## 3 Results and Discussion
### Effect of Bubbles on Displacement of Elastic Cylinder
#### 3.3.3 Results on the Displacement of the Elastic Cylinder  
In the previous section, we derived the relationship between the displacement of the elastic cylinder and the dimensionless distance when the \(\gamma\) value is different, where \(\gamma=1\) is used as the boundary; that is, whether the bubble and the elastic cylinder are in contact is used to distinguish between the two conditions. The following will show and analyze the effects of dimensionless distance and tension on the displacement of the elastic cylinder. Whether near a rigid wall or an elastic wall, dimensionless distance is an important factor affecting the bubble morphology and boundary state. Figure 17 shows the relationship between dimensionless distance and the maximum displacement of elastic cylinders of different sizes. The blue circle in the figure is the maximum displacement of the elastic cylinder during bubble expansion. In this paper, the displacement direction of the elastic cylinder is defined as positive; the red circle is the maximum displacement of the elastic cylinder during bubble collapse. The maximum displacements of the expansion and collapse of elastic cylinders of the same size are plotted on the same \(\gamma-\gamma_{S}\) figure. The curve shown in Figure 17 is the result drawn according to Formula (25), where the values are \(A=0.4\), \(B=0.5\), \(C=\pi/7\), \(C_{d}=0.16\), and \(\rho_{rod}^{*}=1\). It can be seen that the larger the \(\gamma_{rod}\), the better the maximum displacement during the expansion, which is consistent with the data points in the figure. When the dimensionless distance \(\gamma>4\), the bubble basically has no effect on the displacement of the elastic cylinder, which also means that the elastic cylinder does not affect the morphology of the bubble as a boundary. In addition, the derivation of the maximum displacement of the elastic cylinder in the above text is correct, and it can describe the trend and size of the maximum displacement of the elastic cylinder. The dimensionless distance range of the elastic cylinder displacement is mainly \(0<\gamma<4\), among which the maximum displacement occurs in the range of \(0.5<\gamma<1\). The reason why the elastic cylinder produces the maximum displacement in this range is worthy of further study. The concept of impulse will be used for analysis and discussion below. For the discussion in the previous section, there exists \(R_{ref}\), which is shown as \(R_{ref}=4R_{max}\) in Figure 17. When \(d<R_{ref}\), the movement of the bubble will push or attract the elastic cylindrical boundary to produce displacement. Due to the difference in dimensionless distance, the time when the bubble contacts the elastic cylindrical boundary is also different. The premise of this study assumes that when the bubble contacts or wraps the elastic cylinder boundary, the elastic cylinder will no longer continue to displace. When the dimensionless distance is small, the bubble has a relatively large initial velocity at the beginning of its expansion but contacts the elastic cylinder quickly. The accumulation of movement time is small, so the elastic cylinder boundary will not produce a large displacement. When the dimensionless distance is large, the elastic cylinder receives a small thrust from the bubble at the position, and there is basically no speed transmission. Even if the time of the entire bubble expansion cycle is accumulating, it still will not produce a large displacement. However, when the dimensionless distance is appropriate, the bubble has both a higher velocity and a longer time to accumulate displacement, and the displacement  
Figure 17: Relationship between dimensionless distance and maximum displacement of elastic cylinders of different sizes. The red color is for the expansion process, the blue is for the collapse process. of the elastic cylinder is large at this time. This setting can theoretically explain why the maximum value of the maximum displacement is in the interval \(0<\gamma<1\).