Ultrasonics Sonochemistry 23 (2015) 16–20

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Short Communication

Instability of interfaces of gas bubbles in liquids under acoustic excitation with dual frequency Yuning Zhang a,⇑, Xiaoze Du a, Haizhen Xian a, Yulin Wu b a Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, Ministry of Education, North China Electric Power University, 2 Beinong Road, Chang Ping, Beijing 102206, China b State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 30 March 2014 Received in revised form 1 July 2014 Accepted 29 July 2014 Available online 9 August 2014 Keywords: Acoustic cavitation Gas bubbles Sonochemistry Dual frequency Instability Spherical harmonics

a b s t r a c t Instability of interfaces of gas bubbles in liquids under acoustic excitation with dual frequency is theoretically investigated. The critical bubble radii dividing stable and unstable regions of bubbles under dual-frequency acoustic excitation are strongly affected by the amplitudes of dual-frequency acoustic excitation rather than the frequencies of dual-frequency excitation. The limitation of the proposed model is also discussed with demonstrating examples. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Instability of bubble interfaces in liquids is of great importance to both the fundamental and applied physics of cavitation and bubble dynamics [1–19]. A thorough understanding of this topic is essential for many paramount problems with practical applications, e.g., predictions of bubble sonoluminescence [7,8], bubble growth through mass transfer [4,13,20–23], under water explosions [24], rise velocity of bubbles [14,15], cavitation associated damage to propeller [25,26] and medical diagnostic ultrasound imaging [16]. In reality, asymmetries of the bubble shape can be induced due to various reasons e.g., pressure gradient across bubbles, presence of other bubbles or interfaces and gravity. For example, shape oscillations of bubbles can be parametrically excited by the bubble pulsation under a sound field. The study of surface instability provides the validity of the generally adopted assumption of spherical bubbles [4,21–23,27–29] against paramount parameters (e.g., bubble radius, frequency and amplitude of acoustic excitation). Specifically, for chemical engineering, the study of shape instability can help facilitate the sonochemistry process,

⇑ Corresponding author. Address: No. 6 of Mailbox 102206, Beijing, China. Tel.: +86 (0)1061773958. E-mail address: [email protected] (Y. Zhang). http://dx.doi.org/10.1016/j.ultsonch.2014.07.021 1350-4177/Ó 2014 Elsevier B.V. All rights reserved.

design effective sonochemical reactor and improve the efficiency of the whole system. In present paper, a brief summary of the previous works on the instability of bubble surfaces in the literature is given. For classical review papers on this topic, readers are referred to Plesset and Prosperetti [1] and Feng and Leal [2]. Instability of bubble interfaces has been theoretically studied by many researchers e.g., Plesset [3], Eller and Crum [6], Prosperetii [5], Brenner et al. [7], Shaw [9,10] and Harkin et al. [19]. Plesset [3] established a well-known framework to study the fluid flows with spherical symmetry (e.g., bubbles). In Plesset [3], the small departure from spherical form is expressed in terms of an infinite sum of spherical harmonics. Hsieh and Plesset [4] followed Plesset’s analysis to predict the threshold of shape instability of a growing bubble. The inclusion of viscous effects was done by Proseperetti [5] and experimental measurement of threshold of shape instability was provided by Eller and Crum [6]. Brenner et al. [7] distinguished two different mechanisms (Rayleigh–Taylor instability and parametric instability respectively) leading to deviation of bubble interfaces from spherical shapes. The existence of several striking phenomena (e.g., bubble sound emission and erratic bubble dancing problem) provided a strong impetus on the study of coupling between different shape modes, interactions between shape distortions and translational motions, and energy transfer between shape and volume modes. Shaw [9,10] studied the nonlinear mutual interactions

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Y. Zhang et al. / Ultrasonics Sonochemistry 23 (2015) 16–20

