Experimental and theoretical analysis of secondary Bjerknes forces between two bubbles in a standing wave
Introduction
The secondary Bjerknes force is a time-averaged interaction force between two bubbles driven in an acoustic field, named after C.A. Bjerknes and his son V.F.K. Bjerknes [1]. These authors derived a theoretical expression for this force with the following assumptions: (i) The surrounding medium is incompressible; (ii) the gas behaves adiabatically; (iii) the bubble size is smaller than their distance of separation and the bubbles remain spherical; (iv) any higher order scattering from the bubbles is negligible compare to the primary scattering and (v) the driving pressure is low and the bubbles oscillate linearly with the driving frequency. Under these conditions, the secondary Bjerknes force is repulsive if the driving frequency lies between the resonance frequencies of the two bubbles, otherwise the force is attractive [2]. This has been verified experimentally by Kuznetsov [3] and Crum [4]. However, this linear theory fails to describe experimental observations of stable bubble clusters, which can only be explained by a reversal in the secondary Bjerknes force from attractive to repulsive. Oguz and Prosperetti [5] were able to theoretically demonstrate the sign reversal in the secondary Bjerknes force by considering nonlinear bubble oscillations at high driving pressures. However, their analysis was only applicable for bubbles that are far apart.
When the distance of separation is comparable to the bubble size or when the driving pressure is high, assumptions (i), (iii) and (iv) are no longer valid. Zabolotskaya [6], [7] modified Bjerknes’ classical linear theory to account for twofold scattered waves from bubbles, this was later extended by Doinikov and Zavtrak [8] to consider multiple scattering between the bubbles. Both concluded that when bubbles, driven below resonance, move closer to each other, the natural frequency of the multibubble system shifts. This phase shift in the bubble oscillation can change the sign of the secondary Bjerknes force. A similar result was reported by Ida [9], [10], [11], and by Pelekasis and Tsamopoulos [12], [13] who allowed for shape deviations of the bubbles from sphericity, caused by subharmonic resonances. Doinikov [14] derived an expression to account for translational oscillations of the bubbles, the vorticity of the linear scattered field and acoustic streaming, and demonstrated that viscous effects can cause repulsion between bubbles driven below resonance. Barbat et al. [15], [16] modified the linear theory for the secondary Bjerknes force by introducing a model for the coupling between the pulsations of the bubble interfaces. This model allowed a more general evaluation of the dynamic behavior of bubble motion.
Although there are many theoretical studies in the literature, these are mainly focused on the conditions which cause the sign change in the secondary Bjerknes force and are often reported in a set of contour plots [14], [17], [18] showing regions of attraction and repulsion in a range of bubble radii, driving frequencies and pressures. There are in fact small variations in the magnitude of the secondary Bjerknes force within the regions of attraction and repulsion that are rarely discussed. Experimental evidence of the secondary Bjerknes force is limited due to the practical difficulties in measuring these forces in an acoustic field [8], especially at short separation distances where the secondary Bjerknes force is the main dominant force [19]. Therefore, existing experimental studies are either qualitative [20], [21] or restricted to long bubble-bubble separations (>2 mm) and large bubble sizes in the order of millimeters [4], [16]. Recently, trajectory and sign change in secondary Bjerknes force for bubbles with sizes in the micrometer range and with shorter distances have been reported [17], [22], [23], however, these experiments were conducted with the bubbles near or adhered to a surface. In addition, due to difficulties associated with these experiments, usually one ultrasound frequency and power level alone is studied.
In this study, the trajectories of two micrometer size bubbles, trapped in an acoustic standing wave field, were measured experimentally to obtain the relative acceleration and secondary Bjerknes force. Different acoustic frequencies, powers, bubble sizes and separations were investigated. The experimental data were compared against theoretical predictions of secondary Bjerknes forces using coupled equations of radial and translational motions reported by Doinikov [24] and explained via a different visualisation of the simulation. The results will be of interest for understanding and modeling bubble dynamics in an acoustic field.
