## 3 Results and Discussion
### Effect of Bubbles on Displacement of Elastic Cylinder
#### 3.3.2 Force Analysis of Elastic Cylinder
In order to simplify the expression of the formula, the thrust coefficient applied by the bubble to the elastic cylinder is set to \(K\), \(K=\pi C_{D}\rho_{L}r_{rod}\); for a length of \(\Delta x\), the force \(F_{t}\) on the elastic cylinder boundary at time \(t\) is  
\[F_{t}=K\frac{\dot{R}^{2}R^{4}(d+S)^{2}}{((d+S)^{2}+x^{2})^{3}}\Delta x-F_{D} \frac{2S}{5^{2}+\frac{1}{4}L^{2}}\Delta x \tag{19}\]  
For the intersection of the elastic cylinder boundary perpendicular to the bubble (\(x=0\)), the length is \(\Delta x\), the mass of this section of the elastic cylinder \(M_{rod}=\pi r_{rod}^{2}\rho_{rod}\Delta x\), and the tension \(F_{D}=0\). Under this condition, \(S_{0t}\) is the displacement of the elastic cylinder boundary from the initial moment of the bubble to the contact time \(t\) between the bubble and the elastic cylinder boundary. According to the previous assumption, no displacement will occur after the elastic cylinder is wrapped by the bubble. Therefore, \(S_{0t}\) is the displacement of the elastic cylinder under the \(\gamma\leqslant 1\) state:  
\[S_{0t}=\frac{1}{2}\frac{F_{t}}{M_{rod}}t^{2} \tag{20}\]  
Figure 16: Schematic of the effect of dimensionless distance on the elastic cylinder wrapped by the bubble. (**a**) small \(\gamma\), (**b**) large \(\gamma\). Substitute the asymptotic velocity and time \(t\). At this time, \(t\) is taken as \(\gamma\leqslant 1\), and \(d=d+S_{0t}\). At this time, we obtain a quadratic equation about \(S_{0t}\). Solve this quadratic equation; the low-order infinitesimals about \(d\) within the square root are omitted, and the solution of \(S_{0t}\) is  
\[S_{0t}=\frac{A^{2}C_{D}\rho_{L}}{r_{rod}\rho_{rod}}d^{2} \tag{21}\]  
The parameter used for the dimensionless length in this paper is the maximum radius \(R_{max}\) of the bubble. The dimensionless elastic boundary displacement \(\gamma_{S}\) is  
\[F_{L}=\int_{-l}^{l}\frac{1}{2}C_{D}\rho_{L}u_{y}^{2}2\pi r_{rod}\mathrm{d}x \tag{15}\]0  
When \(\gamma>1\), the distance from the initial position of the bubble to the boundary of the elastic cylinder is greater than the maximum radius of the bubble after expansion. Since the bubble generates thrust on the bubble during expansion, the bubble continues to maintain a spherical shape during expansion and does not contact the boundary of the elastic cylinder. At this time, \(R=R_{max}\). In addition, \(d\gg S\) is \(d+S\approx d\) in this case, and \(t\) is like in the expression when \(\gamma>1\) in Equation (12), and the displacement \(S_{0t}\) at this time is obtained as follows:  
\[F_{L}=\int_{-l}^{l}\frac{1}{2}C_{D}\rho_{L}u_{y}^{2}2\pi r_{rod}\mathrm{d}x \tag{15}\]1  
When \(\gamma>1\), the dimensionless elastic boundary displacement \(\gamma_{S}\) is  
\[F_{L}=\int_{-l}^{l}\frac{1}{2}C_{D}\rho_{L}u_{y}^{2}2\pi r_{rod}\mathrm{d}x \tag{15}\]2  
The dimensionless elastic cylinder boundary displacement obtained after the above discussion on dimensionless distance is sorted out as follows:  
\[\gamma_{S}= \begin{cases}\frac{A^{2}C_{D}}{\gamma_{rod}\rho_{rod}^{\ast}}\gamma ^{2}&(\gamma\leqslant 1)\\ \frac{C_{D}CB^{2}}{2\gamma_{rad}\rho_{rod}^{\ast}}\gamma^{-4}&(\gamma>1)\end{cases}  \tag{25}\]