# Study on Bubble Cavitation in Liquids for Bubbles Arranged in a Column Bubble Group
## 5 Conclusions and Discussion  
In this paper, based on the dynamic equation of bubbles in a bubble group, the resonance frequency of bubbles in a columnar bubble group has been obtained by linear simplification of the bubbles' dynamic equation. The correction coefficient between the resonance frequency of bubbles in a columnar bubble group and the Minnaert frequency of a single bubble has been given. The vibration, growth, and collapse process of five and seven linear bubbles under the action of an ultrasonic field have been simulated by using hydrodynamics software. The results show that:  
1. There is a weak coupling interaction between bubbles in the bubble group which can change the resonance frequency of the bubbles. The resonance frequency of bubbles is affected by many factors such as the initial radius of the bubbles, the number of bubbles in a bubble group, and the distance between bubbles. With regard to these parameters, the initial radius of the bubbles  
Figure 11: The evolution of bubble radius with time in a columnar bubble group. (**a**) shows different driving ultrasonic frequencies; the driving ultrasonic frequency is 26 kHz and 40 kHz, 46 kHz. (**b**) shows different ultrasonic intensities the driving ultrasonic intensity is 1.2 Bar and 1.5 Bar.  
and the distance between bubbles have the most significant effects on the resonance frequency of the bubbles. For the same number of bubbles, when the distance between the bubbles is larger, the interaction between the bubbles becomes weaker. When the distance between the bubbles increases to 20 times the initial radius of the bubbles, the resonance frequency of the bubbles returns to that of a single bubble. In this case, the interactions between bubbles can be ignored.
2. The first bubble is affected by other bubbles behind it, and the resonance frequency reduces. The larger the number of bubbles, the stronger the interaction force between them, and the smaller the resonance frequency of the bubbles. However, when the number of bubbles increases to twenty and then continues to increase, the number of bubbles and the resonance frequency of the bubbles tends towards a fixed value.
3. Through Fluent simulation results it was found that due to the mutual restriction between bubbles in the expansion process, the energy ratio of the two bubbles at the edge reaches its maximum volume first in the expansion process. Later, the bubbles begin to contract. Through contracting, the bubbles slowly become, and remain, ellipsoid in nature. After this, the two bubbles begin to collapse, forming a micro jet which points to the back of the bubble and along the axis of the center of the linear bubble group, breaking the bubble until it is completely torn apart.
4. The maximum volume and collapse times of bubbles driven by sound parameters are different. The larger the driving frequency and the sound pressure are, the larger the maximum volume of the bubbles.  
Conceptualization, S.-y.L.; methodology, P.-l.Z.; software, P.-l.Z.; writing, P.-l.Z.; writing--review & editing, S.-y.L.  
This research was funded by the National Natural Science Foundation of China (grant nos. 50875132 and 60573172) and the Industrial Public Relation Project of Shaanxi Technology Committee, China (grant no. 2015 GY182).  
The authors declare no conflict of interest.