# NewApproach to Predict the Motion Characteristics of Single Bubbles in Still Water
## 3 Results
### Prediction Model for the Shape of Single Bubbles
Figure 14 shows the sensitivity of each variable in the model developed using the random forest algorithm, where P is the degree of importance. The formula of P is given by:  
\[VIM_{jm}^{(Gini)}=GI_{m}-GI_{i}-GI_{r} \tag{16}\]  
\[VIM_{j}^{(Gini)}=\sum\
olimits_{i=1}^{n}VIM_{ij}^{(Gini)} \tag{17}\]  
where \(GI\) is Gini index, \(m\) is the number of features, \(GI_{i}\) and \(GI_{r}\) are Gini index of the two new nodes after branching. Basically, in accordance with the increase of Re, Eo, and We numbers, the bubbles experience a change from elliptical shape to either spherical shape or other irregular shapes. Due to the experimental constraints, only the limited range of the Re, Eo, and We numbers was applied in this study. Nonetheless, the relevant analyses were obtained via bubble phase diagram as proposed by [PERSON] [37]. In general, the single bubble motion in still water is mainly affected by viscous, gravitational, surface tension, and inertial forces. According to previous studies, the level of significance of different forces exerted on the bubbles varied with the size of bubbles. For instance, surface tension force exhibits the most \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Predicted label**} \\ \hline \multirow{3}{*}{**Actual label**} & **Label** & **1** & 2 & 3 & Total sample \\ \cline{2-6}  & 1 & 102 & 15 & 0 & 117 \\ \cline{2-6}  & 2 & 21 & 130 & 2 & 151 \\ \cline{2-6}  & 3 & 0 & 6 & 30 & 36 \\ \hline \multicolumn{6}{c}{Total} & 123 & 151 & 30 & 304 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Confusion matrix of the random forest algorithm prediction model. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Predicted label**} \\ \hline \multirow{3}{*}{**Actual label**} & **Label** & **1** & 2 & 3 & Total sample \\ \cline{2-6}  & 1 & 99 & 18 & 0 & 117 \\ \cline{2-6}  & 2 & 17 & 132 & 2 & 151 \\ \cline{2-6}  & 3 & 0 & 2 & 34 & 36 \\ \hline \multicolumn{6}{c}{Total} & 116 & 152 & 36 & 304 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Confusion matrix of the logistic regression algorithm prediction model. significant impact on the bubbles with a diameter of less than 6 mm, meanwhile, if the size is larger than 6 mm, the bubble is mainly affected by inertial force [1; 11; 39]. As displayed in Figure 13, the effect of Eo number to the determination of the shape of single bubbles is the most significant, recording at P-value of 0.407. It is followed by Re number with a value of 0.34, and the least significant parameter is We number with a value of 0.253. This is mainly because Eo number is the parameter, which represents the effect of surface tension on motion characteristics of the single bubbles. The finding is in line with [1; 11; 39], where the diameter of bubbles in this study ranges from 2 to 7.5 mm (mostly below 6 mm) and thereby surface tension plays the role as the main force that affects the bubbles. The surface tension is particularly reflected through the dynamic pressure difference. In other words, when the dynamic pressure difference increases, the bubble surface tension changes drastically and complicates the changes of bubble shape. Figure 15 shows the distribution of bubble shapes with respect to different Re number and Eo number conditions as predicted by KNN, random forest, and logistic regression prediction models. Observing the shape distribution in Figure 15, when Eo falls within the range of 0.5 to 2 and the Re number is in between 600-2000, the bubble is elliptical in shape. When the Eo number ranges from 2 to 5 and the Re number is at a value of 900 to 2000, the bubble is in spherical-cap shape. Meanwhile, when the Eo number falls within the range of 4.5 to 8 and the Re number is recorded at a value of 2000 to 5000, the bubble is irregular in shape. On the other hand, from Figure 15, it is noticed that, when Eo number is ranged from 0.7 to 2, the prediction results of the KNN algorithm show an overlap scenario. The same phenomenon happens when the Eo number ranges from 1 to 2 for the random forest algorithm.