# Experimental Investigation on the Dynamic Characteristics of Bubble-in-Chain Near a Vertical Wall
## 3 Results
### The Energy Variation of the Bubble-in-Chain  
The motion of bubbles near a wall is affected by an additional mass effect of the surrounding fluid, which causes the bubble to carry the surrounding fluid along with it. During this process, the conversion of kinetic energy to surface energy of the bubble plays a crucial role in its dynamic behavior, especially in cases where the wall effect is significant,  
Figure 11: The correlation between the average drag of bubbles at different generation frequencies and Reynolds number. Figure 12: The bubble drag coefficient varies with \(H^{*}\). (**a**) Bubble colliding with the wall; (**b**) bubble not colliding with the wall. such as during bubble-wall collisions [27]. To quantitatively describe the kinetic energy and surface energy of the bubble, it can be calculated using the following equations [1]:  
\[E_{k}=\frac{\pi(\rho_{g}+C_{m}\rho_{l}){d_{\alpha q}}^{3}{V_{b}}^{2}}{12} \tag{9}\]  
\[E_{s}=\sigma A \tag{10}\]  
\[A=\frac{\pi{d_{h}}^{2}}{2}\bigg{(}1+\frac{1-e^{2}}{e}{\tanh}^{-1}e\bigg{)} \tag{11}\]  
\[e^{2}=1-1/\chi^{2} \tag{12}\]  
The coefficient of the additional mass force, denoted as \(C_{m}\), can be determined by the following equation [29]:  
\[C_{m}=\frac{\alpha}{2-\alpha} \tag{13}\]  
\[\alpha=\frac{2\chi^{2}}{\chi^{2}-1}\bigg{(}1-\frac{1}{\sqrt{\chi^{2}-1}}\cos^{ -1}(1/\chi)\bigg{)} \tag{14}\]  
Figure 13 illustrates the dynamic process of energy change in bubbles within pure water and glycerol aqueous solutions. Specifically, Figure 13a,b represent the energy changes of bubbles in pure water, while Figure 13c,d correspond to bubbles in glycerol-water solutions. Figure 13a,c demonstrate the energy change when the bubble collides with the wall, whereas Figure 13b,d depict the energy state when the bubble is not in direct contact with the wall. The gray curve represents the dimensionless path of the bubble, which directly reflects its trajectory under different conditions. Figure 13: Energy variations of bubbles in pure water and glycerol aqueous solutions under two typical modes. (**a**,**b**) represent energy changes of bubbles in pure water, while (**c**,**d**) correspond to bubbles in glycerol-water solutions. (**a**,**c**) demonstrates the energy change when a bubble collides with a wall, whereas (**b**,**d**) depicts its energy state when it is not in direct contact with a wall. The gray curve represents a dimensionless path of a bubble that directly reflects its trajectory under different conditions. In the case of a bubble-wall collision, the change in energy is closely related to the change in trajectory. The peak of the energy curve aligns with the peak of the bubble's trajectory. Specifically, in pure water, there is a secondary peak occurring before and after reaching the kinetic energy peak, indicating a significant local increase in kinetic energy before and after colliding with the wall. However, in a glycerol aqueous solution, this secondary peak is significantly inhibited due to increased viscosity, demonstrating a notable impact of viscosity on bubble dynamics. When the bubble does not collide with the wall, its energy changes show a stronger level of randomness. This increased randomness is primarily attributed to the weakening of the wall effect and reduced fluid dynamics constraints on the bubbles. In this case, the surface energy of the bubble remains almost constant in the glycerol aqueous solution when there is no collision with a wall. This indicates that external environmental influences on bubble surface energy need to be exerted through direct contact. Figure 14 illustrates the impact of bubble generation frequency on changes in bubble energy. Specifically, Figure 14a,c,e depict the energy distribution of bubbles when they collide with the wall, whereas Figure 14b,d,f represent cases where bubbles do not collide with the wall. Figure 14a,b correspond to a bubble formation frequency denoted as \(F1\), Figure 14c,d correspond to a frequency denoted as \(F2\), and Figure 14e,f correspond to a frequency denoted as \(F3\).