## 3 Results and Discussion
### Effect of Bubbles on Displacement of Elastic Cylinder
#### 3.3.2 Force Analysis of Elastic Cylinder  
The forces on the orange dots are shown in Figure 15. The displacement of the elastic cylinder boundary is mainly caused by the thrust or pull of the bubble and the pull on the elastic cylinder boundary itself. First, it is subjected to the internal pull \(F_{D}\), and the direction of the pull is tangent to this circle. The force caused by the bubble movement on the elastic cylinder is analyzed below. In this paper, the elastic cylinder is assumed to be subjected to the lift force given by the bubble in the \(y\) direction. The lift formula is \(F_{L}=\frac{1}{2}C_{D}V^{2}W\). This formula is used to calculate the lift generated by a fluid on the surface of an object. In this formula, the lift coefficient \(C_{D}\) is a very important parameter. It represents an object's efficiency in generating lift, and its value depends on many factors, such as the shape of the object, the angle, the properties of the fluid, and the speed. \(V\) is the velocity of the fluid, usually the velocity of air, because in most aviation applications, the fluid is air. \(V\) in this paper is the velocity of the water flow. \(W\) represents the surface area of the object in the fluid, usually the area of the obstacle. In this paper, the water velocity is \(u_{y}\), and the surface area is the area of the local elastic cylinder \(2\pi r_{rod}\mathrm{d}x\). Substitute it into the lift formula:  
\[F_{L}=\int_{-l}^{l}\frac{1}{2}C_{D}\rho_{L}u_{y}^{2}2\pi r_{rod}\mathrm{d}x \tag{15}\]  
According to Formula (15), the length of the elastic cylinder is \(\Delta x\), and the force exerted by the bubble on this section of the elastic cylinder is  
\[F_{L}=\pi C_{D}\rho_{L}u_{y}^{2}r_{rod}\Delta x \tag{16}\]From the approximate circle \(r^{2}=(d+s)^{2}+x^{2}\) of the elastic cylinder and \(u_{r}=\dot{R}(R^{2}/r^{2})\), the component of the elastic cylinder's velocity in the \(y\) direction can be obtained:  
\[u_{y}=u_{r}sin\theta_{1}=\dot{R}R^{2}(d+s)/((d+s)^{2}+x^{2})^{3/2} \tag{17}\]  
First, the displacement during the bubble expansion process is discussed. During this process, the elastic cylinder boundary is subjected to the thrust caused by the bubble expansion and the tension \(F_{D}\) exerted on the elastic cylinder itself. At this time, the force \(F_{t}\) exerted on the elastic cylinder boundary in the \(y\) direction at time \(t\) is  
\[F_{t}=\int_{-\frac{t}{2}}^{\frac{t}{2}}\pi C_{D}\rho_{L}r_{rod}\frac{\dot{R}^{ 2}R^{4}(d+S)^{2}}{((d+S)^{2}+x^{2})^{3}}\mathrm{d}x-\int_{-\frac{t}{2}}^{\frac {t}{2}}F_{D}\frac{S}{\sqrt{S^{2}+\left(\frac{t}{2}-x\right)^{2}}}\mathrm{d}x \tag{18}\]  
Regarding the force \(F_{t}\) on the boundary of the elastic cylinder, we will continue to discuss its several classifications using the dimensionless distance proposed in the previous section. The main situations discussed are shown in Figure 16. Figure 16a shows the dimensionless \(\gamma\leqslant 1\), that is, where the bubble contacts or even wraps the elastic cylindrical boundary after expansion movement. Figure 16b shows \(\gamma>1\). That is, the dimensionless distance is long, and the bubble will expand in a spherical state until the maximum radius. When \(\gamma\leqslant 1\), the bubble will wrap around the elastic cylinder boundary during the expansion process. The radius of the bubble when wrapping around the elastic cylinder boundary is \(R=d+S\), which is the sum of the distance from the initial position of the bubble to the elastic cylinder and the displacement of the elastic cylinder boundary caused by the bubble expansion.