between the axisymmetric shape oscillations, the axial translational motion, and the volume oscillations of the gas bubble in the liquid. Harkin et al. [19] studied the bidirectional transfer of oscillation energy between shape and volume modes. Recently, bubble dynamics under acoustic excitation with multiple frequencies (e.g., dual and triple frequency) has been intensively investigated by researchers [23,30–39] with the background of sonochemistry and bubble sonoluminescence. Comparing with bubbles excited by single-frequency acoustic waves, cavitation effects are much stronger when bubbles are induced by multi-frequency acoustic waves. For example, it was found that the bubble sonoluminescence under dual-frequency excitation can be boosted up to 3 times of those under single-frequency excitation [33]. Pandit and collaborators conducted a series of work both theoretically and experimentally investigating cavitation and bubble dynamics within multi-frequency sonochemical reactor [32,33]. Feng et al. [30] found that cavitation yield can be greatly enhanced by multi-frequency ultrasonic irradiation. Moholkar et al. [40] theoretically studied the influences of several paramount parameters (e.g., frequency and amplitude ratio, phase difference between waves) on the bubble dynamics and spatial distribution of cavitation events in a dual-frequency ultrasonic processor. A more advanced model was further developed by Moholkar et al. [41] to understand energy transformation chain in ultrasonic processor. Moholkar [42] optimized the dual-frequency ultrasonic processor numerically based on spatially averaged energy transmitted and spatial uniformity of the acoustic pressure amplitude. Recently, Zhang [23] studied the effects of mass transfer on the oscillations of bubbles induced by dual-frequency excitation and found that there exists a threshold of acoustic pressure amplitude for enhancing cavitation effects when using dual-frequency acoustic excitation. From literature review, it was found that the assumption of spherical bubbles is widely adopted when studying bubble dynamics under dual-frequency excitation while the valid regions of this assumption have not been shown yet. In present paper, instability of gas bubble interfaces oscillating in liquids under dual-frequency acoustic excitation is analytically studied following a well-known Plesset’s framework using spherical harmonics to represent shape distortions. An expression of critical bubble radius dividing stable and unstable oscillating regions is derived and influences of paramount parameters (e.g., acoustic pressure amplitude) are shown with several demonstrating examples together with valid regions of present work. 2. Theoretical analysis Here, we assumed that the bubble interface has a distortion from the spherical shape with bubble radius R as follows,

rs ¼ R þ

1 X

an Y n :

ð1Þ

with 2

A¼

ð3Þ Here t is the time; n is the order of shape oscillations of bubbles; r is the surface tension coefficient; ql is the density of the liquid outside the bubbles; qg is the density of the gas inside the bubbles. For the problems of gas bubbles oscillating in liquids to be discussed in present paper, it is safe to assume that qg  ql. Then Eq. (3) reduces to [3]: 2

A¼

ðn  1Þ d R r  ðn  1Þðn þ 1Þðn þ 2Þ : 2 R ql R3 dt

2

d an dt

2

þ

3 dR dan  Aan ¼ 0; R dt dt

ð2Þ

ð4Þ

For convenience, Eqs. (2) and (4) are transformed into [4]: 2

d bn dt

2

þ Gbn ¼ 0;

ð5Þ

with

bn ¼ R3=2 an ;

G ¼ ðn  1Þðn þ 1Þðn þ 2Þ



2

r 3 dR  ql R3 4R2 dt



ðn þ 12Þ d2 R : 2 R dt

ð6Þ

With viscosity and liquid compressibility ignored, the equation of bubble wall can be described by [1]:

€ þ 3 R_ 2 ¼ pext ðR; tÞ  ps ðtÞ ; RR 2 ql

ð7Þ

where

  2r 2r pext ðR; tÞ ¼ P0 þ ðR0 =RÞ3j  ; R0 R

ð8Þ

ps ðtÞ ¼ P 0 þ PA1 cosðx1 tÞ þ P A2 cosðx2 tÞ:

ð9Þ

Here j is the polytropic exponent; P0 is the ambient pressure; R0 is the equilibrium bubble radius; PA1 and PA2 are the amplitudes of acoustic excitation with angular frequencies x1 and x2, respectively. In present paper, for convenience, we assumed that P A1 and PA2 are of the same order and x1 P x2. Ignoring the high order terms (e.g., harmonics, excitation with sum and difference of two frequencies), the solution of Eqs. ((7)– (9)) is [23]:

R ¼ R0 ½1 þ d1 cosðx1 t þ /1 Þ þ d2 cosðx2 t þ /2 Þ;

ð10Þ

with

d1 ¼ 

n¼1

Here rs is the vector of bubble interface; R is the radius of aforementioned spherical bubble interface; Yn is nth order spherical harmonics; an is one of a series of unknown coefficients and is assumed to be independent with each other. For convenience, we assumed that the two fluids inside and outside bubbles are immiscible, nonviscous and incompressible and also assumed that jan j  R. Based on Plesset’s analysis [3], one can obtain the differential equation of an:

2

½nðn  1Þql  ðn þ 1Þðn þ 2Þqg d R=dt  ðn  1Þnðn þ 1Þðn þ 2Þr=R2 : ½nql þ ðn þ 1Þqg R

d2 ¼ 

x20 ¼



P A1

;

