spherical bubbles in an ultrasonic field.In the study of CCl 4 sono-pyrolysis, Merouani et al. [7][8][9] used modified Keller-Miksis equations to analyze the thermal effect and the interaction between bubbles.Shima and Fujiwara [10] investigated the behavior of two spherical bubbles with different sizes near a solid wall and derived the motion equations for two bubbles.However, these models are unsuitable for two interacting nonspherical bubbles in the presence of an electric force.The variation of nonspherical radius components of bubbles cannot be obtained by simply modifying the models for spherical bubbles.Liang et al. [11] developed a dynamic model for two native nonspherical bubbles in an incompressible liquid.However, their derivation includes some important inaccuracies, which leads to incorrect mathematic equations.Furthermore, in the calculation of the total velocity potential near the interface of a bubble, they did not consider the perturbing velocity potential induced by the interaction between bubbles, which is an additional term of the total velocity potential.In addition, none of the above-mentioned studies considered the liquid's compressibility.If the liquid's compressibility is neglected in the numerical calculation, the maximum bubble-radius can easily be larger, and the minimum bubbleradius can be smaller, compared to the actual situation.
The present study aims to develop a dynamic model for nonspherical double-bubbles in a compressible liquid under the coupling effect of ultrasound and an electrostatic field.We used perturbation analysis, the sphere theorem, and the basic fluid-mechanics equations to derive the required model.We mainly analyze the effect of the interaction force on the investigated bubble using numerical solutions to the theoretical model.The variable parameters are: the distance between bubble centers, and the initial radius of the adjacent bubble.Furthermore, in the derivation, we offer an explanation why the effect of the liquid's compressibility cannot transfer from the area near the interface of one bubble to the one of another bubble, when the two bubbles are far apart from each other.Finally, we applied approximate equations to investigate temporal evolutions of the gas pressure and the temperature inside a bubble.These results can provide better insights into the oscillation mechanism of interacting nonspherical bubbles in a compressible liquid.
2 Model for double nonspherical bubbles

Interface equations for double bubbles
To describe a double-bubble system, we introduce a nonspherical function S(h, t) with rotational symmetry, which is independent of the rotation angle /.In spherical coordinates, the interface equations for double bubbles can be expressed as where The small parameter indicates that, for both bubbles, the nonspherical components are much smaller than the spherical components.
Figure 1 shows the sections of two nonspherical bubbles, which are elliptic.The distance L is set to hundreds of microns in this study, namely L ) S a + S b .The reason is that if two bubbles are too close, the coalescence of these two bubbles probably occurs.This is out of the scope of our discussion.

Total pressure-difference on the whole liquid region outside a bubble
The direction of the electric field is set to parallel to the line of bubble centers.For an insulating bubble, the expression for electric stress f s [12,13] is as follows: The electric stress f s exerts on the bubble wall, stretching the bubble in the electric-field direction and compressing the bubble in the direction vertical to the electric field.
If a bubble undergoes adiabatic oscillation, the total pressure-difference on the whole liquid region outside this bubble can be written as where For a nonspherical bubble, R = a 0 (t) or b 0 (t).To simplify the calculation, the bubble is considered to be approximately spherical for the force analysis (Eqs.( 6)-( 8)).The rationality of this simplification is based on that the nonspherical term is much smaller than the spherical term for an elliptical bubble.

