## 4 Results and Discussion
### Comparison of the Present Results with the Earlier Numerical and Experimental Results
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(B\) & **Authors** & \(\mathbf{X_{c}}\) & \(\mathbf{Y_{c}}\) & \(\mathbf{\varphi_{c}}\) & \(\mathbf{R_{0}}\) & \(\mathbf{V}\) & \(\mathbf{H}\) \\ \hline \multirow{2}{*}{50} & Medrow & 0.0026045 & 0.07970 & 178.13 & 0.07975 & 0.0002672 & 0.07921 \\  & **Present** & 0.002604455 & 0.07958193 & 178.1285 & 0.07974769 & 0.000267229 & 0.07909536 \\ \hline \multirow{2}{*}{20} & Toba & 0.016 & 0.194 & 175.3 & 0.196 & 0.0041 & 0.184 \\  & Medrow & 0.016011 & 0.19481 & 175.32 & 0.1963 & 0.004108 & 0.1897 \\  & **Present** & 0.01601056 & 0.1947944 & 175.3211 & 0.1962769 & 0.004107305 & 0.1896872 \\ \hline \multirow{2}{*}{8} & Princen & 0.09101 & 0.44341 & 168.443 & 0.45428 & 0.05846 & - \\  & **Present** & 0.09100837 & 0.4434090 & 168.4433 & 0.4542754 & 0.05846040 & 0.4026157 \\ \hline \multirow{2}{*}{7} & Toba & 0.117 & 0.493 & 166.5 & 0.502 & 0.0863 & 0.433 \\  & Medrow & 0.11520 & 0.49307 & 166.88 & 0.5076 & 0.08463 & 0.4401 \\  & **Present** & 0.1152068 & 0.4930890 & 166.8816 & 0.5075995 & 0.08463203 & 0.4401145 \\ \hline \multirow{2}{*}{4} & Toba & 0.284 & 0.722 & 158.4 & 0.772 & 0.3680 & 0.592 \\  & Princen & 0.28217 & 0.72309 & 158.574 & 0.77245 & 0.36408 & - \\  & Medrow & 0.28217 & 0.72310 & 158.57 & 0.7724 & 0.3641 & 0.5892 \\  & **Present** & 0.2821735 & 0.7230909 & 158.5740 & 0.772442 & 0.3640805 & 0.5891839 \\ \hline \multirow{2}{*}{2} & Toba & 0.655 & 0.989 & 145.9 & 1.169 & 1.67 & 0.711 \\  & Princeton & 0.655999 & 0.98789 & 146.014 & 1.17354 & 1.67520 & - \\  & Medrow & 0.65598 & 0.98782 & 146.02 & 1.174 & 1.675 & 0.7042 \\  & **Present** & 0.6559949 & 0.9878898 & 146.0140 & 1.173535 & 1.675189 & 0.7042526 \\ \hline \multirow{2}{*}{1} & Toba & 1.159 & 1.161 & 134.2 & 1.621 & 5.51 & 0.734 \\  & Medrow & 1.1603 & 1.15854 & 134.56 & 1.629 & 5.505 & 0.7281 \\  & **Present** & 1.160338 & 1.158526 & 134.5617 & 1.628556 & 5.505488 & 0.7280821 \\ \hline \multirow{2}{*}{0.5} & Princen & 1.71400 & 1.23904 & 125.860 & 2.11488 & 13.7728 & - \\  & Medrow & 1.7140 & 1.23890 & 125.87 & 2.115 & 13.77 & 0.6956 \\  & **Present** & 1.714005 & 1.239067 & 125.8599 & 2.114874 & 13.77346 & 0.6956826 \\ \hline \multirow{2}{*}{0.4} & Princen & 1.89485 & 1.25155 & 123.611 & 2.27520 & 17.6919 & - \\  & Medrow & 1.8948 & 1.25152 & 123.61 & 2.275 & 17.69 & 0.6790 \\  & **Present** & 1.894856 & 1.251574 & 123.6100 & 2.275215 & 17.69363 & 0.6790378 \\ \hline \multirow{2}{*}{0.001} & Medrow & 6.5244 & 1.14954 & 101.91 & 6.668 & 565.2 & 0.2994 \\  & Present & 6.524375 & 1.149610 & 101.9076 & 6.667858 & 565.2013 & 0.2994464 \\ \hline \multirow{2}{*}{4\(\times 10^{-8}\)} & [PERSON] & 13.947 & 1.07121 & 95.66 & 14.02 & 5527 & 0.1427 \\  & **Present** & 13.94652 & 1.071357 & 95.6803 & 14.01534 & 5523.744 & 0.1427007 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the present numerical results with those of [PERSON] [29], [PERSON] [30], and [PERSON] and [PERSON] [31] for the selected bubble parameters. The presently computed profiles for bubbles of various sizes (red outlines) are compared with the experimental photographs obtained by [PERSON] et al. [37] and [PERSON] et al. [36] in Figure 8. In addition, the present predictions for the angle of the junction point (\(\theta_{c}\)) are compared with those measured by [PERSON] et al. [37] in Table 2. [PERSON] et al. [37] used a soap solution (\(\sigma=0.0282\) N/m), and the images were taken in the range of \(X_{c}=1\) to 32.5 mm (Figure 8a). Here, the calculated bubble shapes are seen to match the experimental images well, although the results in Table 2 indicate slightly different values for the angle of the junction point (\(\theta_{c}\)). Specifically, the present computational work predicts \(\theta_{c}\) to be about \(1-2\) degrees higher than that measured by [PERSON] et al. [37] Meanwhile, the experimental investigations of [PERSON] et al. [36] were performed on various liquids (e.g., water and ethanol), and the photographs were taken for \(Bo=0.004\) to 2.46 (Figure 8b).