## 1 Introduction
[PERSON] [3] experimentally studied the complex dynamics of underwater explosion bubbles between two parallel plates. As the distance increases, the bubble splits into two sub-bubbles before collapsing, and the reverse jet penetrates the sub-bubbles. The first pulsation period of an underwater explosion bubble between two parallel plates is 30% to 50% larger than that of a free-field bubble. [PERSON] [4] studied the interaction between bubbles generated by electric sparks and a double-layer composite beam (constrained at both ends) consisting of an aluminum sheet coated with an elastic layer. Numerical simulations achieved good agreement with experimental observations, indicating that the bubble collapse time was greatly affected by the nearby two-layer composite beam. When two cavitation bubbles exist in a limited space, the interaction between the bubbles significantly affects the dynamic behavior characteristics of the bubbles. [PERSON] [5]  
Figure 1: Images of experiments studying bubbles in a confined area: (**a**) Between the two boards, (**b**) In a grid tube, (**c**) In a soft tube. presented a numerical study of the dynamic behavior of vapor bubbles in narrow channels filled with viscous liquids. Numerical results show that the lifetime of the bubble increases with increased the viscosity of the liquid surrounding the vapor bubble. [PERSON] [6] reported experimental and numerical studies of the expansion and collapse of cavitation bubbles in narrow gaps. High-speed recordings and numerical simulations show an unexpected enhancement of the jet velocity, a translation of the center of mass, and a sharp increase in the wall shear stress. [PERSON] [7,8] and his team studied the dynamics of cavitation bubbles and induced jets in a liquid gap bounded by two rigid walls. The impact velocity of the liquid jet on the wall can reach more than 200 m/s and depends strongly on the gap height and the bubble position. [PERSON] [9] experimentally and numerically studied the jet behavior and migration characteristics of laser-induced cavitation bubbles in rectangular channels. Numerical calculations show that the maximum velocity reached by the liquid jet directed toward the channel sidewall is 100 m/s, while the peak velocity of the liquid jet directed toward the channel end wall is about 55 m/s. The conclusions of the study on the restrictive region between parallel rigid walls can be analogized to the study of bubble dynamics in tubes. Cavitation in tubes is a common phenomenon in engineering. The safety and reliability of pipelines are crucial for the stable operation of complex systems [10,11]. Cavitation in pipelines is a basic problem in the fields of hydraulic engineering and bioengineering. In particular, cavitation in rigid pipelines has a non-negligible impact on fluid systems and system components such as hydraulic systems and water conservancy systems. Cavitation bubble pulsation under restricted conditions has complex dynamic characteristics and wide applications in various fields, such as liquid pumping, underwater propulsion, and clinical applications. [PERSON] and [PERSON] [12,13] simulated the growth and collapse of one or more bubbles in a finite tube connecting two liquid reservoirs. The results show that, within certain parameter ranges, the system is capable of a net pumping action, moving liquid from one reservoir to another, even in the presence of unfavorable pressure differentials. [PERSON] [14] simulated the growth, expansion, and collapse of bubbles in narrow tubes through experimental and numerical studies. The jet was generated by the attraction of the Bjerknes force on the tube wall and was directed towards the tube wall. [PERSON] et al. [15] used a high-speed camera to study the interaction phenomenon between two oscillating steam bubbles in a rigid, narrow tube. The bubbles were generated by low-voltage electric sparks, and the bubbles occurred in glass tubes of different inner diameters.