Derivation of the theoretical model
Let coordinates a be the spherical coordinates with the origin at the center of bubble a, and coordinates b be the spherical coordinates with the origin at the center of bubble b.
In coordinates a, the velocity potential of bubble a can be written as where Here, U a0 (r a, t) and U a2 (r a , h a, t) are the spherical component and the nonspherical component of the velocity potential of bubble a. k is the wave number, k = x/c, and c is the speed of sound in the liquid.A 0 (t) and A 2 (t) are two functions of time.The derivations of equations ( 10) and ( 11) (see Ref. [3]) are based on the assumption that ultrasonic wave propagates within an infinitely large liquid region.Hence, no standing wave occur in the liquid.This means that translational motions of bubbles cannot occur.Thus, the distance L can be considered constant.
In the dynamic model for bubble a, the interaction terms reflect the effects of bubble b on bubble a.The effect of the liquid's compressibility only occurs in the vicinity of the interface of a bubble, which cannot transfer to the liquid region far away from the interface of a bubble.Importantly, this effect cannot transfer to the vicinity of the interface of another bubble, when the two bubbles are far apart from each other.The reason for this is as follows: In the radial direction, as the distance from a bubble center to a point in the liquid increases, the fluid velocity decreases sharply.The effect of the liquid's compressibility is positively correlated with the fluid velocity.For a liquid region far away from the bubble interface, the fluid velocity is very small, the effect of the liquid's compressibility can therefore be neglected.Hence, in the derivation of the dynamic model for bubble a, the liquid's compressibility needs not to be considered when calculating the velocity potential of bubble b.
In coordinates b, the velocity potential of bubble b is where Here, U b0 (r b , t) and U b2 (r b , h b , t) denote the spherical component and the nonspherical component of the velocity potential of bubble b.B 0 (t) and B 2 (t) are two functions of time.Equations ( 11) and ( 12) can be obtained from a study by Chen et al. [14].
To calculate the total velocity potential for a point in the liquid, we need to convert the expression of the velocity potential of bubble b from coordinates b to coordinates a.When r a is near the interface of bubble a, namely r a ( L, we have the following relations [15][16][17][18][19]: where and P i (cosh a ) is the ith-order Legendre function.Substituting these relations into equations ( 11) and ( 12), we obtain the velocity potentials of bubble b in coordinates a: In addition, according to the sphere theorem by Weiss [20], when bubble a exists in an arbitrary irrotational flow, it can cause a disturbance to this flow.Thus, an extra perturbing velocity potential can occur in the vicinity of the interface of bubble a.When the flow is undisturbed by bubble a, the unperturbed velocity potential of the flow should be U b .Hence, under the disturbance of bubble a, we denote the perturbing velocity potential as U b?a (r a , h a , t), which is also the additional velocity potential induced by bubble b in the vicinity of the interface of bubble a.
The perturbing velocity potential can be calculated by the following equality [20]: Substituting equations ( 15) and ( 16) into the above equality, we obtain Similarly, U a?b (r a , h a , t) represents the perturbing velocity potential induced by bubble a in the vicinity of the interface of bubble b.However, we assume that L ) S a + S b .Thus, the effect of U a?b (r a , h a , t) on the vicinity of the interface of bubble a is extremely small.By calculation, we can formulate U a?b (r a , h a , t) = O(1/L 10 ) in coordinates a.Hence, the term U a?b (r a , h a , t) can be neglected for the calculation of the total velocity potential.
In addition, the total velocity potential U(r a , h a , t) should be consistent with the radius function of bubble a (see Eqs. (1a)) in form.Thus, in the nonspherical term of U(r a , h a , t), only the terms about the second-order Legendre function P 2 (cosh a ) can be retained.The expression for the total velocity potential is: At the interface of bubble a, the total velocity potential satisfies the following two equations: where the expression of DP see equation (4).Equation ( 20) is obtained when the gravity of the liquid is negligible and the liquid flow is irrotational.Substituting equations (1a) and ( 18) into equations ( 19) and ( 20), using the Taylor series expansion of , and only retaining the first two order terms for , we obtain At the interface of bubble b, the total velocity potential satisfies the following equation [20]: Substituting equations (1b) and ( 10)-( 12) into equation (25), using the Taylor series expansion of , and neglecting the Peano remainder O( 2 ), we obtain Here, the liquid's compressibility is not considered.Let R a0 and R b0 be initial radii of bubble a and bubble b, namely R a0 = a 0 (0) and R b0 = b 0 (0).After eliminating A 0 (t), A 2 (t), B 0 (t), B 2 (t), and their derivatives from the equations consisted by equations ( 21)- (24) and equations ( 26)-( 27), we obtain the model for bubble a under the action of bubble b: where Equation (28a) is a modified Rayleigh-Plesset equation for a 0 (t).The term kxa 2 0 ðtÞ _ a 0 ðtÞ is related to the oscillation resistance induced by the liquid's compressibility.Equation (28b) can be regarded as a damped vibration equation for a 2 (t) with external forces, without consideration of timevarying coefficients.In addition, all terms for L are related to the forces that bubble b exerts on bubble a.
Following similar steps, we can also obtain the model for bubble b under the action of bubble a, which has a similar form as the model for bubble a. where : When c ?1, namely k ?0, equations (28a)-(28b) and equations (29a)-(29b) are reduced to the equations for an incompressible liquid, When L ? 1, equations (30a) and (30b) can be further reduced to the equations for a single nonspherical bubble, see equations (32a) and (32b).
3 Numerical solutions and analysis

