The secondary Bjerknes force between two gas bubbles under dual-frequency acoustic excitation

Ultrasonics Sonochemistry 29 (2016) 129–145

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The secondary Bjerknes force between two gas bubbles under dual-frequency acoustic excitation Yuning Zhang a,⇑, Yuning Zhang b,c,⇑, Shengcai Li a a

School of Engineering, University of Warwick, Coventry CV4 7AL, UK Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, Ministry of Education, North China Electric Power University, Beijing 102206, China c School of Power, Energy and Mechanical Engineering, North China Electric Power University, Beijing 102206, China b

a r t i c l e

i n f o

Article history: Received 25 March 2015 Received in revised form 26 July 2015 Accepted 29 August 2015 Available online 29 August 2015 Keywords: The secondary Bjerknes force Gas bubbles Acoustic cavitation Dual-frequency excitation

a b s t r a c t The secondary Bjerknes force is one of the essential mechanisms of mutual interactions between bubbles oscillating in a sound field. The dual-frequency acoustic excitation has been applied in several fields such as sonochemistry, biomedicine and material engineering. In this paper, the secondary Bjerknes force under dual-frequency excitation is investigated both analytically and numerically within a large parameter zone. The unique characteristics (i.e., the complicated patterns of the parameter zone for sign change and the combination resonances) of the secondary Bjerknes force under dual-frequency excitation are revealed. Moreover, the influence of several parameters (e.g., the pressure amplitude, the bubble distance and the phase difference between sound waves) on the secondary Bjerknes force is also investigated numerically. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Driven by acoustic waves, bubbles in a liquid will oscillate, named as ‘‘acoustic cavitation” [1,2]. Because of its unique physical complexity [3], chemical applications [4] and biomedical significance [5], effects of acoustic cavitation are being employed intensively, e.g., to measure bubble size distributions [6–8], to facilitate the chemical reactions [9–15], and to perform noninvasive therapy [16–18]. When bubbles oscillate in acoustic field, the radiation pressure generated by other cavitation bubbles can cause the mutual attraction or repulsion between bubbles. This well-known phenomenon is termed as ‘‘the secondary Bjerknes force’’ [19,20]. The secondary Bjerknes force, leading to the translational motion of bubbles, determines the agglomeration or dispersion of bubbles [21–25], which is essential for understanding the dynamics of bubble clouds and their applications (e.g., sonochemistry). The existence of the secondary Bjerknes force could influence the effect of cavitation in sonochemical reactors through increasing the fraction of energy transfer of the combined collapsing bubbles [13]. After agglomeration due to the secondary Bjerknes force, bubbles may deform at the early stage of collapse, which

⇑ Corresponding authors at: Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, Ministry of Education, North China Electric Power University, Beijing 102206, China (Y. Zhang). E-mail addresses: [email protected] (Y. Zhang), zhangyn02@gmail. com (Y. Zhang). http://dx.doi.org/10.1016/j.ultsonch.2015.08.022 1350-4177/Ó 2015 Elsevier B.V. All rights reserved.

decreases the efficiency of energy converting of the cavitation as well as the efficiency of the sonochemical reactions [26,27]. The mutual interaction between bubbles can also affect the pattern of the erosion of the equipment in the ultrasonic fields [14] and affect the consequent of the acoustic cleaning [15]. The direction of the secondary Bjerknes force (i.e., the bubbles whether attracting or repulsing each other) is a paramount topic of cavitation systems. According to the linear theory [19,28,29], when the driving frequency is between the linear resonance frequencies of the two bubbles, the secondary Bjerknes force between the two bubbles is repulsive. Otherwise, the secondary Bjerknes force is attractive. The linear theory has been extended by many researchers. Zabolotskaya [21] proposed that when the two bubbles approaching each other, their resonance frequencies will be effectively increased. As a result, even though the two bubbles are both driven above their resonances, the secondary Bjerknes force could change from attractive force to repulsive force. Harkin et al. [30], Doinikov and Zavtrak [23] and Ida [31,32] have also found that the distance between the bubbles could affect the direction of the secondary Bjerknes force. The nonlinearity of the bubble oscillators is one of the primary factors leading to sign reversals of the Bjerknes force. Oguz and Prosperetti [33] reported that in particular cases, the sign of the secondary Bjerknes force is opposite to the direction predicted by the linear theory, even there is slightly nonlinear oscillation when the driving pressure amplitude is below 0.5 bar. Mettin et al. [28] found similar phenomenon in the cases of the oscillating bubbles with strong collapse under the acoustic field with a high

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Nomenclature Roman letters speed of sound in the liquid cl e12 unit vector pointing from bubble 1 to bubble 2 FB the secondary Bjerknes force the secondary Bjerknes force coefficient fB fs frequency of external single-frequency sound field f1 frequency of external sound field of frequency x1 f2 frequency of external sound field of frequency x2 L separation distance between the centres of two bubbles P0 ambient pressure Pe total input power (Pa) radiation pressure generated by the oscillations of bubprad1 ble 2 at the centre of bubble 1 prad2 radiation pressure generated by the oscillations of bubble 1 at the centre of bubble 2 rp2 pressure gradient generated by bubble 2 at the centre of bubble 1 rP 2 complex pressure gradient generated by bubble 2 at the centre of bubble 1 Rj instantaneous bubble radius of bubble j first derivative of the instantaneous bubble radius of R_ j bubble j €j R second derivative of the instantaneous bubble radius of bubble j Rj0 equilibrium bubble radius of bubble j Rrs corresponding resonance bubble radius of the driving frequency of the single-frequency excitation (m) Rr1 corresponding resonance bubble radius of one component driving frequency of the dual-frequency excitation (m) Rr2 corresponding resonance bubble radius of the other component driving frequency of the dual-frequency excitation (m) t time T period of bubble oscillation (s) vj instantaneous volume of bubble j

pressure amplitude (e.g., over 1.0 bar). Doinikov [29] and Pelekasis et al. [24] investigated the effects of the harmonics of bubble oscillations on the secondary Bjerknes force. Barbat et al. [34] identified a ‘‘periodic motion pattern” when the bubbles of equal sizes are forced near their resonance frequency. There are also other parameters influence the mutual interactions between bubbles, such as the viscosity [35] and the compressibility [36] of the liquid. In past decades, the bubble dynamics under multi-frequency (e.g., dual- or triple-frequency) acoustic excitation have attracted much attention of researchers. Compared to the single-frequency approach, the multi-frequency approach could promote the acoustical scattering cross section [37] and the mass transfer through the bubble–liquid interface [38–39], to enhance the intensity of sonoluminescence [40–43], to increase the efficiency of sonochemical reactions [27,44–47], to improve the accuracy of ultrasound imaging [48–50] and tissue ablation [51,52]. Many parameters affect the sonochemical effects of the multi-frequency approach, such as the frequencies of the ultrasound [47,53], the amplitude ratio and phase difference between component sonic waves [54–57]. It was found that the enhancement of the bubble–bubble interaction through the Bjerknes force may be one of the underlying mechanisms of the effects of multi-frequency excitation [26,27,47]. As far as we know, the secondary Bjerknes force under multifrequency excitation has not been revealed yet. In the present paper, the secondary Bjerknes force between two gas bubbles in liquids excited by dual-frequency acoustic waves is

