## 3 Results and Discussion
### Effect of Jet Direction
#### 3.2.1 Effect of Elastic Cylinder Tension on Jet Direction  
Figures were plotted according to the classification of the jet direction to study the relationship between the elastic cylinder's tension and the bubble jet's direction. Figure 11 shows the functional relationship of \(\gamma-F\), where the horizontal axis is the elastic cylinder tension in N, and the vertical axis is the dimensionless distance \(\gamma\) between the initial position of the bubble and the elastic cylinder boundary. The black asterisk represents the state without jets, the blue inverted triangle represents the direction of the jet pointing to the elastic cylinder wall, and the pink equilateral triangle represents the jet away from the elastic cylinder wall. The figure can be divided into three parts: region I, region II, and region III.  
The black asterisks basically exist in region I (yellow shading). When the vertical coordinate value is greater than the blue boundary line, the bubbles appear to have no jets; that is, the dimensionless distance can be regarded as infinite. This dimensionless distance is shorter than the dimensionless distance without jets near the rigid cylinder, which is related to the size of the elastic cylinder and the properties of the material. From the horizontal direction, the no jet situation in region I has little to do with the tension of the elastic cylinder. Even if the tension is increased, the size of the dimensionless distance is still the key factor in determining whether a jet is generated. Almost all pink equilateral triangles are contained in region III (pink shading). After the bubble collapses in this region, the bubble cloud moves away from the elastic cylinder, which is defined in this paper as the jet moving away from the elastic cylinder wall. It can be observed that this is similar to a jet near a rigid wall. That is, when the dimensionless distance is short, a jet moving away from the cylindrical wall is produced. For the elastic cylindrical boundary, the position of the boundary is not fixed but displaced as the bubble expands. Due to different tensions, at the same dimensionless distance, the displacement caused by the bubble pushing the elastic cylindrical boundary will be different, and the interaction between the bubble and the elastic cylindrical boundary will also be different. In region II (blue shading), almost all blue inverted triangle symbols exist. The jet pointing to the cylinder wall is represented by a blue inverted triangle, which includes bubbles with jets pointing only to the wall and bubbles with jets pointing in both directions, but the jets pointing to the wall are dominant. In summary, after the bubble cycle ends, the bubble cloud's direction is toward the cylindrical wall. This section defines it as a jet directed toward the elastic cylindrical wall. In the vertical direction, region II sandwiched between the blue boundary line and the pink boundary line in Figure 11 is the jet generated by the bubble, and the direction of the jet is the jet directed toward the elastic cylindrical wall.  
To summarize the jet direction in the figure, the relationship between the jet direction and the dimensionless distance is as follows:  
\[\left\{\begin{array}{ll}\gamma>3&\text{no jets},\\ 3>\gamma>0.8&\text{jet toward the wall}\\ \gamma<0.8&\text{jet away from the wall}\end{array}\right. \tag{1}\]  
Figure 11: Effect of elastic cylinder tension on bubble jet direction.  
The above discussion discusses the influence of elastic cylinder tension on the direction of the bubble jet and gives the corresponding division of two boundaries and three regions. In the process of studying bubble morphology, it was found that the relative radius of the elastic cylinder also affects the bubble jet. Then, the following will discuss how the dimensionless radius of the elastic cylinder affects the jet direction near the boundary of the elastic cylinder.