# Study on Bubble Cavitation in Liquids for Bubbles Arranged in a Column Bubble Group
## 1 Introduction  
Cavitation is a typical hydrodynamic phenomenon that exists in many fields, such as hydraulic engineering [1], shipping [2], ultrasonic cleaning [3], underwater explosions [4], and the medical industry [5]. There are many reports on single and two cavitation bubble dynamics and the interaction between bubbles near a wall [6; 7; 8]. In fact, bubbles in liquids always exist in the form of a bubble group. The shape of bubble groups in liquids is very complicated and the vibration, translation, size, shape, and life of each bubble in a bubble group are not the same. Moreover, there is a weak mutual coupling between the bubbles in a bubble group [9]. It is of a great significance to study the utilization of cavitation and coupling between multiple bubbles [10; 11; 12]. [PERSON] et al. [13] have investigated the dynamic behavior of cavitation bubbles on solid surfaces via experiments and numerical simulations. They believed that the pressure generated by each individual bubble should be considered in the Rayleigh-Plesset equation, and that the outer bubble movement period was short. [PERSON] et al. [14] have studied the interaction of two bubbles of the same size in a free-field. [PERSON] [15] has investigated the dynamic equation of bubbles in a columnar bubble group and analyzed the kineticbehavior of bubbles and the characteristics of multi-bubble sonoluminescence. The study of bubble dynamics is beneficial to regulating and controlling the intensity of ultrasonic cavitation. [PERSON] [16] and [PERSON] [12] have pointed out that the velocity of a bubble micro-jet is in the range 20-150 m/s. [PERSON] et al. [17] have observed through high-speed photography that when small bubbles are near a wall, these small bubbles under the action of a shock wave directly hit the wall, which may cause a certain degree of denudation of the structure. [PERSON] et al. [18] have conducted experimental and numerical studies on the interaction of two bubbles in a free field. [PERSON] et al. [19] have found that injecting large bubbles in liquids can inhibit the occurrence and development of cavitation, and their research has mainly discussed the influence of the interactions between bubbles near a wall. The shape distribution of bubbles in water is very complicated, but with the modulation of ultrasonic waves, bubbles will aggregate into stable special structures such as \"bubble grapes\", \"cone-like bubble structures\", \"Acoustic lichtenberg figures\", and \"spherical\" structures [20; 21; 22; 23; 24]. Studies have shown that the secondary Bjerkens force between bubbles is a major factor in bubble clusters [25; 26; 27]. Classical Bjerknes force theory states that two interacting bubbles can only exhibit the following behaviors: if the ultrasonic frequency is between the resonance frequency of two bubbles, the bubbles will repel each other; if the ultrasonic frequency is beyond the resonance frequency of two bubbles, the bubbles will attract each other. Hence, it is very important to learn about the resonance frequency of bubbles. [PERSON] et al. [28] have simulated the coupling of two horizontally arranged bubbles under a free surface. However, the number of bubbles in a bubble group is very large, and the shape of the bubbles is very complicated. In order to analyze coupling effects between bubbles, the spherical and columnar bubble group models are widely used. [PERSON] et al. [29] concluded that the radius of bubbles in a champagne glass should be 0.94 mm according to the Minnaert frequency relationship by monitoring the radial acoustic frequency of bubbles at the bottom of a champagne glass. In fact, there is a weak coupling interaction between bubbles in a bubble group. This coupling effect is a kind of inhibitory effect on bubble growth, oscillation, and collapse; after this effect the resonance frequency of the bubble decreases [30; 31].