**Author details**

Ce Guo

analyzed. The results of numerical analysis illustrate that the bubble under the rigid boundary has a lower maximum radius and a longer collapse time than the bubble under the free boundary, which indicates that the rigid boundary has an inhibition effect for ultrasonic cavitation. The velocity of the bubble collapse decreases with the increase of the initial radius of the bubble, and it rises with the increase of the dimensionless distance from the bubble to the solid wall. Especially when the bubble reaches a certain value away from the solid wall, the bubble motion near the solid boundary can be approximated as the bubble motion under the free boundary. Whatever for the solid boundary and the free boundary, when the acoustic pressure amplitude

velocity of the bubble collapse rises approximately linearly, as the acoustic pressure amplitude

can increase first and then decrease. Thus, the optimal acoustic pressure amplitude can be obtained, at which the velocity of the bubble collapse can be up to maximum and cavitation effect is most violent. In addition, the velocity of the bubble collapse under the free boundary decreases faster than that under the rigid boundary, and then it can decrease as the ultrasonic frequency increases. Based on that, the relationship between the velocity of the bubble collapse and the microjet is established. It can be determined that the analysis for the collapse velocity of the bubble is contributed to seek the optimal value of the microjet and furthermore to achieve the purpose of indirect judgment and control microjets. Moreover, the velocity of the microjet obtained in the research is in the range of tens of micrometers, which is nearly the same magnitude with the experiments measured by Brujan and other scholars. Therefore, it can be considered that the bubble model and its relationship with the microjet have a certain reference value in theory, which provides an implication for further understanding the dynamics of

The research was primarily supported by the National Natural Science Foundation of China (51275490 and 50975265), the open foundation of Shanxi Key Laboratory of Advanced Manufacturing Technology (XJZZ201601-06) and the school fund of Taiyuan University of Technology (2016QNOZ). I would like to express my gratitude to all those who helped me during the writing of this work. Specially, I would like to express my gratitude to my students

Yang Fengyu and Yao Binting, who helped me to revise the English translation.

cavitation bubbles on the solid wall induced by the ultrasonic field.

, the ultrasonic cavitation cannot form in the liquid. For the free boundary, the

. However, the velocity of the bubble collapse under the rigid boundary

is less than 1 *p*<sup>0</sup>

86 Cavitation - Selected Issues

is greater than 1 *p*<sup>0</sup>

**Acknowledgements**

**Appendices and nomenclature**

*f* ultrasonic frequency

*c* speed of sound in the liquid *D* distance between two bubbles Address all correspondence to: guoce1027@163.com

College of Mechanical Engineering, Taiyuan University of Technology, Taiyuan, China
