# Experimental Investigation on the Dynamic Characteristics of Bubble-in-Chain Near a Vertical Wall
## 3 Results
### Dynamic Characteristics of the Bubble-in-Chain  
The motion of bubbles is intricately linked to the resistance they encounter. The resistance experienced by a bubble can be determined by the equilibrium between its buoyancy and the resistance [26]:  
\[C_{D}=4\frac{d_{eq}(\rho_{L}-\rho_{G})g}{3\rho_{L}{V_{b}}^{2}} \tag{8}\]  
Figure 11 illustrates the correlation between the drag coefficient of bubbles and the Reynolds number at various generation frequencies. The observed trend aligns with previous studies [27; 28]. As the Reynolds number increases, the drag coefficient of bubbles decreases across all generation frequencies. Although there is a similarity in the drag coefficient extremum among different generation frequencies, high-frequency bubbles exhibit a wider range of distribution for their drag coefficients. This finding corresponds  
Figure 9: The bubble velocity components at different initial wall distances. **(a,c,e)** show the cases where bubbles collide with the wall, whereas **(b,d,f)** show the cases where bubbles do not collide with the wall. **(a,b)** depict the initial wall distance of \(L1\), **(c,d)** illustrate the initial wall distance of \(L2\), and **(e,f)** demonstrate the initial wall distance of \(L3\). to the trend observed in average velocity distribution of bubbles, indicating that bubble generation frequency significantly influences their drag characteristics. The average drag is insufficient to fully capture the influence of the wall on bubbles. The wall effect primarily relies on the distance (\(H\)) between the bubble and the wall. By dimensionless treatment of this distance using the equivalent radius of the bubble (\(H^{*}=H/r_{eq}\)), a more precise analysis of the wall effect can be achieved. Figure 12 illustrates, based on motion mode of the bubble-in-chain, the corresponding relationship between bubble drag coefficient and dimensionless distance \(H^{*}\). Irrespective of collision with or without collision against the wall, under all generation frequencies, an increase in \(H^{*}\) leads to a decrease in bubble drag coefficient. When the bubble does not collide with the wall, the drag coefficient increases with \(H^{*}\), indicating that wall effects reduce the resistance experienced by the bubble to some extent. Conversely, when the bubble collides with the wall, the drag coefficient decreases with an increase in \(H^{*}\). It is worth noting that for  
Figure 10: The variation of bubble velocity components at different generation frequencies. (**a**,**c**,**e**) depict cases where bubbles collide with the wall, whereas (**b**,**d**,**f**) illustrate cases where bubbles do not collide with the wall. The bubble generation frequencies in (**a**,**b**) are denoted as \(F1\), those in (**c**,**d**) are denoted as \(F2\), and those in (**e**,**f**) are denoted as \(F3\). bubbles generated at a frequency of \(F3\), when \(H^{*}\) exceeds 3, there is a slight increase in drag coefficient with an increase in \(H^{*}\). The mechanism of variation in the bubble drag coefficient differs across different modes. The upward flow field induced by the bubble is formed near the wall, which affects the resistance experienced by the bubble. Bubbles that do not collide with the wall move within the main flow area of the induced flow field. As \(H^{*}\) increases, the liquid phase velocity decreases and leads to an increase in viscous resistance, resulting in an increase in the resistance coefficient. For bubbles that collide with the wall, when \(H^{*}\) is small (less than 2.5), energy conversion and frictional resistance during collision cause an increase in the resistance coefficient. Additionally, a thin liquid film similar to a boundary layer induced by the flow field near the wall experiences rapid decay as \(H^{*}\) decreases, leading to a sharp increase in viscous resistance for bubbles and consequently causing an increase in their resistance coefficient [29].