V1 xj x_ j €xj

complex instantaneous volume of bubble 1 non-dimensional perturbation of the instantaneous bubble radius of bubble j first time derivative of xj of bubble j second time derivative of xj of bubble j

Greek letters total damping constant of bubble j bj bacj acoustic damping constant of bubble j thermal damping constant of bubble j bthj bv j viscous damping constant of bubble j es non-dimensional amplitude of single-frequency driving sound field e1 non-dimensional amplitude of external sound field of frequency x1 e2 non-dimensional amplitude of external sound field of frequency x2 j polytropic exponent ll viscosity of the liquid lth effective thermal viscosity ql density of the liquid r surface tension coefficient x angular frequency of the driving sound field with singlefrequency x1 one angular frequency of the driving sound field of dualfrequency x2 another angular frequency of the driving sound field of dual-frequency x0j natural frequency of bubble j Symbols overdot time derivative overbar complex conjugate hi time average

studied both analytically and numerically. The paper is organized as follows. In Section 2, the basic equations are introduced together with the details of the analytical solution and the numerical simulation method. In Section 3, firstly, the results obtained by the analytical and numerical simulations are compared and the valid region of the analytical solution are shown; secondly, the general features of the secondary Bjerknes force under dual-frequency excitation are investigated based on the numerical simulations; thirdly, the influence of several parameters, such as the pressure amplitude, the bubble distance and the phase difference between the two sound waves, on the secondary Bjerknes force is discussed. Section 4 concludes the main findings of the present paper. Section 5 discusses the limitations of the present work. 2. Equations and solutions In this section, the basic equations for calculating the secondary Bjerknes force under dual-frequency excitation are introduced. Then, the analytical solution and the numerical simulation for solving these equations are given. 2.1. Basic equations Here, the mutual interaction force (e.g., the secondary Bjerknes force) between two oscillating bubbles (numbered as ‘‘bubble 1” and ‘‘bubble 2” respectively) in liquids under the dual-frequency

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acoustic excitation is considered. According to the literature (e.g., Ref. [28]), the radiation pressure generated by bubble 2 at the centre of bubble 1 (prad1) or vice versa (prad2) can be expressed as

prad1 ¼

where

ql d _ 2 ðR2 R2 Þ;

ð1Þ

L dt

q d prad2 ¼ l ðR_ 1 R21 Þ:

Here, prad1 is the radiation pressure generated by the oscillations of bubble 2 at the centre of bubble 1; prad2 is the radiation pressure generated by the oscillations of bubble 1 at the centre of bubble 2; ql is the density of the liquid; L is the separation distance between the centres of the two bubbles; t is the time; R1 and R2 are the instantaneous bubble radii of bubbles 1 and 2 respectively; overdot denotes the time derivative. Then, with involving the radiation pressure generated by other bubbles, the equations of bubble motion [58] for two interacting bubbles should be (Ref. [28], Eq. (7)) !

!

1 Mj q

_ _ R_ 1 € 1 þ 3 1  R1 R_ 2 ¼ 1 þ R1 p1 ðR1 ; tÞ  ps ðtÞ R1 R 2 cl 3cl 1 cl ql

!

nj ¼

!

ð3Þ

!

þ

R2 d½p2 ðR2 ; tÞ  ps ðtÞ 1 d _ 2  ðR1 R1 Þ; dt L dt

ql c l

ð4Þ

where

  2r 2r 4ðll þ lth Þ _ R1 ; ðR01 =R1 Þ3j  p1 ðR1 ; tÞ ¼ P 0 þ  R1 R01 R1

ð5Þ

  2r 2r 4ðll þ lth Þ _ ðR02 =R2 Þ3j  R2 ;  p2 ðR2 ; tÞ ¼ P 0 þ R2 R02 R2

ð6Þ

ps ðtÞ ¼ P0 ð1 þ e1 eix1 t þ e2 eix2 t Þ:

2.2. Analytical solutions

ðN21 e1 eix1 t þ N22 e2 eix2 t Þ;

2ðll þ lth Þ

ql R20j Mj

þ

aj ¼

ð11Þ

ð12Þ

R0j 2 x ; 2cl 0j

ð13Þ

ð14Þ

P0

ql R20j

Mj ¼ 1 þ

;

ð15Þ

R0j 4ll ; cl ql R20j

ð16Þ

xk R0j cl

ð17Þ

i:

Here, j = 1, 2; k = 1, 2; x0j is the natural frequency of the bubble j; bj, bvj, bthj and bacj are the total, viscous, thermal and acoustic damping constants of the bubble j respectively. In this section, as only linear oscillations of bubbles (up to the first order of e) are considered, the solution of x1 and x2 can be expressed as

x1 ¼ A11 eix1 t þ A12 eix2 t ;

ð18Þ

x2 ¼ A21 eix1 t þ A22 eix2 t :

ð19Þ

Then with substituting Eqs. (18) and (19) into Eqs. (10) and (11), one can obtain

ðx201  x21 þ 2b1 x1 iÞA11 

ðx202  x21 þ 2b2 x1 iÞA21 

ðx201  x22 þ 2b1 x2 iÞA12 

ðx202  x22 þ 2b2 x2 iÞA22 

R202 R201 R201 R202 R202 R201 R201 R202

n2 x21 A21 ¼ 

n1 x21 A11 ¼ 

n2 x22 A22 ¼ 

n1 x22 A12 ¼ 

a1 M1

a2 M2

a1 M1

a2 M2

N11 e1 eix1 t ;

ð20Þ

N21 e1 eix1 t ;

ð21Þ

N12 e2 eix2 t ;

ð22Þ

N22 e2 eix2 t :

ð23Þ

By solving Eqs. (20)–(23), we got

In this section, a traditional perturbation method is employed to solve the equations in Section 2.1. For details of solving procedure, readers are referred to Doinikov [29] and Zhang [59]. The framework of Zhang [59] has been extended to investigate the secondary Bjerknes force under dual-frequency excitation. Here, for convenience, most of the nomenclatures in Zhang [59] are adopted. We assume the solutions of Eqs. (3) and (4) as

