Dynamics of two interacting bubbles in an acoustic field

This paper contains theoretical and experimental results on the relative motion of two pulsating spherical bubbles along their line of centres, in a liquid subjected to an acoustic field. The motion is caused only by the secondary Bjerknes forces. The linear theory for the secondary Bjerknes forces is modified by introducing a model for the coupling between the pulsations of the interfaces. The secondary effects introduced by this model are determined by the frequency indices of the bubbles, defined as the ratio of the forcing frequency to the resonance frequency of each bubble. The equations of motion are set up with the conservative Lagrangian formalism. This approach allows an analytical study of all the possible patterns of motion and the identification of the set of governing parameters: total energy and interaction coefficient. A pair of bubbles driven far from their resonance frequencies may attract or repel, depending on whether their frequency indices are respectively on the same side or on either side of unity. For forcing frequencies close to resonance, the proposed model predicts a new pattern of relative motion, namely a periodic motion (oscillations) around an equilibrium bubble separation. The experimental study identifies this new periodic pattern of motion, for acoustically levitated bubbles of nearly equal sizes, forced near their resonance frequency. A quantitative study on the variation of the relative velocity with the separation between the bubbles shows that the conservative model for the motion holds for large and moderate separations. The following information is reported: (a) a classification of the pairs of bubbles, based upon their phase difference in oscillations; (b) a model for the coupling of the pulsations of two bubbles; (c) formulas for the interaction force field of two pulsating bubbles, for all of the categories; (d) a study of all possible patterns of relative motion (collisions, scattering and oscillations), with their conditions of occurrence; (e) experimental data for two attracting bubbles; (f) experimental data for two oscillating bubbles.