## 2 Methods and Data
### Input Variables Using Dimensionless Numbers or Primitive Parameters
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\pi\) & **Set1** & **Set2** & **Set3** \\ \hline  & \(\pi_{1}\) & \(\frac{h_{sat}}{W_{sat}}\) & \(\frac{h_{sat}}{W_{sat}}\) & \(\frac{\mu_{d}}{\mu_{c}}\) \\ \cline{2-5}  & \(\pi_{2}\) & \(W_{sat}\) & \(\frac{h_{c}}{W_{sat}}\) & \(\frac{\mu_{c}}{\mu_{c}}\) \\ \cline{2-5}  & \(\pi_{3}\) & \(\frac{h_{c}}{W_{sat}}\) & \(\frac{h_{c}}{W_{sat}}\) & \(\frac{\mu_{c}\sigma X}{\mu_{c}^{2}}\left(St_{c}\right)\) \\ \cline{2-5}  & \(\pi_{4}\) & \(\frac{h_{c}}{W_{sat}}\) & \(\frac{h_{c}}{W_{sat}}\) & \(\frac{h_{c}h_{c}}{\sigma}\left(C_{dc}\right)\) \\ \cline{2-5}  & \(\pi_{5}\) & \(\frac{d_{c}}{W_{sat}}\) & \(\frac{d_{c}}{W_{sat}}\) & \(\frac{\mu_{d}\sigma}{\sigma^{2}}\left(Cat_{d}\right)\) \\ \cline{2-5}
**Independent** & \(\pi_{6}\) & \(\frac{\chi}{W_{sat}}\) & \(\frac{\chi}{W_{sat}}\) & \(\alpha\) \\ \cline{2-5}  & \(\pi_{7}\) & \(\alpha\) & \(\alpha\) & \(\frac{\rho_{c}\sigma W_{sat}}{\mu_{c}^{2}}\) \\ \cline{2-5}  & \(\pi_{8}\) & \(\frac{\rho_{c}}{\rho_{c}}\) & \(\frac{\mu_{d}}{\mu_{c}}\) & \(\frac{\rho_{c}\sigma W_{sat}}{\mu_{c}^{2}}\) \\ \cline{2-5}  & \(\pi_{9}\) & \(\frac{h_{c}}{\sqrt{W_{sat}}}\left(Oh_{c}\right)\) & \(\frac{\rho_{d}}{\rho_{c}}\) & \(\frac{\rho_{c}\sigma W_{c}}{\mu_{c}^{2}}\) \\ \cline{2-5}  & \(\pi_{10}\) & \(\frac{\mu_{d}}{\sqrt{W_{sat}}}\left(Oh_{d}\right)\) & \(\frac{W_{sat}\rho_{c}\kappa_{c}}{\mu_{c}}\left(Re_{c}\right)\) & \(\frac{\rho_{c}\sigma H_{c}}{\mu_{c}^{2}}\) \\ \cline{2-5}  & \(\pi_{11}\) & \(\frac{W_{sat}\rho_{c}\mu_{c}^{2}}{\sigma}\left(W_{bc}\right)\) & \(\frac{W_{sat}\rho_{d}\mu_{d}}{\mu_{d}}\left(Re_{d}\right)\) & \(\frac{\rho_{c}\sigma W_{d}}{\mu_{c}^{2}}\) \\ \cline{2-5}  & \(\pi_{12}\) & \(\frac{W_{sat}\rho_{d}\mu_{d}^{2}}{\sigma}\left(W_{cb}\right)\) & \(\frac{W_{sat}\rho_{d}\mu_{d}^{2}}{\sigma}\left(W_{cb}\right)\) & \(\frac{\rho_{c}\sigma d_{d}}{\mu_{c}^{2}}\) \\ \hline \multirow{2}{*}{**Target Variables**} & \(\pi_{13}\) & \(\frac{L_{d}}{W_{sat}}\) & \(\frac{L_{d}}{W_{sat}}\) & \(\frac{\rho_{c}\sigma L_{d}}{\mu_{c}^{2}}\) \\ \cline{2-5}  & \(\pi_{14}\) & \(\sqrt{\frac{W_{sat}^{3}\rho_{c}}{\sigma}}f\) & \(\frac{W_{sat}f}{\mu_{c}}\left(St\right)\) & \(\frac{b_{c}^{2}f}{\rho_{c}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Three typical solutions of dimensionless numbers (\(\pi\)). The ML model brought the different input variables with the primitive parameters and dimensionless numbers to compare the recognition accuracy of flow patterns in the two-group experiments. How to choose the appropriate dimensionless numbers to descript the generalized system was also discussed by the prediction of droplet characteristics. The four flow patterns were identified to be slug, dripping, jetting, and others in both Experiment 1 and 2 (Figure 2). In fact, the droplet was the gas bubble in Experiment 1 and the liquid droplet in Experiment 2. Although the droplet was a dissimilar material, it had similar features in the generalized variable system. The slug, dripping and jetting were the important flow patterns due to their formation generating the highly consistent droplets [39; 40; 41]. The ML model was applied to classify the four flow patterns and predict the droplet characteristics in the slug and dripping flow patterns due to enough data in experiments.