R1 ¼ R01 ð1 þ x1 Þ;

ð8Þ

R2 ¼ R02 ð1 þ x2 Þ:

ð9Þ



A11 ¼ 

n2 €x2 ¼ 

a1 M1

ðN 11 e1 eix1 t þ N12 e2 eix2 t Þ;

a1 e1 N21 X1

M2 

A21 ¼ 

a2 e1 N11 X1

M1

 N11 2 ðx02  x21 þ 2b2 x1 iÞ ; M1

ð24Þ

n1 x21 þ

 N21 2 ðx01  x21 þ 2b1 x1 iÞ ; M2

ð25Þ

X 1 ¼ ðx201  x21 þ 2b1 x1 iÞðx202  x21 þ 2b2 x1 iÞ  n1 n2 x41 : and

A12 ¼ 



a1 e2 N22 X2

M2 

ð10Þ

n2 x21 þ

with

Then based on Eqs. (3) and (4), one can obtain

R201

a2 M2

    2r 2r  ; 3j P 0 þ R0j R0j

ð7Þ

Here, cl is the undisturbed speed of sound in the liquid; P0 is the ambient pressure; ql is the density of the liquid; r is the surface tension coefficient; R01 and R02 are the equilibrium bubble radii of bubbles 1 and 2 respectively; j is the polytropic exponent; ll is the viscosity of the liquid; lth is the effective thermal viscosity; e1 and e2 are the non-dimensional amplitudes of the driving sound wave; x1 and x2 are the angular frequencies of the driving sound waves.

R202

n1 €x1 ¼ 

R0j ; L

Njk ¼ 1 þ

_ _ R_ 2 € 2 þ 3 1  R2 R_ 2 ¼ 1 þ R2 p2 ðR2 ; tÞ  ps ðtÞ R2 R 2 cl 3cl 2 cl ql

€x1 þ 2b1 x_ 1 þ x201 x1 þ

2 l R0j

R202

bj ¼ bv j þ bthj þ bacj ¼

!

R1 d½p1 ðR1 ; tÞ  ps ðtÞ 1 d _ 2  ðR2 R2 Þ; þ dt L dt ql c l 1

x20j ¼

R201

ð2Þ

L dt

1

€x2 þ 2b2 x_ 2 þ x202 x2 þ

A22 ¼ 

a2 e2 N12 X2

M1

ð26Þ

n2 x22 þ

 N12 2 ðx02  x22 þ 2b2 x2 iÞ ; M1

ð27Þ

n1 x22 þ

 N22 2 ðx01  x22 þ 2b1 x2 iÞ ; M2

ð28Þ

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Fig. 1. Predictions of the secondary Bjerknes force coefficient fB versus the equilibrium radius of bubble 2 under single-frequency excitation by the analytical solution (dashed line) and the numerical simulations (solid line). (a) Pe/P0 = 0.01. (b) Pe/P0 = 0.03. (c) Pe/P0 = 0.05. (d) Pe/P0 = 0.2. The subplot in (d) shows the variations of fB for R02 between 10 lm and 20 lm. fs = 100 kHz. R01 = 10 lm. Rrs corresponds to the resonance radius of the driving frequency. The horizontal line indicates where fB = 0.

Fig. 2. Predictions of the secondary Bjerknes force coefficient fB versus the equilibrium radius of bubble 2 under dual-frequency excitation by the analytical solution (dashed line) and the numerical simulations (solid line). (a) Pe/P0 = 0.01. (b) Pe/P0 = 0.03. (c) Pe/P0 = 0.05. (d) Pe/P0 = 0.2. The subplot in (d) shows the variations of fB for R02 between 10 lm and 15 lm. f1 = 100 kHz. f2 = 200 kHz. R01 = 10 lm. Rr1 and Rr2 indicate the corresponding resonance radii of the driving frequencies respectively. The horizontal line indicates where fB = 0.

with

X 2 ¼ ðx201  x22 þ 2b1 x2 iÞðx202  x22 þ 2b2 x2 iÞ  n1 n2 x42 :

ð29Þ

As the secondary Bjerknes force is a mutual interaction between two interacting bubbles, only the force on bubble 1 needs to be

determined. If the bubble is small compared to the wavelength of the sound wave generated by the other bubble, then the translational force on the bubble is equal to the negative gradient of the radiation pressure times the bubble volume. The secondary Bjerknes force caused by bubble 2 on bubble 1 is defined as the time average of the translational force [19]

Y. Zhang et al. / Ultrasonics Sonochemistry 29 (2016) 129–145

133

Fig. 3. The variations of the secondary Bjerknes force coefficient fB in the R01–R02 plane under single-frequency [fs = 100 kHz (a), fs = 200 kHz (b)] and dual-frequency [f1 = 100 kHz and f2 = 200 kHz (c)] excitation. Pe/P0 = 0.03. Rr1 and Rr2 indicate the corresponding resonance radii of the driving frequencies respectively. The repulsive forces (i.e., fB < 0) are represented by red areas while the attractive forces (i.e., fB > 0) are represented by grey scales. The scale bars are located at the bottom right corner. The white points and the arrows indicate the two-bubble system with R01 = 16 lm and R02 = 32 lm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

F B ¼ hv 1 rp2 i;

ð30Þ

with

4 3

v 1 ¼ pR31 ; rp2 ¼

ð31Þ

ql d _ 2 ðR2 R2 Þe12 L2 dt

ð32Þ

Here, FB denotes the secondary Bjerknes force; v1 is the instantaneous volume of the bubble 1; rp2 is the pressure gradient generated by bubble 2 at the centre of bubble 1; h i denotes the time average during one oscillation period; e12 is the unit vector pointing from bubble 1 to bubble 2. Here, following the framework of Doinikov [29], one can define v1 = Im(V1) and rp2 = Im(rP2). For linear cases, Eqs. (31) and (32) can be represented as

V 1 ¼ V C þ V 11 þ V 12 

rP2 ¼ rP21 þ rP22  ¼

ql L2

4 3 pR ð1 þ 3A11 eix1 t þ 3A12 eix2 t Þ; 3 01

ql L2

ð33Þ

R302 €x2 e12

R302 ðx21 A21 eix1 t þ x22 A22 eix2 t Þe12

ð34Þ

In Eq. (33), as VC is a constant, hVCrP2i = 0 and this term will not contribute to the secondary Bjerknes force. Then after time averaging, he2ix1 t i ¼ 0 and he2ix2 t i ¼ 0. The secondary Bjerknes force can be expressed as

1 F B ¼  ReðV 11 rP21 þ V 12 rP22 Þ: 2

ð35Þ

Here, a bar over the symbol denotes the complex conjugate. By substituting Eqs. (33) and (34) into Eq. (35), the final expression of the secondary Bjerknes force under dual-frequency excitation is obtained as

h i F B ¼ 2pql R201 R202 n1 n2 x21 ReðA11 A21 Þ þ x22 ReðA12 A22 Þ :

ð36Þ

Hence, up to the second order of e1 (or e2), based on Eq. (36), the secondary Bjerknes force under dual-frequency approach can be considered as the linear combination of those under the two component single-frequency approaches. With the generations of harmonics, the solution will be very complex, causing problems for physical interpretation. For details, readers are referred to Doinikov [29].