Effect of the distance between bubble centers on the radius of bubble a
By solving equations (28a)-(28b) and equations (29a)-(29b) numerically, we obtain the temporal evolutions of the spherical coefficient a 0 (t) and the nonspherical coefficient a 2 (t) of bubble a at different distances between bubble centers (see Figs. 2 and 3).In these two figures, the condition L ? 1 represents the oscillation of an isolated nonspherical bubble.The other parameters see Table 1.
In Figure 2, the variations of a 0 (t) with time, at different distances L, have a similar pattern.Within one period, a 0 (t) first increases rapidly to its maximum during the rarefaction cycle, then, it decreases sharply to its minimum in a rather short time.This is followed by a series of rebounds.Furthermore, as the distance between two bubbles increases, the maximum of a 0 (t) increases, and, correspondingly, the minimum of a 0 (t) decreases.When L ? 1, a 0 (t) reaches its highest maximum and lowest minimum.In Figure 3, a 2 (t) shows exponential increases with time during the first period.Theoretically, a 2 (t) shows a larger increase at a longer distance L. In other words, when the distance between bubble centers is longer, bubble a is subject to lower oscillation resistance from bubble b, and appears to be more ellipsoidal.The similar time-varying curves of a 2 (t) were also reported by Liang et al. [11,21,22] and Chen et al. [14].

Effect of the initial radius of bubble b on the radius of bubble a
When the distance is fixed at L = 350 lm, and the value of R b0 is variable, we can solve the equations (28a)-(28b) and equations (29a)-(29b) numerically and obtain timevarying curves of two radius components of bubble a.All parameters are the same as in Section 3.1.As depicted in Figure 4, when R b0 3.5 lm, the wave peaks of a 0 (t) increase, while R b0 is increasing.However, as the initial radius R b0 increases from 3.5 lm to 6 lm, the wave peaks of a 0 (t) decrease significantly.Hence, the highest maximum and the lowest minimum of a 0 (t) occur when R b0 = 3.5 lm.Moreover, the spherical component a 0 (t) undergoes faster oscillation damping for smaller R b0 values.Figure 5 reveals that, when bubble b has a higher initial radius R b0 , the coefficient a 2 (t) shows a lower increase during one oscillation period.This means that the nonspherical feature of bubble a is more significant for smaller R b0 values.

