## 3 Results and Discussion
### Effect of Bubbles on Displacement of Elastic Cylinder
#### 3.3.3 Results on the Displacement of the Elastic Cylinder
This involves the accumulation of force given to the elastic cylinder by the bubble, and the accumulation of force over time is also the definition of impulse. Therefore, the relationship between the displacement of the elastic cylinder can be proved by the impulse received by the elastic cylinder. That is, there is a dimensionless distance that makes the impulse have a maximum value, which is manifested as the maximum displacement of the elastic cylinder boundary having a maximum value. In classical mechanics, the impulse of the resultant external force on an object is equal to the increment in its momentum (i.e., the final momentum minus the initial momentum), which is called the momentum theorem. Impulse is a process quantity that represents the cumulative effect of a force on an object over a period of time. \[I=\int_{0}^{t}F_{t}\mathrm{d}t \tag{26}\]  
For the dimensionless conversion of impulse, impulse is equal to the increment in momentum; that is, the units of impulse and momentum are the same. In this paper, the momentum of bubble drainage is used to make the impulse of elastic cylinder dimensionless, where the momentum of bubble drainage \(I_{b}\) is \(\gamma<1\):  
\[I_{b}=M_{b}V_{b}=\frac{1}{2}\rho_{L}*\frac{4}{3}\pi d^{3}*A\dot{R} \tag{27}\]  
So, the dimensionless impulse \(I^{*}\) of the elastic cylinder is  
\[I^{*}=\frac{I}{I_{b}}=\frac{3 QC_{D}\gamma_{rad}}{8A^{2}}\gamma^{-2} \tag{28}\]  
where \(\Delta x=QR_{max}\). For the case where the dimensionless distance is slightly larger (\(\gamma\geqslant 1\)), by substituting the time and the force into the Equation (26), we obtain the following expression:  
\[I=K\Delta x\dot{R}^{2}\rho_{L}\frac{R_{max}^{4}}{d^{4}}BR_{max}\sqrt{\frac{ \rho_{L}}{p_{V}-p_{\infty}}} \tag{29}\]  
At this time, the impulse of the bubble is the product of the maximum volume of the bubble and the average velocity:  
\[I_{b}=M_{b}V_{b}=\frac{2\rho_{L}*\frac{4}{3}\pi R_{max}^{3}*R_{max}}{BR_{max} \sqrt{\frac{\rho_{L}}{p_{V}-p_{\infty}}}} \tag{30}\]  
In this case, the dimensionless impulse \(I^{*}\) is  
\[I^{*}=\frac{I}{I_{b}}=\frac{3 QC_{D}B^{2}\gamma_{rad}}{2}\gamma^{-4} \tag{31}\]  
To summarize, there are two types of dimensionless impulse:  
\[I= \begin{cases}\frac{3 QC_{D}\gamma_{rad}}{8A^{2}}\gamma^{-2}&(\gamma\leqslant 1 )\\ \frac{3 QC_{D}B^{2}\gamma_{rad}}{\gamma^{-4}}&(\gamma>1)\end{cases}  \tag{32}\]Substitute \(A=0.2,B=1,C_{d}=0.16,\gamma^{\star}_{rod}=0.1\), and \(Q=0.01\) into formula (32) to plot the graph of the dimensionless distance and dimensionless impulse of the elastic cylinder, as shown in Figure 18.  
The blue curve in the figure is \(\gamma\leqslant 1\), and the orange curve is \(\gamma>1\). The trend in the image is consistent with the findings of this article. When the dimensionless distance is small, the elastic cylinder boundary is subjected to a large impulse, but because \(\gamma\) is small, the impulse accumulation time is short, as shown in point A in the figure; the elastic cylinder does not necessarily have the maximum value of the maximum displacement. When the dimensionless distance is large, as shown in point B in the figure, although the elastic cylinder has a long impulse accumulation time, the impulse has decayed, and the maximum displacement value will also be smaller than the maximum value. Therefore, there is a maximum value of \(\gamma_{S}\) in this interval, and the maximum value is related to the impulse size and the dimensionless distance. This is a complex process and is also closely related to the properties of the boundary conditions themselves. It is worth discussing in depth in future scientific research. The main variables for the elastic cylinder in this paper are the dimensionless distance between the bubble and the elastic cylinder, and the variables of the elastic cylinder itself are the dimensionless size and the tension received. The dimensionless distance and dimensionless size were discussed in the previous discussion.