Acoustic response of bubbles inside a cylindrical cavitationbubble cluster generated by low-frequency ultrasound
Authors: Wang Cheng-Hui  , Mo Run-Yang  , Hu Jing  ,   Shaanxi Key Laboratory of Ultrasonics, Shaanxi Normal University, Xi'an 710062, China
Published: December 2020

An ultrasonic horn can radiate a strong ultrasonic wave into viscous liquid contained in a tank or cylindrical cup, and bubble clusters could be generated by the high-intensity ultrasound in the liquid. In the bubble clusters, interaction of bubbles exists because of the secondary radiation of bubbles. Therefore, the oscillations of bubbles are coupled. On the other hand, the surrounding liquid pressure of the bubbles in the cluster is influenced by the oscillations of the bubbles, which induces a pressure gradient on the boundary of the cluster. Therefore, the oscillation of a bubble inside a cluster is contracted by the formation of the cluster and its structure evolution. In this paper, a cylindrical cavitation bubble cluster is considered as a mixture drop of bubbles and liquids, and the motion of the cluster boundary is proposed with a second two-dimensional (2D) Rayleigh equation related to the difference between the inner mixture pressure and the outside liquid pressure on the boundary. Based on the bubble cluster boundary dynamical equation, a new mathematical model is developed to describe the motion of cavitation bubbles inside a cylindrical cluster when the effects of coupled oscillation are included. Comparing the new model equation with the Rayleigh-Plesset equation of single bubble in unbounded liquid, it is easy to draw the conclusion that the contraction of oscillating bubbles is strengthened by the coupled oscillation of bubbles and the boundary motion. In the cylindrical cluster, the oscillation of bubbles is suppressed, and the natural frequency of bubbles is reduced. The proposed model is used as a basis for the numerical investigation of the nonlinear acoustic response of bubbles. The suppression of the bubble oscillation is strengthened by increasing the number density of bubbles. Comparing numerical curves of the maximum radius of the oscillating bubble, it is shown that there are local peaks which are related to the resonance response of bubbles. In some unstable parameter regions, the maximum radius of the oscillating bubble varies sensitively with the tiny change of the parameters. The parameter space distribution of the unstable regions is related to the initial bubble radius and driving frequency of ultrasound. According to the numerical results related to the parameters, such as bubble number density, initial radius, driving frequency and pressure amplitude of ultrasound, it is found that the unstable acoustic response could be amplified for bubbles of smaller initial radius driven by a low-frequency ultrasound. For cavitation bubbles of initial radii ranging from 1 m to 10 m in low-frequency ultrasonic field, the unstable regions of parameter spaces related to the evolution of maximum radius become broader with the decrease of bubble initial radius and driving frequency of ultrasound. Therefore, the tiny bubbles inside cylindrical clusters have stronger nonlinear properties and the change of the parameters in the dynamical model equation has greater influence on the tiny bubbles.