Analyses of the cavitation of bubble a using approximate equations
These rapid increases of the coefficient a 2 (t), as shown in Figures 3 and 5, are unlikely to occur physically.In fact, it is impossible for a 2 (t) to increase without limits.This can be interpreted as follows: A nonspherical bubble has already collapsed before its nonspherical coefficient a 2 (t) reaches such a large value.The variation curves of a 2 (t) are only the results of the numerical solution to dynamic equations.Thus, to further investigate the effect of bubble b during the cavitation of bubble a, we introduce approximate equations.The method is to take the mean of electric stress f s in half period, and approximately transform a nonspherical bubble into a spherical bubble.The reason for adopting this method is as follows: the presence of electric stress reduces     the surface tension at a bubble wall, which causes a bubble to expand more during the rarefaction cycle.Therefore, it is more suitable to take the mean of electric stress f s in half period, rather than the mean of the bubble radius S(h, t) in half period.A nonspherical bubble should be approximately treated as a spherical bubble with smaller surface tension, instead of a spherical bubble with a larger initial radius.
The mean of f s in half period is Let R a and R b be radii of approximate spherical bubbles of bubble a and bubble b.By inserting the term f s into the right sides of equations ( 28a) and (29a) and replacing a 0 (t) and b 0 (t) with R a and R b , we can rewrite equations (28a) and (29a) as where The equations (34a) and (34b) are the approximate equations that describe the radius variations of bubble a and bubble b with the interaction between two bubbles.To simplify the calculation, we assume that = 0.1 in the following analysis.As long as > 0, all conclusions below do not change with the value.

Effect of bubble b on the radius of bubble a
In Section 3.2, we concluded that, when R b0 = 3.5 lm, a 0 (t) reaches its highest maximum and lowest minimum.Thus, the numerical solutions of equations (34a) and (34b) are performed for the bubble pair with initial radii R a0 = 4.5 lm and R b0 = 3.5 lm.All other parameters are the same as in Section 3.1.Figure 6 shows the temporal evolution of R a at different distances between two bubbles.The condition L ? 1 reflects the oscillation of an isolated bubble.With increasing distance between bubble centers, the oscillation amplitude of the radius of bubble a increases.This can be explained as follows: A longer distance between bubble centers leads to less oscillation resistance and higher expansion during the rarefaction cycle for bubble a.During the compression cycle, bubble a with a larger maximum radius undergoes a more violent collapse, and it therefore reaches a lower minimum radius.Hence, the highest maximum radius and the lowest minimum radius are obtained during the oscillation of an isolated bubble.

Effect of bubble b on the gas pressure and the temperature inside bubble a
When the van der Waals hard-core radius h is into account, the expressions for the gas pressure P g and the temperature T g in the center of bubble a are given by [23,24] where T 0 is the ambient temperature, T 0 = 298.15K.For bubble a with an initial radius R a0 = 4.5 lm, h = R a0 /8.54 % 0.53 lm.As shown in Figures 7 and 8, under all conditions, the maximum gas-pressure and temperature inside bubble a occur at the end of the collapse, when bubble a reaches its minimum radius.Furthermore, during the negative pressure phase of the ultrasound wave, the maximum internal pressure and temperature of an isolated bubble are significantly higher, compared to a bubble with another adjacent bubble.In general, regardless of the distance between two bubbles and the initial radius of an adjacent bubble, the interaction between two bubbles always acts as resistance to the oscillation of a bubble, when parameters of ultrasound and an electrostatic field are constant.

Conclusion
In the present study, we developed a model for two interacting nonspherical bubbles in a compressible liquid under the coupling effect of ultrasound and electrostatic field.When the distance between bubble centers is shorter, bubble a is subject to higher oscillation resistance from bubble b, and appears to be more ellipsoidal.When the initial radius R b0 equals to 3.5 lm, a 0 (t) reaches its highest maximum and lowest minimum.For a larger R b0 value, the nonspherical feature of bubble a becomes more significant after several periods.By solving approximate equations numerically, we found that, at the end of the collapse, the maximum pressure and temperature inside an isolated bubble are significantly higher than a bubble with an adjacent bubble.However, this obtained model can only depict the interaction between two adjacent bubbles.In fact, when a number of bubbles undergo ultrasonic cavitation simultaneously, the interaction between them can be more complicated.Therefore, a subject of our future research could be the oscillation of a nonspherical bubble cluster, as well as the effect of the interaction on the resonance frequency of the investigated bubble.