## 4 Dynamic Characteristics of a Bubble in Short Tubes
### Pressure in the Tube  
In the study of the 100 mm tube, experiments and numerical simulations were carried out under the conditions of different tube diameters (\(\gamma_{t}\)) and different initial bubble occurrence positions (\(\eta_{b}\)). Figure 10a shows the pressure change over time at the center of the bubble occurrence position when the tube length is 100 mm and the tube diameter is 10 mm at \(\eta_{b}=0.5\). As shown in the figure, the pressure increases instantly upon the initial generation of the bubble. The bubble bursts at about 8 ms, and the flow field is instantly disturbed at this time. The pressure is weaker than the initial pressure peak, then decays and disappears. The pressure at the center of the initial bubble when the tube diameter is 6 mm is shown in Figure 10b. Compared with the 10 mm tube, the initial pressure is reduced, and the bubble cycle is prolonged. When the diameter exceeds a certain value, the effect of the wall is weak and can be basically ignored, and the movements form of the bubble and the jet are basically consistent with the free field. Whether it is a horizontal wall, a curved wall, or a cylindrical boundary, the distance parameter (\(\gamma_{t}\)) has a great influence on the bubble form and jet behavior. The smaller \(\gamma_{t}\) is, the greater the obstruction of the fluid by the boundary and the longer the bubble pulsation period. In addition, when the initial bubble occurs at different locations, the first thing that comes is the difference in bubble form, which causes changes in the surrounding fluid and the pressure in the tube. Figure 10c,d show the pressure changes at the center of the initial bubble under the conditions of \(\eta_{b}=0.2\) and \(0.3\). The first thing that can be observed is the sudden increase in pressure at the initial moment. However, unlike the bubble occurring at the symmetric center, there is no trend of pressure increase and decrease at the moment of bubble collapse as in the condition of \(\eta_{b}=0.5\). This is because the bubbles are generated at an eccentric position, and the expansion and collapse of the bubbles on both sides are not symmetrical. When the bubbles collapse, the bubbles on the right collapse earlier than the bubbles on the left. The bubbles on the left have not yet collapsed, and the collapsed bubbles on the right have moved to the inside of the bubbles on the left. For the phenomenon of a bubble bursting in the cylinder, the initial position of the bubble (\(\eta_{b}\)) also greatly affects its dynamic behavior; the smaller the \(\eta_{b}\), the longer the length of the cylinder on the right, the stronger  
Figure 9: Images of a bubble in a tube under the conditions of \(\eta_{b}=0.2\), \(\gamma_{t}=0.411\), \(\gamma_{L}=6.85\), and \(\Lambda=16.67\). the obstruction of the fluid, the longer the corresponding bubble period, and the smaller the generated jet velocity, guided by the jet of the first collapsed bubble on the left to move. Figure 11 shows the pressure curves at several typical moments on the center line and generatrix of the tube when the initial bubble occurs at the symmetric center position, the tube length is 100 mm, and the diameter is 10 mm. This result is obtained by numerical simulation. Images A, B, and C on the left above are bubble graphics during expansion, and images D, E, and F on the right are bubble graphics during collapse. Figure 11a,b are the pressures at the center line of the tube. It can be seen from Figure 11a that at time A, the pressure at the bubble boundary is the largest, and due to the initial formation of the bubble, it causes pressure fluctuations in the tube. Then, the bubble gradually expands, the pressure fluctuation is no longer obvious, and this state lasts for a long time. In Figure 11b, during bubble collapse, the pressure at the center line gradually increases, and the wall pressure does not return to the initial pressure until the bubble collapses.