Investigation of transition frequencies of two acoustically coupled bubbles using a direct numerical simulation technique

The theoretical results regarding the ``transition frequencies'' of two acoustically interacting bubbles have been verified numerically. The theory provided by Ida [Phys. Lett. A 297 (2002) 210] predicted the existence of three transition frequencies per bubble, each of which has the phase difference of $\pi /2$ between a bubble's pulsation and the external sound field, while previous theories predicted only two natural frequencies which cause such phase shifts. Namely, two of the three transition frequencies correspond to the natural frequencies, while the remaining does not. In a subsequent paper [M. Ida, Phys. Rev. E 67 (2003) 056617], it was shown theoretically that transition frequencies other than the natural frequencies may cause the sign reversal of the secondary Bjerknes force acting between pulsating bubbles. In the present study, we employ a direct numerical simulation technique that uses the compressible Navier-Stokes equations with a surface-tension term as the governing equations to investigate the transition frequencies of two coupled bubbles by observing their pulsation amplitudes and directions of translational motion, both of which change as the driving frequency changes. The numerical results reproduce the recent theoretical predictions, validating the existence of the transition frequencies not corresponding to the natural frequency.