## 4 Conclusions  
In this study, the relationship between bubble morphology and elastic cylinder displacement was investigated through electric spark experiments. The focus was on examining the effects of dimensionless distance, tension, and cylinder size on bubble behavior and cylinder displacement. Additionally, the accuracy and practical applicability of the results were evaluated in conjunction with a reliability analysis.  
First, the morphological changes in bubbles at different dimensionless distances were studied. At a larger dimensionless distance (\(\gamma\)), the bubble behaves in a free-field state, is almost unaffected by the elastic cylinder, and maintains a spherical shape. When \(\gamma\) becomes slightly smaller, the bubble may contact the cylinder but still maintains a spherical shape during expansion. This is closely related to the characteristics of the elastic boundary. In the contraction stage, the bubble morphology is significantly influenced by \(\gamma\). When \(\gamma\) is small, the bubble will be divided into two hemispheres by the elastic cylinder after contacting it during expansion, and it will take on a mushroom shape during contraction. Tension has almost no effect on the bubble morphology. Regarding the effect of elastic cylinder size, when \(\gamma_{rod}\) is small, the bubble mostly maintains a spherical shape during both expansion and contraction. As \(\gamma_{rod}\) increases, the bubble remains spherical during expansion, but the side closest to the cylinder becomes drop-shaped during contraction, while the side away from the cylinder remains spherical. When the bubble contacts or wraps around the cylinder, slight deformation occurs during the expansion process. The interaction between the bubble and the cylinder during the contraction stage causes the side near the cylinder to collapse significantly slower, resulting in the bubble presenting a bulb or mushroom shape. In terms of elastic cylinder displacement, a cylinder force model is established in combination with the bubble motion model, and the displacement formula for the cylinder is calculated. The calculation results align well with the experimental data trend. An impact  
Figure 19: Therelationship between the dimensionless maximum displacement and distance of the expansion and collapse of some experimental data.  
force experiment verified the displacement change and revealed the dimensionless distance at which the maximum displacement occurs based on the cumulative force action time. Additionally, tension has little effect on the displacement of the cylinder.  
This study ensures the reliability of the results through repeated experiments and numerical simulations, confirming the consistency between experimental data and model calculations. The stability and accuracy of the conclusions were validated across variables such as dimensionless distance, tension, and cylinder size. The experimental and theoretical predictions of displacement and bubble morphology align closely, particularly in the design of elastic materials and the control of bubble dynamics. The findings contribute to a deeper understanding of bubble-elastic cylinder interactions, with wide applications in fluid mechanics, engineering design, and material reliability. This research also supports bubble control, the optimization of elastic structures, and reliability assessments in engineering practices.  
Conceptualization, Y.G. and D.S.; methodology, Y.G.; software, Y.G. and J.W.; validation, J.W.; formal analysis, Y.G. and D.S.; resources, Y.G.; writing--original draft preparation, Y.G.; writing--review and editing, D.S.; funding acquisition, D.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 12172100).  
Not applicable.  
Data Availability Statement: Data is contained within the article.  
The authors declare no conflicts of interest.