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Fig. 4. The variations of the secondary Bjerknes force coefficient fB versus equilibrium bubble radius of bubble 2 when the radius of bubble 1 is fixed as: (a) R01 = 10 lm, (b) R01 = 25 lm, (c) R01 = 40 lm. The bubbles are driven by single-frequency [fs = 100 kHz (dashed line), fs = 200 kHz (dotted line)] and dual-frequency [f1 = 100 kHz and f2 = 200 kHz (solid line)] excitation respectively. Pe/P0 = 0.03. The horizontal line indicates fB = 0.

2.3. Numerical simulations Substituting Eqs. (31) and (32) into Eq. (30), one can obtain

  q d F B ¼ hv 1 rp2 i ¼  v 1 2l ðR_ 2 R22 Þ e12 L dt   2 ql d v ¼ v 1 22 e12 : 4pL2 dt

ð37Þ

Integrating Eq. (37) over a period of the volume oscillations by partial integration, one can obtain the formula for the secondary Bjerknes force,

FB ¼

ql

4pL2

hv_ 1 v_ 2 ie12 ;

ð38Þ

period of bubbles may become two or four times of the driving period. The time selected for averaging is determined with the aid of the quantitative analysis of the period of the oscillating curves. No chaotic oscillation was observed in the discussed parameter zones. With the amplitude increasing, the Eq. (38) may lose its validity due to the non-periodical pulsations of bubbles (e.g., chaotic oscillations). Such examples were not found in our cases. For more violent cases (e.g. with amplitude up to 1.32 bar in Mettin et al. [28]), in certain parameter zones, no chaotic oscillation was found either. For numerical simulations, Eqs. (3) and (4) are solved by an explicit Runge–Kutta formula [60]. Then Eqs. (38)–(40) are employed to obtain the secondary Bjerknes force. And the results are represented by the secondary Bjerknes force coefficient [28]

fB ¼

where

v_ 1 ¼ 4pR21 R_ 1 ;

ð39Þ

v_ 2 ¼ 4pR22 R_ 2 :

ð40Þ

As mentioned in Eq. (32), e12 is the unit vector pointing from bubble 1 to bubble 2. Therefore, if the value of FB is positive, the force on bubble 1 pointing from bubble 1 to bubble 2, i.e., the secondary Bjerknes force between two bubbles is attractive and vice versa. In Eq. (38), it is essential to ensure that the studied bubbles pulsations are periodical. In the present paper, several measures were employed to ensure the quality of the time averaging in Eq. (38). Before performing the time averaging, the oscillation time was chosen to be long enough hence the transient term decays totally. For linear oscillations (corresponding to limited values of acoustic amplitude), the period of the bubble oscillations is quite clear (i.e., the lowest common multiple of the two driving periods). For some cases with big bubble radii (e.g., R01 = R02 = 50 lm), the oscillation

ql _ _ hv 1 v 2 i:

ð41Þ

4p

Therefore, according to Eq. (38), if fB > 0, then FB > 0, the bubbles attract each other; if fB < 0, then FB < 0, the bubbles repulse each other. The corresponding resonant bubble radii of the driving frequencies are calculated based on Eq. (12), and are represented as Rrs (single-frequency approach) and Rr1 and Rr2 (dual-frequency approach). The constants employed in analytical and numerical methods are: P0 = 101,300 Pa; ll = 1.0 mPa s; lth = 0; r = 0.0725 N/m; cl = 1486 m/s; j = 1.4. If not specified, the distance between the centres of the bubbles is L = 1  103 m. For comparison, the total input power, P e =P 0 ¼ ðe21 þ e22 Þ , remains constant in both the single-frequency excitation and dual-frequency excitation cases. 1=2

3. Results and discussions In this section, demonstrating examples of the secondary Bjerknes force coefficient generated by two oscillating gas bubbles

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under dual-frequency acoustic excitation within a wide range of parameters are shown. For comparison, cases with singlefrequency acoustic excitation are also provided. Firstly, the predictions of the secondary Bjerknes force by the analytical solution [Eq. (36)] are compared with those predicted by the numerical simulation to illustrate the validity of the analytical solution in Section 3.1. Secondly, based on the numerical simulations, the characteristics of the secondary Bjerknes force under dualfrequency excitation are investigated in Section 3.2. Thirdly, the influence of the pressure amplitude on the secondary Bjerknes force is investigated numerically in Section 3.3. 3.1. Comparison between the analytical solution and the numerical simulations For comparison, the equilibrium radius of bubble 1 is fixed (R01 = 10 lm) and the variations of fB with the change of the radius of bubble 2 are calculated. Figs. 1 and 2 show the curves of fB predicted by the analytical and numerical approaches under the single-frequency excitation (fs = 100 kHz) and the dual-frequency excitation (f1 = 100 kHz and f2 = 200 kHz) respectively. The resonance bubble radii corresponding to the driving frequencies are marked in the figures as Rrs, Rr1 and Rr2. As shown in Fig. 1, for single-frequency excitation, the value of fB reaches maximum when R02  Rrs. Then the sign of fB changes from positive to negative at R02  Rrs, which can also be concluded from the well-known formula [19,29]:

FB ¼

2pjAj2 x2 R01 R02

qL ðx201  x2 Þðx202  x2 Þ 2

:

