## 4 Results and Discussion
### Correlations and Asymptotic Behaviors of the Bubble Parameters  
The log-log plots of the various bubble parameters with respect to the Bond number for the interval of \(5\times 10^{-5}\leq\mathit{Bo}\leq 5000\) are presented in Figure 10. Each plot can be divided into the following three regions: (I) \(5\times 10^{-5}\leq\mathit{Bo}\leq 0.1\), (II) \(0.1\leq\mathit{Bo}\leq 100\), and (III) \(100\leq\mathit{Bo}\leq 5000\). The parameters for each interval are fitted with a fifth-order polynomial on the log-log scale according to Equation (28):  
\[\log_{10}\Pi\approx\sum_{i=0}^{5}c_{i}\left(\log_{10}\mathit{Bo}\right)^{i}, \tag{28}\]  
where \(\Pi\) denotes the dimensionless bubble parameters. The coefficients \(c_{i}\) are presented in Table 5.  
Figure 10: Correlation and two asymptotes of the computed bubble parameters as a function of the Bond number on a log-log scale. The correlation is given by a fifth-order polynomial function in each of the three Bond number regions (I, II, and III). The two asymptotes are given by a power law. Interestingly, \(X_{c}\) can be well expressed even with a single \"bending function\" for \(5\times 10^{-5}<Bo<5000\), as given by Equation (29):  
\[\log_{10}X_{c}\approx 0.01864+\log_{10}\biggl{(}\frac{Bo}{1.2787}\biggr{)}-0.4889 \ \log_{10}\biggl{[}1+\biggl{(}\frac{Bo}{1.2787}\biggr{)}\biggr{]}. \tag{29}\]  
The curves fitted by the three fifth-order polynomials and by the single bending function are compared in Figure 11, where both correlations are seen to fit the numerical data well. Figure 11: Comparison of the fitting results of the fifth-order polynomials and bending function for \(X_{c}\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Parameter** & **Interval** & \(\epsilon_{0}\) & \(\epsilon_{1}\) & \(\epsilon_{2}\) & \(\epsilon_{3}\) & \(\epsilon_{4}\) & \(\epsilon_{5}\) \\ \hline \multirow{3}{*}{\(X_{c}\)} & \(5\times 10^{-5}<Bo<0.1\) & \(-0.1735\) & \(0.8713\) & \(-0.07803\) & \(-0.02357\) & \(-0.003523\) & \(-0.0002078\) \\  & \(0.1<Bo<100\) & \(-0.2107\) & \(0.7763\) & \(-0.1426\) & \(-0.001089\) & \(0.02162\) & \(-0.004812\) \\  & \(100<Bo<5000\) & \(-0.1349\) & \(0.6468\) & \(-0.09254\) & \(0.02825\) & \(-0.004265\) & \(0.0002566\) \\ \hline \multirow{3}{*}{\(Z_{c}\)} & \(5\times 10^{-5}<Bo<0.1\) & \(-0.5639\) & \(0.6232\) & \(-0.3526\) & \(-0.09327\) & \(-0.01357\) & \(-0.000821\) \\  & \(0.1<Bo<100\) & \(-0.5699\) & \(0.6353\) & \(-0.2707\) & \(0.01518\) & \(0.02576\) & \(-0.006322\) \\  & \(100<Bo<5000\) & \(-0.5019\) & \(0.5271\) & \(-0.2434\) & \(0.06047\) & \(-0.007931\) & \(0.0004324\) \\ \hline \multirow{3}{*}{\(\theta_{c}\)} & \(5\times 10^{-5}<Bo<0.1\) & \(1.561\) & \(0.5498\) & \(0.02396\) & \(0.005715\) & \(0.0006714\) & \(0.00003092\) \\  & \(0.1<Bo<100\) & \(1.517\) & \(0.4145\) & \(-0.1141\) & \(-0.02333\) & \(0.01714\) & \(-0.002079\) \\  & \(100<Bo<5000\) & \(1.529\) & \(0.4392\) & \(-0.1993\) & \(0.04873\) & \(-0.006298\) & \(0.0003392\) \\ \hline \multirow{3}{*}{\(R_{0}\)} & \(5\times 10^{-5}<Bo<0.1\) & \(0.03884\) & \(0.3443\) & \(-0.08807\) & \(-0.02505\) & \(-0.003558\) & \(-0.000201\) \\  & \(0.1<Bo<100\) & \(0.05469\) & \(0.4101\) & \(0.00479\) & \(0.01815\) & \(-0.00413\) & \(-0.0004402\) \\  & \(100<Bo<5000\) & \(0.04825\) & \(0.3953\) & \(0.04915\) & \(-0.01243\) & \(0.001658\) & \(-0.0000919\) \\ \hline \multirow{3}{*}{\(H\)} & \(5\times 10^{-5}<Bo<0.1\) & \(-0.1509\) & \(0.1079\) & \(-0.2103\) & \(-0.05746\) & \(-0.007922\) & \(-0.0004377\) \\  & \(0.1<Bo<100\) & \(-0.1561\) & \(0.1188\) & \(-0.1691\) & \(-0.03095\) & \(-0.0007063\) & \(0.006116\) \\  & \(100<Bo<5000\) & \(0.2356\) & \(-0.3661\) & \(-0.0688\) & \(0.01897\) & \(-0.002739\) & \(0.0001626\) \\ \hline \multirow{3}{*}{\(H_{0}\)} & \(5\times 10^{-5}<Bo<0.1\) & \(-0.6643\) & \(1.432\) & \(-0.0471\) & \(-0.01573\) & \(-0.002533\) & \(-0.000158\) \\  & \(0.1<Bo<100\) & \(-0.7411\) & \(1.215\) & \(-0.2392\) & \(-0.02608\) & \(0.03433\)Two asymptotes appear in Figure 10 as the Bond number goes to zero or infinity.