# Experimental Investigation on the Dynamic Characteristics of Bubble-in-Chain Near a Vertical Wall
## 3 Results
### Motion Mode of the Bubble-in-Chain  
The motion of a single bubble near a wall exhibits stochastic behavior and does not consistently adhere to a predetermined migration mode, even under identical operational conditions [24]. The ascent of the bubble-in-chain is influenced by adjacent bubbles, leading to a more intricate migration mode and an increased occurrence of stochasticity within individual bubbles, and it is imperative to analyze their motion modes. The chain of bubbles near the wall exhibits two typical modes of movement: one involves bouncing along the wall, and the other entails forming a spiral upward path near the wall. This differs from the zigzag motion observed in a single bubble during free rising and grants the bubble chain more freedom. The wake induced by the preceding bubble disturbs the bubble-in-chain, causing not only vertical movement towards the wall but also lateral displacement parallel to the wall. This results in a three-dimensional spiral upward trajectory. The bubble behavior of the two typical modes mentioned above in aqueous glycerol and pure water is depicted in Figure 6. Figure 6a,b illustrate the results in aqueous glycerol solutions, while Figure 6c,d demonstrate the results in pure water. The frequency of bubble generation is \(F2\), and the initial distance from the wall is \(L1\). It is evident that there is a slight increase in bubble size (equivalent diameter from 3.2 mm to 3.6 mm), accompanied by a more significant degree of deformation due to the lower viscosity of pure water. Notably, in glycerol aqueous solutions, a more pronounced bubble-wall collision process occurs with observable close contact between the bubble and the wall, distinguishing it from previous studies on single bubbles [25]. This phenomenon can be attributed to the interaction among adjacent bubbles in the bubble chain and an increase in bubble velocity. Figure 5: The average velocity distribution of bubbles. (**a**,**b**) consider the effect of different frequencies on bubble velocity, while (**c**,**d**) consider the impact of varying initial wall distances. The impact of generation frequency on the bubble rise process is illustrated in Figure 7, particularly when considering the wall effect, where the bubble exhibits a bouncing motion along the wall. The initial distance between the bubble and the wall is set as \(L1\). Figure 7a-c depict the ascending behavior of bubbles under different formation frequencies: \(F1\), \(F2\), and \(F3\), respectively. Upon colliding with the wall, the wavelength of the bubble trajectory increases proportionally to the generation frequency of bubbles, and the frequency of bubble-wall collisions decreases. Even in the absence of collision with the wall, the bubbles maintain a three-dimensional spiral trajectory. During the motion of bubbles, especially during bubble-wall collisions, substantial alteration in shape occurs. These changes can be quantified by considering the aspect ratio (\(\chi=d_{v}/d_{h}\)) and Weber number (\(We=\rho V_{b}{}^{2}d_{q}/\sigma\)). The relationship between the aspect ratio and Weber number of bubbles at different generation frequencies is depicted in Figure 8. As the Weber number increases, the aspect ratio of the bubble decreases. In the absence of collisions, the distribution of bubble aspect ratios becomes more concentrated with a smaller range of variation, indicating that wall collisions lead to more pronounced shape changes in bubbles. However, it should be noted that even without wall collisions, bubbles formed at high frequencies (e.g., \(F3\)) exhibit  
Figure 6: Two typical modes of near-wall bubble motion in pure water and glycerol aqueous solution. (**a**,**b**) illustrate the results in aqueous glycerol solutions, whereas (**c**,**d**) demonstrate the results in pure water. The frequency of bubble generation is \(F2\), and the initial distance from the wall is \(L1\). Figure 7: Typical rising mode of bubble-in-chain near the wall at different generation frequencies.