ð42Þ

where x01 and x02 are the natural frequencies of bubble 1 and bubble 2 respectively. Therefore, according to Eq. (42), if the driving frequency lies between the two linear resonance frequencies (i.e., R01 < Rrs < R02 or R02 < Rrs < R01), the bubbles repulse each other; otherwise (i.e., ‘‘Rrs < R01 and Rrs < R02” or ‘‘R01 < Rrs and R02 < Rrs”) they attract each other. The curves under the dual-frequency excitation in Fig. 2 show two peaks at R02  Rr2 and R02  Rr1 respectively. The characteristics of the secondary Bjerknes force under dual-frequency excitation will be discussed in detail in Section 3.2. When the acoustic pressure is low [as shown in Figs. 1(a) and 2(a)], e.g., Pe/P0 = 0.01, the analytical solution and the numerical simulation agree well under both single- and dual-frequency excitation. With increasing values of Pe, the values of fB near resonances increase dramatically. The values of fB near resonances predicted by the analytical solution are higher than those predicted by the numerical simulation and the difference between the two approaches increases with increasing Pe. The positions of the peaks of the curves predicted by the numerical simulation move towards smaller bubble radii. The influence of the acoustical pressure will be discussed in Section 3.3. When Pe/P0 is up to 0.2, as the subplot in Fig. 1(d) shows, another peak appears at the resonance radius corresponding to 2fs (the first harmonic, see Section 3.2 for details) in the numerical approach. Similarly, in the subplot in Fig. 2(d), a peak appears at

Fig. 5. The variations of the secondary Bjerknes force coefficient fB in the R01–R02 plane under single-frequency [fs = 100 kHz (a), fs = 150 kHz (b)] and dual-frequency [f1 = 100 kHz and f2 = 150 kHz (c)] excitation. Pe/P0 = 0.3. The repulsive forces (i.e., fB < 0) are represented by red areas while the attractive forces (i.e., fB > 0) are represented by grey scales. The scale bars are located at the bottom right corner. The resonances marked with white circles are harmonics (solid line), subharmonics (dashed line) and combination resonance (dash-dotted line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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the resonance radius corresponding to f1 + f2 (the combination resonance, see Section 3.2 for details) in the numerical approach. The analytical approach fails to predict these phenomena. Therefore, the analytical solution is only valid when the pressure amplitude is low. 3.2. The basic features of the secondary Bjerknes force under dual-frequency excitation Fig. 3 shows the variation of the secondary Bjerknes force coefficient fB in the R01–R02 plane under low sound pressure amplitude (Pe/P0 = 0.03). The repulsive forces (i.e., fB < 0) are represented by red areas while the attractive forces (i.e., fB > 0) are represented by grey scales. The darker the colour is, the higher the absolute value of fB is. Fig. 3(a) and (b) are the predictions of fB under single-frequency excitation (fs = 100 kHz and fs = 200 kHz respectively). As shown in these figures, there are four regions in R01–R02 planes, divided by the ‘‘boundaries” corresponding to the resonance radius of the driving frequency: (a) Repulsive regions (fB < 0): R01 < Rrs < R02; R02 < Rrs < R01. (b) Attractive regions (fB > 0): Rrs < R01 and Rrs < R02; Rrs > R01 and Rrs > R02.

Fig. 3(c) shows the fB between two bubbles under the dualfrequency excitation (f1 = 100 kHz and f2 = 200 kHz). Divided by the resonance bubble radii corresponding to the component driving frequencies, there are nine regions in the R01–R02 plane, which could be classified into three categories: R02 < Rr2 < Rr1 < R01; (a) Repulsive regions (fB < 0): R01< Rr2 0): R01 < Rr2 and R02 < Rr2; Rr2 < R01 < R02 < Rr1 and Rr2 < R02 < R01 < Rr1; Rr1 < R01 and Rr1 < R02. (c) Uncertain regions (where fB could be positive or negative): R01 < Rr2 < R02 < Rr1; R02 < Rr2 < R01 < Rr1; Rr2 < R02 < Rr1 < R01; Rr2 < R01 < Rr1 < R02. This classification could also be explained by Eq. (36). In regions (a) and (b), the values of fB under the two component singlefrequency excitation have the same sign. According to Eq. (36), the values of fB under the dual-frequency excitation can be considered as a linear combination of the values of fB under the two component single-frequency excitation if the value of Pe is limited. Hence, in regions (a) and (b), the sign of fB remains unchanged under dual-frequency excitation. In region (c), fB could be enforced or suppressed by adding the second acoustic excitation, leading to the sign change, which depends on the relative values of fB

Fig. 6. The variations of the secondary Bjerknes force coefficient fB in the R01–R02 plane under single-frequency [fs = 100 kHz (a), fs = 200 kHz (b)] and dual-frequency [f1 = 100 kHz and f2 = 200 kHz (c)] excitation. Pe/P0 = 0.1. Rr1 and Rr2 indicate the corresponding resonance radii of the driving frequencies respectively. The repulsive forces (i.e., fB < 0) are represented by red areas while the attractive forces (i.e., fB > 0) are represented by grey scales. The scale bars are located at the bottom right corner. The white points and the arrows indicate the two-bubble system with R01 = 16 lm and R02 = 32 lm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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corresponding to the two component frequencies. According to this classification, for typical sonochemical applications with bubble radii ranging from 2 to 10 lm and the driving frequencies equalling to several kilo Hertz, the bubbles will oscillate far below the resonances hence they will attract each other (large pressure amplitude may change this conclusion owing to the strong nonlinearity of bubble oscillations). For further illustration, the values of fB versus the equilibrium radius of bubble 2 (R02) is shown in Fig. 4 for three typical cases: R01 < Rr2 < Rr1 (R01 = 10 lm), Rr2 < R01 < Rr1 (R01 = 25 lm) and Rr2 < Rr1 < R01 (R01 = 40 lm). For the cases under single-frequency excitation, the absolute values of fB rise significantly near the resonance. Furthermore, the sign of fB would change near the resonance. For dual-frequency excitation, there are peaks near both resonance bubble radii corresponding to the two frequencies, which means that the sign of fB may change two or three times in the full range of R02 (10–50 lm). Moreover, in the region away from the resonance radius, the values of fB under dual-frequency excitation are between the corresponding values of fB under two component singlefrequency excitation. However, the positions of the peaks of dualfrequency approach are slightly different from the positions of single-frequency approaches. Taking the case in which R01 = 10 lm [Fig. 4(a)] as an example. When R02 < Rr2 < Rr1, all the fB under single- and dual-frequency excitation are positive, corresponding to the ‘‘attractive regions” in Fig. 3. When R02 >

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Rr1 > Rr2, all the fB under single and dual-frequency excitation are negative, corresponding to the ‘‘repulsive regions” in Fig. 3. When Rr2 < R02 < Rr1, fB in the low-frequency approach (100 kHz) is positive while fB in the high-frequency approach (200 kHz) is negative. Therefore, fB under the dual-frequency excitation varies from negative to positive, corresponding to the ‘‘uncertain regions” in Fig. 3(c). When the pressure amplitude is higher, the influence of the nonlinearity will be important. Fig. 5 shows the variations of fB in the R01–R02 plane under a relatively high sound pressure amplitude (Pe/P0 = 0.3). Here, the driving frequencies are 100 kHz and 150 kHz respectively. Comparing these results with the cases under the low pressure amplitude, there are still boundaries near the resonances radii corresponding to the driving frequencies. Furthermore, new peaks of fB and new ‘‘repulsive regions” appear in the original ‘‘attractive regions”. These phenomena are marked by white circles in Fig. 5. They can be classified as below: (a) Harmonics (marked by solid lines) occur near the corresponding resonance radii corresponding to the frequency nf1 or mf2, where n = 2,3 and m = 2 in Fig. 5. (b) Subharmonics (marked by dashed lines) occur near the resonance radii corresponding to the frequency f1/n or f2/m. Limited by the range of the bubble radii, only the subharmonic of the high frequency component (150 kHz) is shown. In Fig. 5(b) and (c), m = 2.

