## 3 Results and Discussion
### Effect of Bubbles on Displacement of Elastic Cylinder
#### 3.3.1 Model of Elastic Cylinder Displacement  
The influence of bubbles on the elastic rope can be divided into several stages. In the first period, when the bubbles are active, the bubbles will push the elastic cylinder to move, and the elastic cylinder will produce the maximum displacement along the expansion direction. Since the two ends are fixed, the elastic cylinder is arc-shaped as a whole. This state continues until the bubble expands to the maximum radius. The bubble undergoes the first cycle of collapse motion. When the elastic cylinder is not wrapped, the bubble is clearly pulling it to move in the collapse direction. When the elastic cylinder is in contact with or wrapped by the bubble, the boundary is not pulled to displace until the bubble is about to break away from the boundary of the elastic cylinder in the late stage of collapse. After this period of movement, the elastic cylinder will make periodic fluctuations. Due to  
Figure 14: Trends in dimensionless maximum displacement and dimensionless distance during expansion and collapse for some experiments. the complexity of its movement and the importance of the maximum displacement, this paper only focuses on the study of the maximum displacement of the bubble's expansion and collapse in the first movement cycle. In order to study the displacement of the elastic cylinder, a bubble motion model and an elastic cylinder force model are proposed, and then the maximum displacement is derived and verified. Assume that the bubble radius moves in a spherical motion in the form of \(R=R(t)\), and set the cylinder velocity surface to a sphere with a radius of \(r\) and the same center as the bubble, with a velocity of \(u_{r}\). According to mass conservation in the incompressible flow field,  
\[\rho(4\pi r^{2})u_{r}=\rho(4\pi R^{2})\dot{R} \tag{4}\]  
The velocity \(u_{r}\) on the concentric sphere of the bubble is obtained as follows:  
\[u_{r}=\dot{R}(R^{2}/r^{2}) \tag{5}\]  
The velocity in the \(y\) direction is decomposed at the position where the concentric sphere contacts the boundary of the elastic cylinder, and the velocity in the \(y\) direction at the position where the concentric sphere contacts the elastic cylinder is obtained:  
\[u_{y}=u_{r}sin\theta=\dot{R}(R^{2}/r^{2})d/r \tag{6}\]  
When \(R\gg R_{0}\), the asymptotic growth rate of the cavitation bubble is  
\[\dot{R}\rightarrow\sqrt{\left(C*\frac{p_{V}-p_{\infty}}{\rho_{L}}\right)} \tag{7}\]  
The MRG algorithm yields \(C=2/3,7/\pi\). The acceleration at the initial stage of cavitation growth is  
\[\ddot{R}(0)=\frac{p_{\infty}(0)-p_{\infty}^{*}}{\rho_{L}R_{0}} \tag{8}\]  
The initial growth time of the bubble can be calculated as  
\[t=\left\{C*\frac{\rho_{L}R_{0}^{2}(p_{V}-p_{\infty})}{(p_{\infty}(0)-p_{ \infty}^{*})^{2}}\right\}^{1/2} \tag{9}\]  
This paper proposes the existence of a distance reference value \(R_{ref}\), where \(R_{ref}\) is the limit distance at which the elastic cylinder boundary can produce deformation. When \(d\geqslant R_{ref}\), that is, \(\gamma>R_{ref}/R_{max}\), the elastic cylinder boundary will not be displaced because the distance is long enough. When \(R_{ref}\geqslant d>R_{max}\), that is, \(R_{ref}/R_{max}\geqslant\gamma>1\), at this distance, according to the discussion in the previous section, since the size of the elastic cylinder is not large enough, and the two ends of the boundary are fixed but elastic, the expansion movement of the bubble can push the elastic cylinder to displace. The bubble can continue to maintain a spherical shape during the expansion process, and this state conforms to the Rayleigh-Plesset equation. At this time, it can be concluded that the time for the bubble to move to the maximum radius is  
\[t=0.915R_{max}\sqrt{\frac{\rho_{L}}{p_{\infty}-p_{V}}} \tag{10}\]  
When \(R\leqslant R_{max}\), that is, \(\gamma\leqslant 1\), the bubble in the expansion process will make contact with the elastic cylindrical boundary.