ð11Þ

;

ð12Þ

ql R20 x20  x21  

P A2

ql R20 x20  x22 

    2r 2r  : 3 j P þ 0 R0 R0 ql R20 1

ð13Þ

Here x0 is the natural frequency of gas bubble oscillations in liquids. The expressions of the phases /1 and /2 are not used in present paper hence their expressions are not given here. For details, readers are referred to Zhang [23]. Substituting Eq. (10) into Eq. (6) and omitting terms with order higher than the first order, one can obtain

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Y. Zhang et al. / Ultrasonics Sonochemistry 23 (2015) 16–20

G ¼ ðn  1Þðn þ 1Þðn þ 2Þ "

r ql R30





#





#

r 1 þ 3ðn  1Þðn þ 1Þðn þ 2Þ þ nþ x21 d1 3 2 ql R0  cosðx1 t þ /1 Þ "

r 1 þ 3ðn  1Þðn þ 1Þðn þ 2Þ þ nþ x22 d2 2 ql R30  cosðx2 t þ /2 Þ:

ð14Þ

Generally, it is safe to assume that

3ðn  1Þðn þ 1Þðn þ 2Þ









r 1 1  nþ x22  n þ x21 : 3 2 2 ql R0

In Section 3, we will re-visit this assumption. Hence, Eq. (14) can be simplified as

G ¼ ðn  1Þðn þ 1Þðn þ 2Þ





r 1 þ nþ x21 d1 2 ql R30

  1  cos ðx1 t þ /1 Þ þ n þ x22 d2 cosðx2 t þ /2 Þ: 2

ð15Þ

According to stability theory of Eq. (5), the solution is stable if G > 0. Hence, based on Eq. (15), we obtain,

ðn  1Þðn þ 1Þðn þ 2Þ

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 1 > nþ x41 d21 þ x42 d22 : 3 2 q l R0

ð16Þ

Substituting Eqs. (11) and (12) into Eq. (16), the stable region of bubble is

R0 < Rcrit ¼



ðn  1Þðn þ 1Þðn þ 2Þr sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : x41 P2A x42 P2A 1 1 2 nþ2 2 þ 2 jx20 x21 j jx20 x22 j

ð17Þ

Here Rcrit is the critical bubble radius dividing the stable and unstable regions. Owing to the natural frequency (x0) is also a function of bubble radius, Eq. (17) can be only solved numerically. If x1 P x2  x0, one can find that Eq. (17) reduces to

Rcrit ¼

ðn  1Þðn þ 1Þðn þ 2Þr  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : n þ 12 P2A1 þ P2A2

ð18Þ

Hence for this region, values of angular frequency have very limited effects on the critical bubble radius. In Hsieh and Plesset [4], a rough estimation of Rcrit was provided based on assumed constant values of d1 and d2 (e.g., d1 = d2 = 102). As d1 or d2 is also a function of bubble radius as shown in Eqs. (11) and (12), a more general treatment is given in present paper by replacing d1 and d2 using functions of R0 (referring to Eq. (17)).

Fig. 1. Influences of the order of spherical harmonics on the values of critical bubble radius (Rcrit). P A1 ¼ P A2 ¼ 1:01  103 Pa; f1 = 1 MHz; f2 = 500 kHz. Solid line represents Rcrit under dual-frequency excitation while dashed line represents Rcrit under single-frequency excitation.

not be discussed in present paper. Based on Fig. 1, one can find that Rcrit increases with the increase of n. If R0 is smaller than Rcrit with n = 2, the bubble will be absolutely stable. Therefore, in the following discussions, we only need to predict Rcrit for n = 2. Comparing with Rcrit under single-frequency excitation, the values of Rcrit under dual-frequency excitation are much lower. In Fig. 1, the predicted values of Rcrit for single-frequency excitation with f1 = 1 MHz and f2 = 500 kHz are nearly the same and cannot be distinguished in the Fig. 1. Hence, as shown in Fig. 1 and Eq. (18), the frequency of acoustic excitation has very limited effects on the predictions of critical bubble radius. According to Eq. (18), one can find that the values of critical bubble radius (Rcrit) for dual-frequency excitation are highly dependent on the input energy represented by the total acoustic qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pressure amplitude Pe (P e ¼ P2A1 þ P2A2 ). Fig. 2 shows the influences of the total acoustic pressure amplitude (Pe) on the critical bubble radius (Rcrit) for dual-frequency excitation. From Fig. 2, it is clear that Rcrit will decrease with the increase of the total acoustic pressure amplitude. For example, bubbles with radius beyond 3.46 mm are unstable for Pe = 100 Pa while bubbles larger than