Fig. 7. The variations of the secondary Bjerknes force coefficient fB in the R01–R02 plane under single-frequency [fs = 100 kHz (a), fs = 200 kHz (b)] and dual-frequency [f1 = 100 kHz and f2 = 200 kHz (c)] excitation. Pe/P0 = 0.2. Rr1 and Rr2 indicate the corresponding resonance radii of the driving frequencies respectively. The repulsive forces (i.e., fB < 0) are represented by red areas while the attractive forces (i.e., fB > 0) are represented by grey scales. The scale bars are located at the bottom right corner. The white points and the arrows indicate the two-bubble system with R01 = 16 lm and R02 = 32 lm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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(c) Combination resonances (marked by the dash dotted line) occur near the resonance radii corresponding to the frequency nf1 + mf2. In Fig. 5(c), n = m = 1. It is a unique character of the secondary Bjerknes force under dual-frequency excitation. For detailed definitions of the resonances (e.g., main resonance, harmonics, subharmonics, and ultraharmonics), readers are referred to Lauterborn [61] and Lauterborn and Kurz [3]. Like the main resonances, there are peaks of fB near the harmonics, subharmonics and combination resonances. And the sign of fB changes when the bubble radius crosses over these resonance radii. In particular, for harmonics of higher order [i.e., the second harmonic in Fig. 5(a)], only a peak of fB appears while the sign of fB does not change. Doinikov [29] derived an analytical solution of the secondary Bjerknes force by including the first harmonic of bubble oscillation under the single-frequency excitation (Ref. [29], Eq. (37)). If the second harmonic included, it will be ð1Þ

ð2Þ

ð3Þ

F B ¼ ðF 1 þ F 1 þ F 1 Þe12 :

ð43Þ

ð1Þ

Here, F 1 represents the force induced by the linear component of ð2Þ

bubble oscillation, which is of order e2. F 1 represents the force induced by the first harmonic component of bubble oscillation, ð3Þ

which is of order e4. F 1 represents the force induced by the second harmonic component of bubble oscillation, which is of order e6. Therefore, as shown in Fig. 5, when e = Pe/P0 = 0.3, the effect of the first harmonic will become important and can change the value significantly as well as the sign of fB. However, under this pressure, the effect of the second order harmonic is not strong enough, so it can only change the value of fB but not the sign, as shown in Fig. 5(a). By comparing Fig. 5(a) and (b) with Fig. 5(c), one can conclude that the secondary Bjerknes forces in the R01–R02 plane under dualfrequency excitation involve all the harmonics and subharmonics corresponding to the two component frequencies. Meanwhile,

there are unique combination resonances in the R01–R02 plane under dual-frequency excitation. Therefore, the variation of the sign of fB in the R01–R02 plane under the dual-frequency excitation shows much more complicated patterns. 3.3. Influence of the pressure amplitude Figs. 6 and 7 show the variations of the values of fB in the R01–R02 planes with the total driving pressure amplitude Pe/P0 equalling to 0.1 and 0.2 respectively. The driving frequencies are 100 kHz and 200 kHz. By comparing Figs. 3, 6 and 7, one can conclude that the total pressure amplitude can influence the signs and the values of fB. Obviously, the increasing pressure amplitude will increase the absolute value of fB, leading to enforcement of the mutual interactions between bubbles. With the increase of the pressure amplitude, the boundaries of the regions move toward the smaller bubble radii, which is owing to the resonance frequencies of bubbles getting lower induced by the nonlinearity during bubble oscillation (termed as ‘‘bending phenomenon”). For the dual-frequency approach, the regions covered by the attractive and repulsive forces in the ‘‘uncertain regions” vary with the pressure amplitude as follows: the attractive area increases in regions Rr2 < R02 < Rr1 < R01 and Rr2 < R01 < Rr1 < R02 while it decreases in regions R01 < Rr2 < R02 < Rr1 and R02 < Rr2 < R01 < Rr1. Therefore, for the bubbles with radii near the resonances, the sign of the forces between the bubbles will change with the increase of the pressure amplitude. The system with R01 = 16 lm and R02 = 32 lm is a typical example indicated by white points in Figs. 3, 6 and 7, which could reveal the detailed influence of the added frequency in the dualfrequency approach and the amplitude of the acoustic pressure on the value of fB. Fig. 8 illustrates the variations of the value of fB with the pressure amplitude. For the two-bubble system with R01 = 10 lm and R02 = 14 lm, which is away from the ‘‘boundaries”, fB is positive under both single- and dual-frequency excitation and increases with the pressure monotonically. In contrast, in

Fig. 8. The variations of the secondary Bjerknes force coefficient fB with the sound pressure amplitude Pe/P0 under single-frequency [fs = 100 kHz (dashed line), fs = 200 kHz (dotted line)] and dual-frequency [f1 = 100 kHz and f2 = 200 kHz (solid line)] excitation. In the upper image, R01 = 10 lm and R02 = 14 lm. In the lower image, R01 = 16 lm and R02 = 32 lm. The horizontal line indicates where fB = 0.

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the system with R01 = 16 lm and R02 = 32 lm, which is near the resonances, the sign of fB changes with the pressure amplitude. For single-frequency excitation, the sign of fB changes once at Pe/ P0 = 0.19 for excitation with driving frequency f1 or at Pe/P0 = 0.08 for excitation with driving frequency f2. For dual-frequency excitation, the sign of fB changes twice at Pe/P0 = 0.065 and 0.18 respectively. In particular parameter zone, the values of fB under the dual-frequency excitation are no longer between the values of the two single-frequency cases. Instead, in the region where Pe/P0 is between 0.08 and 0.18, the values of fB for the dual-frequency approach are negative while the values for the single-frequency approaches are all positive.

