The nonlinear characteristics of the pulsations, translations and the secondary Bjerknes force

Dynamics of double bubbles under the driving of burst ultrasound

“… in Eq. (2) can be expressed as [23] , [24] In this paper, the driving is the acoustic pressure of burst ultrasound which can be expressed as with and f being the amplitude and the center frequency of burst, respectively. N decides the number of acoustic cycles in the burst.…”

“…N decides the number of acoustic cycles in the burst. The secondary Bjerknes force exerted by the th bubble to the i th bubble is given by [17] , [24] where is the volume of the i th bubble. The radii and positions of two bubbles can be acquired by solving Eqs.…”

“…The simulation results show if the driving is strong enough, the two bubbles will get close and move with a steady spacing rather than collide and coalesce. Zhang et al [24] introduced the bifurcations of radii and positions of double bubbles based on Doinikov’s model. Bremond et al [25] investigated the combination process of two bubbles with high speed camera, and compared the bubble pictures recorded during experiments to simulation results.…”

Influence of rigid wall on the nonlinear pulsation of nearby bubble

“…As in Refs. [12] , [13] , [14] , [17] , in each driving acoustic cycle, one point is stored for the illustration of bifurcation structure. In order to make sure that the dynamical state of bubble is stable, points are collected within 400 450 cycles.…”

“…The systems having same ratio share significant similarities in the bifurcating behavior and can be classified to the same category. Zhang et al [17] explored the nonlinear pulsations of double moving bubbles. In addition, they discussed the nonlinearity of secondary Bjerknes force with bifurcation diagrams.…”