## 3 Results and Discussion
### Effect of Jet Direction
#### 3.2.2 Effect of Elastic Cylinder Size on Bubble Jet Direction  
Figure 12 shows the effect of the dimensionless size of the elastic cylinder on the direction of the bubble jet. The figure shows the functional relationship of \(\gamma-\gamma_{rad}\), where the horizontal axis is the dimensionless radius of the elastic cylinder, and the vertical axis is the dimensionless distance between the initial position of the bubble and the boundary of the elastic cylinder. The asterisk indicates no jet, the inverted triangle indicates the jet pointing to the wall of the elastic cylinder, and the equilateral triangle indicates the jet away from the wall of the elastic cylinder. The blue, pink, orange, and black colors are elastic cylinders with actual diameters of 0.4 mm, 0.8 mm, 1.2 mm, and 2.0 mm. Similarly to the effect of elastic cylinder tension on the jet direction, Figure 12 can also be divided into three regions: region I, region II and region III. The classification of the jet direction remains consistent in this paper. Region I is a bubble without a jet, region II is a bubble with a jet, where the jet direction points to the elastic cylinder wall, and the working condition in region III is that the jet is away from the elastic cylinder. Regarding the downward vertical direction of Figure 12, the asterisks basically exist in region I (yellow shading). The dimensionless distance \(\gamma>3\) can be regarded as the movement of the bubble on the free surface. The dimensionless distance of the elastic cylinder boundary is equivalent to the influence at infinity, which is consistent with the previous discussion in this section. From the horizontal direction, although the dimensionless size of the elastic cylinder changes, the dimensionless distance of the jet is not affected by its size. In region III (pink shading), all the equilateral triangles are distributed in this area, which is divided by another boundary, below, which indicates the jets away from the elastic cylinder wall. This is similar to the inference of jets near the rigid cylindrical wall and the  
Figure 12: Effect of elastic cylinder size on bubble jet direction. elastic cylindrical boundary. The direction of the jet is related to the dimensionless distance. When the dimensionless distance between the bubble and the wall is small, a jet away from the cylindrical wall will appear. After fitting, the boundary satisfies the following formula:  
\[\gamma=0.13\gamma_{rad}+0.37 \tag{2}\]  
For region III, from the horizontal direction, as the dimensionless size of the elastic cylinder increases, the dimensionless distance required to produce a jet far away from the cylinder boundary becomes larger; that is, when the size of the elastic cylinder is large, a jet far away from the boundary can be produced under a slightly smaller dimensionless distance condition. This situation can be explained by the following analysis. Figure 13 is a schematic of bubbles surrounding the elastic cylinder boundary. The two bubbles in the figure are the same size, and Figure 13a shows the situation when the elastic cylinder is smaller. For the assumptions made in the pink boundary, the larger the elastic cylinder boundary size, the smaller the dimensionless distance required to generate a jet away from the elastic cylinder wall. Based on this schematic, this paper discusses some of the assumptions. Figure 13a,b are compare, where the bubble size and dimensionless distance are the same, and the only different parameter in these two cases is the size of the elastic cylinder. Due to the difference in size, first of all, it was proposed in the previous analysis of bubble morphology that after the elastic cylindrical boundary is wrapped, the bubble will deform and be divided into two spheres along the elastic boundary. The size of the bubble on the side close to the cylindrical boundary is smaller than that on the side far away.