Section snippets
Experimental details
An imaging system (Fig. 1(a)), similar to that described by Jiao et al. [19], was used to collect experimental data. A cylindrical Pyrex cell containing two parallel flat surfaces for viewing and back lighting purposes was used. A hollow cylindrical transducer was attached to the bottom of the cell. The transducer (Z7, American Piezo Ceramics Inc., Mackeyville, PA, U.S.A.) was driven at the required frequency (20–100 kHz) by a frequency generator (HM8131-2, Hameg Instruments GmBH, Mainhausen,
Results and discussion
The trajectories of two approaching bubbles were captured under different frequencies and pressures. Selected trajectories of bubbles with similar sizes, driven under the same acoustic pressure but different frequencies, are shown in Fig. 2. For all frequencies, the approach velocity increases as the distance between the two bubbles is reduced. Although 80 kHz and 100 kHz appear to have similar approach velocities, they are much faster compare to the two lower frequencies. This can be explained
Conclusions
In this study, we have experimentally obtained the trajectories and relative accelerations between two approaching bubbles under a range of acoustic frequencies, pressures, bubble sizes and separations. Under these conditions, the secondary Bjerknes forces were all attractive and only variations in the magnitude of the forces were observed. Using a previously published simulation model involving coupled equations of radial and translational motions, we have demonstrated that the observed
Acknowledgements
We are grateful to Prof. Yos Morsi for facilitating the use of the high-speed camera and for detailed comments on a draft of this paper. The financial support through the Australian Research Council for the DECRA (Discovery Early Career Research Award, DE120101567) is also gratefully acknowledged.
References (30)
A characteristic frequency of two mutually interacting gas bubbles in an acoustic field
Phys. Lett. A
(2002)- et al.
Planar dynamics of two interacting bubbles in an acoustic field
Appl. Math. Comput.
(2004) - et al.
Study on the bubble transport mechanism in an acoustic standing wave field
Ultrasonics
(2011) Fields of force
(1906)Effects of the second harmonic on the secondary Bjerknes force
Phys. Rev. E
(1999)Motion of bubbles in a liquid under the action of Bjerknes forces arising in an acoustic field
Sov. Phys. Dokl.
(1960)Bjerknes forces on bubbles in a stationary sound field
J. Acoust. Soc. Am
(1975)- et al.
A generalization of the impulse and virial theorems with an application to bubble oscillations
J. Fluid Mech.
(1990) Phase conjugation of sound beams in connection with four-phonon interaction in a liquid containing gas bubbles
Sov. Phys. Acoust.
(1984)Nonlinear waves in liquid with gas bubbles
Tr. Inst. Ob. Fiz. Akad. Nauk
(1989)
On the mutual interaction of two gas bubbles in a sound field
Phys. Fluids
Number of transition frequencies of a system containing an arbitrary number of gas bubbles
J. Phys. Soc. Jpn
Alternative interpretation of the sign reversal of secondary Bjerknes force acting between two pulsating gas bubbles
Phys. Rev. E
Bjerknes forces between two bubbles. Part 1. Response to a step change in pressure
J. Fluid Mech.
Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field
J. Fluid Mech.
Cited by (45)
А particle model of interaction between weakly non-spherical bubbles
2024, Applied Mathematical ModellingRefined secondary Bjerknes force equation for double bubbles with pulsation, translation, and deformation
2024, Ultrasonics SonochemistryNumerical investigation of the translational motion of bubbles: The comparison of capabilities of the time-resolved and the time-averaged methods
2023, Ultrasonics SonochemistryCitation Excerpt :The primary Bjerknes force describes the effect of acoustic pressure gradient on single bubbles, and it gathers the bubbles in certain areas, such as pressure nodes or antinodes in the case of a standing sound field [77–79,78]. The bubble–bubble interaction via the emitted pressure is described by the secondary Bjerknes force that can attract or repel neighbouring bubbles [80–82]. In addition, the bubble–bubble interaction has an influence on the radial dynamics as well, which was observed numerically [83,59,84] and experimentally [85].
Ultrasound-assisted extraction of lipids as food components: Mechanism, solvent, feedstock, quality evaluation and coupled technologies – A review
2022, Trends in Food Science and TechnologyCitation Excerpt :The process is illustrated in detail in Fig. 1. As shown in Fig. 1(a), ultrasound is a longitudinal wave that can rarefy and compress the medium on the propagation path, thus mandatorily forming a nonuniform pressure field and forcing uniform distributed medium to rapidly vibrate (Jiao et al., 2013, 2015; Zhang et al., 2016). The rarefaction field has a smaller pressure, and the compression field has a larger one, as shown in Fig. 1(b).