## 3 Results and Discussion
### Bubble Motion
#### 3.1.3 Bubble Motion near an Elastic Cylinder with Different Rod Sizes  
First, the size of the elastic boundary \(r_{rod}\) is dimensionless, \(\gamma_{rod}=r_{rod}/R_{max}\), where \(r_{rod}\) is the actual size of elastic cylinder, and \(R_{max}\) is the bubble radius when the bubble expands to the maximum. This section discusses the interaction between the bubble and elastic cylinder under four different dimensionless radius conditions, where \(\gamma_{rod}=0.02\), \(0.04\), \(0.064\), and \(0.10\). Figure 8 shows bubble morphology images when \(\gamma_{rod}=0.02\) with typical dimensionless distances. In Figure 8a, the elastic cylinder is much smaller the bubble (\(\gamma_{rod}=0.02\)), and the initial position of the bubble is close to the elastic cylinder. At \(0.44\) ms, it can be  
observed that the bubble pushes the elastic cylinder to move in the direction of bubble expansion, and the bubble wraps part of the elastic cylinder. At \(t=0.76\) ms, the elastic cylinder wrapped inside the bubble is no longer affected by the force and continues to displace. This state continues until the bubble expands to its maximum radius. In the collapse stage, the reduction in the bubble radius and the reverse acceleration during collapse will cause the displacement of the elastic cylinder to change, but the elastic cylinder of this size does not change the bubble shape much, and it still basically maintains a spherical shape. In Figure 8b, the initial position of the bubble is slightly far from the elastic cylinder (\(\gamma=0.60\)), and the bubble basically remains spherical during expansion. During the collapse stage, due to the interaction between the elastic cylinder and the bubble, the bubble remains spherical on the side away from the elastic cylinder and becomes teardrop-shaped on the side close to the elastic boundary. It can be clearly observed that at this dimensionless distance, the displacement caused by the bubble collapse is larger than the expansion, which is related to the collapse velocity acceleration. As shown in Figure 8c, the initial position of the bubble is far away from the elastic cylinder (\(\gamma=1.11\)), and the shape of the bubble is no longer affected by the existence of the elastic boundary. Figure 9 shows the bubble morphology at several typical dimensionless distances when \(\gamma_{rod}=0.04\). Figure 9a shows that the initial position of the bubble is close to the elastic boundary, and the bubble will contact the elastic boundary during expansion (\(\gamma=0.40\)). The bubble will push the elastic cylinder during expansion, causing it to remain spherical until the bubble contacts the boundary (\(t=1.01\) ms). When the bubble collapses, it interacts with the elastic cylinder. At \(t=1.81\) ms, the bubble no longer maintains a spherical shape but is similar to a mushroom or light bulb shape. For a rigid cylinder, a mushroom-shaped bubble is likely to cause a jet away from the wall. At \(t=1.53\) ms, the image of the bubble expanding to the maximum radius in the second cycle shows that there is a jet away from the elastic wall. Figure 4 shows bubble morphology images at several typical dimensionless distances when \(\gamma_{rod}=0.06\). The process of changing bubble morphology is similar to that of \(\gamma_{rod}=0.04\). Figure 10 shows bubble morphology images at several typical dimensionless distances when \(\gamma_{rod}=0.10\). This elastic cylindrical boundary was the largest in the experiment, and it is similar to and different from the previous sizes. Figure 10a shows that the initial position of the bubble is close to the elastic boundary (\(\gamma=0.24\)); the bubble contacts the elastic cylinder during expansion. The side away from the elastic boundary still basically maintains a spherical shape, and the side close to the boundary will produce a reduction in radius and no longer maintain a consistent spherical shape. This is similar to the situation shown previously in Figure 4a.