The role of the bubble–bubble interaction on radial pulsations of bubbles

Abstract

Highlights Bubble–bubble interaction can suppress or enlarge the radial pulsations of bubbles. The suppression or enlargement depends on the ultrasound parameters, the radii, the distances and the bubble number. The frequency response curve of expansion ratio decreases and shifts to left due to bubble–bubble interaction. The larger the coupling strength is, the more the left-shifting is.

Nonlinear dynamics and acoustic emissions of interacting cavitation bubbles in viscoelastic tissues

“…Under ultrasound excitation, two bubbles with initial radii (R 10 and R 20 ) and positions (bubble centers x 10 and x 20 ) could occur radial and translational motions (i.e., R 1 (t), R 2 (t) and x 1 (t), x 2 (t)) over time, respectively. It is assumed that both bubbles remain spherical during their oscillations, and the mass exchange (i.e., evaporation/condensation and gas diffusion) at the gas–liquid interfaces and chemical reactions inside the bubbles are neglected for simplicity as demonstrated in earlier works [28] , [34] , [35] , [36] , [39] , [40] , [41] , [42] .

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“…The translational motions of bubbles were always neglected in previous studies [30] , [31] , [32] , [34] , [35] , [36] , [39] , [40] , nevertheless, the radial and translational motions of bubbles, as well as the viscoelastic drag experienced by bubbles while moving in the viscoelastic medium were comprehensively considered herein. The Keller-Miksis equations coupled with the bubble–bubble interactions were used as follows [28] , [42] : where R i (t), R j (t) and x i (t), x j (t) are the radii and center positions of the i th and j th bubble, respectively.…”

“…The overdot denotes the time derivative. The d(t) = |x i (t)-x j (t)| denotes the distance between the centers of bubbles i and j at any time t, c is the speed of sound in the surrounding medium, ρ is the medium density, D e,i is the viscoelastic drag acting on the i th bubble during its translational motion in the viscoelastic medium, p s,i (t) is the pressure at the wall of i th bubble, which is determined by [24] : where p g,i is the pressure inside the i th bubble, which is assumed to obey the van der Waals equation [32] , [35] , [40] : where σ is the surface tension, is the stress in radial direction at r = R i , which was introduced to be able to couple the cavitation model with the viscoelastic models. The p 0 is the atmospheric pressure, R i0 is the initial radius of the i th bubble, h i is the van der Waals hard-core radius for i th bubble, κ is the polytropic exponent of the gas within bubble.…”

“…to describe the cavitation bubble dynamics in soft tissues and to analyze the resultant acoustic emissions, but only focusing on a single bubble for simplicity [24] , [25] , [26] , [27] . For the bubble–bubble interactions, many researches have been focused on the translational motions or radial pulsations of bubbles in Newtonian fluids [28] , [29] , [30] , [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] , [39] , [40] . It has demonstrated that the bubble–bubble interactions make them behave very differently from individual bubbles.…”

“…It has demonstrated that the bubble–bubble interactions make them behave very differently from individual bubbles. Specifically, the expansion ratios of bubbles can be suppressed or enlarged, which largely depends on the ultrasound parameters, the ambient bubble radii, the distances between bubbles and the number of bubbles [35] , [39] , [40] . Recently, several studies have attempted to clarify the impacts of the viscoelasticity of the surrounding medium on the translational and/or radial motions of bubbles with considering bubble–bubble interactions [41] , [42] .…”