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The differences between the single-frequency approach and the dual-frequency approach are illustrated in Fig. 9. Fig. 9(a) shows the variations of the pressure amplitude in one driving period. Fig. 9(b) shows the oscillation curves of the bubbles in one driving period. In Figs. 9 and 10, T equals to the period of the excitation with fs = 100 kHz, i.e., T = 1  105 s. Under the single-frequency excitation, the two bubbles oscillate in phase in most of the time. However, under the dual-frequency excitation, the two bubbles oscillate out of phase in most of the time. Therefore, the time average value of v_ 1 v_ 2 [as shown in Fig. 9(c)] in one period is positive, leading to positive fB under single-frequency excitation while it is negative, leading to negative fB under the dual-frequency excitation.

Fig. 9. (a) Normalized driving pressure Ps/P0 versus normalized time t/T. (b) The instantaneous bubble radii R1 (black solid line) and R2 (blue dashed line) versus normalized time t/T during one driving period. (c) v_ 1 v_ 2 versus normalized time. Pe/P0 = 0.1. R01 = 16 lm. R02 = 32 lm. f1 = 100 kHz and f2 = 200 kHz. T = 1/f1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 10. The bubble oscillations under dual-frequency excitation (f1 = 100 kHz and f2 = 200 kHz) with different pressure amplitudes. (a) The instantaneous bubble radii R1 (black solid line) and R2 (blue dashed line) versus normalized time t/T during one driving period. (b) v_ 1 v_ 2 versus normalized time. Pe/P0 = 0.03, 0.1, and 0.2 respectively. R01 = 16 lm. R02 = 32 lm. T = 1/f1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

For further study of the influence of the pressure amplitude on the bubbles with radii near the resonances, Fig. 10 illustrates the bubble oscillations under dual-frequency excitation with different driving pressure amplitudes. As shown in Fig. 10(a), when the pressure amplitude is low (e.g., Pe/P0 = 0.03), the two bubbles oscillate in phase for the most of the time. As shown in Fig. 10(b), the time average of v_ 1 v_ 2 yields positive fB. With Pe/P0 rising to 0.1, the oscillation phase of the bigger bubble (R02 = 32 lm) shifts while the oscillation phase of the smaller bubble (R01 = 16 lm) remains so that the two bubbles oscillate out of phase. When Pe/P0 rises to 0.2, the phase of the bigger bubble remains the same while those of the smaller bubble shifts so that the two bubbles oscillate in

phase again. Therefore, as shown in Fig. 10(b), the time average value of v_ 1 v_ 2 is negative, leading to negative fB when Pe/P0 = 0.1 while it becomes positive, leading to positive fB when Pe/P0 = 0.2. 3.4. Influence of the separation distance between bubbles For investigating the variation of fB when bubbles approaching each other, the values of fB are calculated with the separation distance of bubbles L = 0.5  103 m. The driving frequencies f1 = 100 kHz and f2 = 200 kHz and pressure amplitude Pe/P0 = 0.03. The R01–R02 planes of fB under both single- and dual-frequency excitation are shown in Fig. 11. Compared Fig. 3 with Fig. 11, for

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single-frequency excitation, one can conclude that the boundaries of regions move toward the bigger bubble radii with reducing the distance between bubbles, which may due to the natural resonance of bubbles getting higher when bubbles get closer [21,23]. So the sign of fB for the cases near the boundaries may changes when the bubbles approaching each other. Take the case with R01 = 25 lm, R02 = 34 lm driven by the acoustic wave with 100 kHz as an example. As shown in Fig. 3(a), when L = 1  103 m, x01 > x while x02 < x, so the sign of fB is negative. With L reducing to 0.5  103 m, the resonance frequency of bubble 2 rises above the driving frequency (i.e., x02 > x), hence the sign of fB reverse to positive [as shown in Fig. 11(a)]. Therefore, for dual-frequency excitation, all the boundaries also move toward the bigger bubble radii. The regions covered by the attractive and repulsive forces in the ‘‘uncertain regions” vary with the pressure amplitude as follows: the area of attractive region increases in regions R01 < Rr2 < R02 < Rr1 and R02 < Rr2 < R01 < Rr1 while it decreases in regions Rr2 < R02 < Rr1 < R01 and Rr2 < R01 < Rr1 < R02. 3.5. Influence of the phase difference between the two driving acoustic waves In the dual-frequency system, the phase difference between the two driving acoustic waves is also an important factor influencing the effects of cavitation. For instance, it can affect the overall

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distribution of acoustic pressure amplitude and the energy dissipation in sonochemical reactors [55,57]. Furthermore, compared with the cases under single-frequency approach, for dual-frequency approach, the phase difference is also a unique parameter which affects the value of fB. If the phase difference is taken into account, Eq. (7) changes to:

 ps ðtÞ ¼ P 0 1 þ e1 eix1 t þ e2 eiðx2 tþDuÞ

ð44Þ

Here, Du is the phase difference between two driving waves. Fig. 12(a) illustrates the variations of the values of fB in the R01–R02 plane with phase difference Du = 1.4p. The driving frequencies f1 = 100 kHz and f2 = 200 kHz, and the pressure amplitude Pe/P0 = 0.1. Compared to the R01–R02 plane with no phase difference [as shown in Fig. 6(c)], when Du = 1.4p, both the value and the sign of fB change little except in the ‘‘uncertain regions”. The attractive area increases in regions Rr2 < R02 < Rr1 < R01 and Rr2 < R01 < Rr1 < R02 while it decreases in regions R01 < Rr2 < R02 < Rr1 and R02 < Rr2 < R01 < Rr1. The sign of fB of the cases near the boundaries change with the phase difference. The system with R01 = 16 lm and R02 = 32 lm is a typical case near the boundaries while the system with R01 = 10 lm and R02 = 14 lm is away from resonances. Fig. 12(b) compares the variation of the value of fB with Du (from 0 to 2p) between the two mentioned cases. For the case with R01 = 10 lm and R02 = 14 lm, the value of fB varies in a small region, i.e., between 0.29 and 0.35, and the sign of fB keeps positive.