3. Results and discussions In this section, stability of interfaces of air bubbles in water is considered as demonstrating examples. The following constants are adopted during simulations: ambient pressure P0 = 1.01  105 Pa; density of the liquid ql = 1000 kg/m3; surface tension coefficient r = 0.0728 N/m; polytropic exponent j = 1.2. For more details about predictions of polytropic exponent, readers are referred to Zhang and Li [28,29,43]. For convenience, the angular frequencies x1 and x2 will be transformed into frequencies f1 and f2. Fig. 1 shows the influence of the order of spherical harmonics on the critical bubble radius (Rcrit) for acoustic excitation with single and dual frequency respectively. n = 1 corresponds to the translational motion of the whole bubbles and hence case with n = 1 will

Fig. 2. Influence of the total acoustic pressure amplitude (Pe) on the values of critical bubble radius (Rcrit). f1 = 1 MHz; f2 = 500 kHz; P A2 =P A1 ¼ 2.

Y. Zhang et al. / Ultrasonics Sonochemistry 23 (2015) 16–20

19

of present work. Our studies found that the critical bubble radii separating the stable and unstable regions decrease with the increase of the input energy (e.g., the total acoustic pressure amplitude) while are independent of the acoustic frequencies. The model developed here can be employed into other models (e.g., [40–43]) to further optimize the highly efficient dual-frequency ultrasonic processor. Acknowledgements This work was financially supported by ‘‘the Fundamental Research Funds for the Central Universities’’ (Project No.: 2014ZD09) and the 111 Project (Project No.: B12034). References

Fig. 3. Determination of critical bubble radius (Rcrit) based on graphical plots. P A1 ¼ P A2 ; f1 = 1 MHz; f2 = 500 kHz.

0.348 mm are unstable for Pe = 1000 Pa. Hence, with the increase of the total acoustic pressure amplitude, bubbles have great tendency to be unstable. Our study further reveal that with Pe fixed, the influences of pressure ratio of two terms in dual-frequency excitation (i.e., PA2 =PA1) on the predictions of Rcrit are rather limited. For even higher amplitude, the assumption of small distortion of bubble oscillations employed in Section 2 will be violated. As a result, there exists a threshold value of acoustic pressure amplitude beyond which there is no solution for Eq. (17). Fig. 3 shows this threshold value (PA1 ¼ P A2 ¼ 1:67  104 Pa for this case) using graphical plots. We denote the right hand side of Eq. (17) as f (R0) for convenience. The solid lines in Fig. 3 represent the lines Y = f (R0) for P A1 ¼ P A2 ¼ 1:0  104 ; 1:67  104 and 3:0  104 Pa respectively and the dashed line in Fig. 3 represent the line Y = R0. The intersection of Y = f (R0) and Y = R0 is the solution for Rcrit. As shown in Fig. 3, for PA1 ¼ PA2 ¼ 1:0  104 Pa, the solution of Eq. (17) exist while for P A1 ¼ P A2 ¼ 3:0  104 Pa, there is no solution. To obtain Eq. (15), we assumed that 3ðn  1Þðn þ 1Þðn þ 2Þr=  ql R30  n þ 12 x22 . Now, we will re-examine this assumption. For case shown in Fig. 1 with n = 2, f2 = 500 kHz and R0 = 2.44  104 m, 3ðn  1Þðn þ 1Þðn þ 2Þr=ql R30 is 1.80  108 while ðn þ 12Þx22 is 2.47  1013. Hence, this assumption can be safely adopted. For other parameters (e.g., surface tension coefficient), their influence on interface stability can be directly inferred from Eq. (17) and will not be discussed separately. For the experimental measurement of critical bubble radius under dual-frequency excitation, according to our knowledge, there is still no available literature for comparison. In present model, the rectified mass diffusion across bubble– liquid interfaces are not included. According to the literature [20–23], the process of rectified mass diffusion is rather slow (i.e., only very limited gas diffused into or out of bubble through the interfaces in each cycle of bubble oscillations). Hence, for the study of transient behavior of bubbles (e.g., their instability and predictions of critical bubble radii), it is safe to ignore the effects of rectified mass diffusion. 4. Conclusions In present paper, theoretical analysis has been performed to study the instability of spherical bubble interface oscillating in the liquids induced by dual-frequency acoustic excitation. The influences of paramount parameters (e.g., acoustic pressure amplitudes) have been shown and discussed together with valid regions

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