# Experimental Investigation on the Dynamic Characteristics of Bubble-in-Chain Near a Vertical Wall
## 3 Results  
### Statistical Characteristics of Bubble-in-Chain Near the Wall  
The influence of wall effects on bubbles can be discerned through the initial wall distance, which denotes the distance from the point of bubble generation to the adjacent wall, i.e., the separation between the bubble-generating needle and the wall [21]. The initial distance is dimensionless, denoted as \(L^{*}=L/r_{eq}\), where req represents the equivalent radius of the bubble. Its calculation formula can be expressed as \(r_{eq}=d_{eq}/2\). In this study, three distinct initial wall distances are chosen (as indicated in Table 3) to represent strong, medium, and weak wall effects, respectively. Although it is commonly believed that bubble-wall collisions primarily contribute to the reduction of bubble velocity in the near-wall region [1], our findings suggest that, under different operational conditions, other factors besides bubble-wall collisions also influence bubble velocity. The trajectories of the near-wall bubbles at different generation frequencies are presented in Figure 4. As the bubble generation frequency increases, both the limit velocity and trajectory amplitude of bubbles increase. In the \(xz\) plane, there is a more pronounced deviation of bubble trajectories from the wall, whereas in the \(yz\) plane, a wider region is covered by bubble trajectories. The bubble velocity is significantly impeded by the wall when the bubble generation frequencies are \(F1\) and \(F2\), respectively. Although the limiting bubble velocity increases as the frequency rises from \(F1\) to \(F2\), it remains low near the wall where the trajectory appears cyan and has a velocity of approximately 0.2 m/s. For the highest frequency, \(F3\), the limiting velocity of bubbles near the wall is larger than that far away from it, indicating that an increase in bubble generation frequency aids in reducing wall-induced velocity inhibition and exerts a positive influence on bubble migration speed. The effect of the wall on bubble velocity was quantified by averaging the rising velocity and position coordinates of each group of bubbles. The position coordinates and velocity of the bubble were normalized using \(R_{eq}\) and \(\overline{V_{b}}\) (\(x^{*}=x/r_{eq}\), \(y^{*}=y/r_{eq}\), \(V_{b}{}^{*}=V_{b}/\overline{V_{b}}\)). The relationship between the average velocity and average position is depicted in Figure 5. The wall effect typically induces a reduction in the velocity of individual bubbles [22; 23]. The bubble velocity near the wall (\(x^{*}<5\)) decreases as \(x^{*}\) decreases when the initial wall distance is \(L1\). The bubble velocity of \(F1\) remains stable at \(x^{*}=5\). As \(x^{*}\) exceeds 8, the velocity of the bubble with frequency \(F2\) gradually decreases, whereas the bubble with frequency \(F3\) starts to decrease at a smaller value of \(x^{*}\) and exhibits a more rapid decline. The distribution of bubble velocity changes in a parabolic manner with respect to \(y^{*}\), resembling that observed for free rising bubbles. When the frequency of bubble generation is fixed, the average velocity of bubbles exhibits a consistent trend at different \(L^{*}\). As \(x^{*}\) increases, \(V_{b}{}^{*}\) gradually rises and stabilizes after reaching a critical value. The \(x^{*}\) required to reach the critical velocity decreases with an increase in the \(L^{*}\), while the bubble velocity also exhibits a slight decrease. Particularly at \(L3\), the velocity of the bubble experiences a slight decrease with increasing \(x^{*}\) after stabilization. Moreover, \(V_{b}{}^{*}\) does not exhibit any noticeable regular changes with an increase in \(y^{*}\); however, the velocity distribution is narrower for smaller \(L^{*}\). This phenomenon can be \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Initial Distance** & \(L1\) & \(L2\) & \(L3\) \\ \hline \(L^{*}\) & 0.5 & 1.5 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Normalized initial wall distance. elucidated by the findings of [PERSON] et al.