Fig. 11. The variations of the secondary Bjerknes force coefficient fB in the R01–R02 plane under single-frequency [fs = 100 kHz (a), fs = 200 kHz (b)] and dual-frequency [f1 = 100 kHz and f2 = 200 kHz (c)] excitation when the separation distance between bubbles L = 0.5  103 m. Pe/P0 = 0.03. Rr1 and Rr2 indicate the corresponding resonance radii of the driving frequencies respectively. The repulsive forces (i.e., fB < 0) are represented by red areas while the attractive forces (i.e., fB > 0) are represented by grey scales. The scale bars are located at the bottom right corner. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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In contrast, for the case with R01 = 16 lm and R02 = 32 lm, the value of fB varies violently and the sign of fB changes four times with Du increasing. Fig. 13 further illustrates the bubble oscillations of the case with R01 = 16 lm and R02 = 32 lm under dual-frequency excitation with the phase difference Du equalling to 0, 0.6p, p, 1.4p, respectively. Obviously, as shown in Fig. 13(a), the phase difference affects the amplitude pressure of the driving wave as well as the wave form. Then in Fig. 13(b), the amplitude and the phase of oscillation of bubble 2 does not change much with the change of Du. In contrast, the oscillation phase of bubble 1 shifts and the oscillation amplitude increases with the increase of Du. As a result, when Du equals to 0 or p, the two bubbles oscillate out of phase while when Du equals to 0.6p, 1.4p, the two bubbles oscillate in phase. Consequently, when Du changes, the corresponding time average of v_ 1 v_ 2 yields fB with different signs, as shown in Fig. 13(c).

4. Conclusions In this work, both the analytical and numerical solutions of the secondary Bjerknes force between two gas bubbles under dualfrequency excitation are obtained. According to the analytical solution, the secondary Bjerknes force under dual-frequency excitation can be considered as the linear combination of those under the two component single-frequency approaches. The analytical solution and the numerical simulation results agree well when the pressure amplitude is quite low, e.g., Pe/P0 = 0.01. The basic features of the secondary Bjerknes force under dualfrequency excitation are investigated numerically. There are peaks and change of the sign of the secondary Bjerknes force near the resonance bubble radii corresponding to the driving frequencies. For single-frequency excitation, the predicted values of fB in the R01–R02 plane could be divided into four regions by the resonance radius corresponding to the driving frequency and the values of fB

Fig. 12. (a) The variations of the secondary Bjerknes force coefficient fB in the R01–R02 plane under dual-frequency (f1 = 100 kHz and f2 = 200 kHz) excitation. Pe/P0 = 0.1. The phase difference Du = 1.4p. Rr1 and Rr2 indicate the corresponding resonance radii of the driving frequencies respectively. The repulsive forces (i.e., fB < 0) are represented by red areas while the attractive forces (i.e., fB > 0) are represented by grey scales. (b) The variations of the secondary Bjerknes force coefficient fB with phase difference Du dualfrequency (f1 = 100 kHz and f2 = 200 kHz) excitation. For the case with R01 = 16 lm and R02 = 32 lm (solid line) and case with R01 = 10 lm and R02 = 14 lm (dashed line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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can be categorized into two groups (i.e., attractive and repulsive regions). For dual-frequency excitation, the fB in the R01–R02 plane can be divided into nine regions by the resonance radii corresponding to the two component frequencies and the values of fB can be categorized into three groups (i.e., attractive, repulsive and uncertain regions). When the pressure amplitude is relatively high, the harmonics and subharmonics become prominent in the R01–R02

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plane under both single- and dual-frequency excitation. There are combination resonances in the R01–R02 plane under the dualfrequency excitation. All these resonances could lead to the change of the sign of the secondary Bjerknes force. With the increase of the pressure amplitude, the boundaries lean over toward smaller bubble radii. And it will also affect both the values and the sign of the secondary Bjerknes force, especially for the cases with bubble radii

Fig. 13. (a) Normalized driving pressure Ps/P0 versus normalized time t/T (b) The instantaneous bubble radii R1 (black solid line) and R2 (blue dashed line) versus normalized time t/T during one driving period under dual-frequency excitation with different phase difference. (c) v_ 1 v_ 2 versus normalized time. In (a) and (c), the phase difference Du = 0 (solid line), 0.6p (dashed line), p (dotted line), 1.4p (dash-dotted line), respectively. Pe/P0 = 0.1. R01 = 16 lm. R02 = 32 lm. f1 = 100 kHz and f2 = 200 kHz. T = 1/f1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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close to the resonance radii. In summary, compared with the single-frequency approach, the sign and the strength of the Bjerknes force in the R01–R02 plane varies in a more complicated pattern under dual-frequency excitation. When the two bubbles get closer, the boundaries of regions in the R01–R02 plane move toward the bigger bubble radii, leading to the change of the direction of the Bjerknes force of the cases near the boundaries. The phase difference between two driving waves does not affect the sign of the Bjerknes force for the cases away from boundaries while it changes both the sign and the value of the Bjerknes force coefficient for the cases near the boundaries significantly. 5. Limitations and future work In the present paper, the time delay caused by the propagation of the sound waves is ignored. Based on the given parameters in the present paper, the ratio between delay time and the driving period of external wave is limited. Hence, effects of time delay has been ignored in the present paper. To incorporate the time delay effects, the basic equations [Eq. (7) and the last terms shown in Eqs. (3) and (4)] should be modified. Time delay could possible change the symmetry of the mutual interaction forces [28], which is an important assumption employed in the present paper and also in many other published papers. The present paper did not account for the translational motion of the bubbles. To obtain the detailed translational motion, a clear description of all kinds of forces on the two bubbles should all be presented including viscous drag force, added mass force, primary Bjerknes force and the present secondary Bjerknes force together with the buoyancy and gravity forces. For details, readers are referred to the framework by Parlitz et al. [62]. The complexity of this problem will also raise noticing that the translational motion is normally in companion with the motion in other directions, leading to a coupled problem. An impressive study on the motion of bubble pair was given by Lauterborn and Kurz [3] (p. 58). When bubbles become closer, Lauterborn and Kurz [3] pointed out that the motion cannot further described by the present theory based on the calculation of the secondary Bjerknes forces. More recent studies include the computational study of the translational motion of two interacting bubbles in a megasonic field by Ochiai and Ishimoto [15] and experimental study by Jiao et al. [63,64]. In this paper, the influence of the phase difference between two driving frequencies is investigated numerically only through several special cases (e.g., four specific values of Du are chosen). However, the influence of the phase difference on the secondary Bjerknes force is complex. For instance, how the phase difference affects the sound wave also depends on the frequency ratio of the two waves, which will further affect the interaction of the bubbles. Therefore, more detailed work (both analytical and numerical) is needed in future for understanding the influence of the phase difference more deeply and systematically. Acknowledgements The authors acknowledge the financial support from China Scholarship Council, the Ph.D. tuition fees from the University of Warwick, the EPSRC WIMRC grant (RESCM 9219), the Fundamental Research Funds for the Central Universities (Project No.: 2014ZD09) and the National Natural Science Foundation of China (Project No.: 51506051). References [1] M.S. Plesset, A. Prosperetti, Bubble dynamics and cavitation, Annu. Rev. Fluid Mech. 9 (1977) 145–185.

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