Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field

Abstract

The motion of two gas bubbles in response to an oscillatory disturbance in the ambient pressure is studied. It is shown that the relative motion of bubbles of unequal size depends on the frequency of the disturbance. If this frequency is between the two natural frequencies for volume oscillations of the individual bubbles, the two bubbles are seen to move away from each other; otherwise attractive forces prevail. Bubbles of equal size can only attract each other, irrespective of the oscillation frequency. When the Bond number, Bo (based on the average acceleration) lies above a critical region, spherical-cap shapes appear with deformation confined on the side of the bubbles facing away from the direction of acceleration. For Bo below the critical region shape oscillations spanning the entire bubble surface take place, as a result of subharmonic resonance. The presence of the oscillatory acoustic field adds one more frequency to the system and increases the possibilities for resonance. However, only subharmonic resonance is observed because it occurs on a faster timescale, O(1/ε), where ε is the disturbance amplitude. Furthermore, among the different possible periodic variations of the volume of each bubble, the one with the smaller period determines which Legendre mode will be excited through subharmonic resonance. Spherical-cap shapes also occur on a timescale O(1/ε). When the bubbles are driven below resonance and for quite large amplitudes of the acoustic pressure, ε ≈ 0.8, a subharmonic signal at half the natural frequency of volume oscillations is obtained. This signal is primarily associated with the zeroth mode and corresponds to volume expansion followed by rapid collapse of the bubbles, a behaviour well documented in acoustic cavitation experiments.

Experimental and theoretical analysis of secondary Bjerknes forces between two bubbles in a standing wave

“…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.…”

Coupled pulsation and translation of two gas bubbles in a liquid

“…In the case of two initially identical bubbles and a step change in the externally applied pressure field, typically either spherical-cap shapes or globally deformed bubbles are observed (Pelekasis 1991;Pelekasis & Tsamopoulos 1993a) in the attractive regime, leading to an eventual breakup of the bubbles. For oscillatory applied fields, Pelekasis (1991) and Pelekasis & Tsamopoulos (1993b) report agreement with the classical linear theory for low forcing amplitudes. By contrast, for oscillatory forcing with large amplitudes and with frequency in between the natural (volume) frequencies, they identify a subharmonic resonance at half the natural frequency of the bubble with smaller period that significantly alters the shape deformations and the interaction force.…”

On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations

“…Over the past decades a significant amount of research has been devoted to the study of single bubble dynamics (see [39] for a review on earlier work on the subject), as well as in the problem of bubblebubble interaction [40]. The theoretical approach to the secondary Bjerknes force assuming inviscid fluids by Pelekasis and Tsamopoulos [41,42] has produced interesting results in a wide range of forcing frequencies, pressure amplitudes and bubble sizes. The surface of the bubbles has been allowed to deform from spherical retaining its axial symmetry.…”