## 3 Results and Discussion  
### Bubble Motion  
#### 3.1.1 Bubble Motion near an Elastic Cylinder with Different Distances  
This section will discuss the effect of the dimensionless distance parameter \(\gamma\) of the elastic cylinder boundary on the bubble morphology. Therefore, the parameters of the elastic cylinder remain unchanged, the tension of the elastic cylinder is taken as 0.49 N, and the dimensionless radius of the elastic cylinder is approximately \(\gamma_{rod}=0.06\). Figure 4a depicts the motion state of the bubble in two cycles under the condition of a dimensionless distance \(\gamma=0.24\). The bubble expands to its maximum size at 0.92 ms and maintains a spherical shape during the expansion process, as shown at 0.39 ms and 0.60 ms. During the collapse process, the position where the elastic cylinder contacts the bubble will influence its shape. When the bubble collapses at 1.62 ms, the bottom contacts the surface of the elastic cylinder, and the bubble no longer maintains a completely spherical shape. At 1.72 ms, the bubble collapses, and the first cycle of the bubble ends. The bubble expands to the maximum in the second cycle at 1.97 ms. It can be observed that the bubble produces a jet away from the elastic cylinder in the first cycle, and the bubble with impurities and \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dimensionless Parameters** & **Based on** & **Letters** & **Corresponding Formula** \\ \hline Elastic cylinder radius & \(R_{max}\) & \(\gamma_{rod}\) & \(r_{rod}/R_{max}\) \\ distance & \(R_{max}\) & \(\gamma\) & \(d/R_{max}\) \\ density & \(\rho_{L}\) & \(\rho^{*}\) & \(\rho/\rho_{L}\) \\ velocity & \(R^{*}\) & \(\sqrt{\Delta P/\rho_{L}}\) & \(\dot{R}/\sqrt{\Delta P/\rho_{L}}\) \\ time & \(t^{*}\) & \(0.915R_{max}\sqrt{\rho_{L}/\Delta P}\) & \(t/0.915R_{max}\sqrt{\rho_{L}/\Delta P}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dimensionless parameters. Figure 3: Schematic of parameter measurement (**a**) \(d\), (**b**) \(D\) and \(d_{1}\), (**c**) \(d_{2}\). expansion in the second cycle produces displacement away from the wall. The situation where the bubble is slightly far from the elastic cylinder is shown in Figure 4b, and the dimensionless distance is \(\gamma=0.58\). Under this condition, the bubble basically maintains a spherical shape before reaching the maximum volume. At this time, the elastic wall has little effect on the bubble morphology. The bubble expands to its maximum volume at \(t=0.94\) ms. At \(t=1.70\) ms, due to the collapse of the bubble and the existence of the elastic cylinder, the bubble is no longer spherical, and a sharp shape appears near the elastic wall. This is caused by the wall being attracted to the bubble. After the 1.99 ms bubble has completely collapsed, it can be observed that when the second cycle of the 2.31 ms bubble expands to the maximum, the centroid of the bubble moves toward the elastic wall. The jet points to the elastic boundary. During the secondary expansion of the bubble, affected by the wall and with its residual velocity before the jet, the bubble moves along the jet direction and eventually collapses near the wall. The situation where the bubble is far away from the elastic cylinder is shown in Figure 4c; the dimensionless distance is \(\gamma=1.04\). Due to the long distance, the elastic wall has basically no effect on the shape of the bubble. The expansion and collapse of the bubble are basically spherical. There is no obvious difference between the bubble shape and the movement of the bubble in the free field. However, when the second cycle expands to the maximum moment (\(t=2.61\) ms), the displacement direction of the bubble's centroid is toward the elastic cylinder. From the above analysis, it is evident that the shape of the bubble is related to the dimensionless distance, \(\gamma\). Figure 4 presents interaction images of two-period bubbles and elastic walls in three typical cases.