## 4 Results and Discussion
### Important Bubble Parameters and Profiles for a Wide Range of the Bond Numbers
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
60 & 0.0331200 & 0.003159008 & 0.000154907 & 0.01394203 & 6.122573 & \(-\)34.367091 \(\times 10^{-2}\) & 7.763199 \(\times 10^{-9}\) \\
30 & 0.0065636 & 0.00842589 & 0.00122227 & 0.00525227 & 7.955407 & \(-\)19.98895 \(\times 10^{-2}\) & 3.993956 \(\times 10^{-9}\) \\
20 & 0.09967012 & 0.0301315 & 0.00410705 & 0.124085 & 8.408457 & \(-\)48.55708 \(\times 10^{-2}\) & 6.72799 \(\times 10^{-9}\) \\
16 & 0.1245074 & 0.00400891 & 0.00704303 & 0.17942436 & 2.364173 & \(-\)11.91305 \(\times 10^{-2}\) & 3.58303 \(\times 10^{-9}\) \\
10 & 0.1974039 & 0.121816 & 0.03109598 & 0.078421 & 9.988182 & \(-\)38.80760 \(\times 10^{-2}\) & 5.99972 \(\times 10^{-9}\) \\
7 & 0.2788360 & 0.2086884 & 0.08462303 & 0.9946970 & 10.87842 & \(-\)26.66992 \(\times 10^{-2}\) & 3.869972 \(\times 10^{-9}\) \\
4 & 0.464155 & 0.427182 & 0.3640805 & 2.483887 & 11.61313 & \(-\)6.331212 \(\times 10^{-2}\) & 5.348903 \(\times 10^{-9}\) \\
2 & 0.8182227 & 0.8158624 & 1.657819 & 6.96844 & 12.27823 & \(-\)6.689889 \(\times 10^{-2}\) & 6.69809 \(\times 10^{-9}\) \\
1 & 1.2729849 & 1.2022202 & 5.558488 & 1.659457 & 12.982490 & \(-\)30.031094 \(\times 10^{-9}\) & 1.56443 \(\times 10^{-9}\) \\
05 & 1.796671 & 1.728766 & 13.773764 & 29.45927 & 13.99496 & \(-\)1.199595 \(\times 10^{-2}\) & 9.49237 \(\times 10^{-9}\) \\
02 & 2.257079 & 2.341705 & 4.846636 & 55.468245 & 14.57318 & \(-\)6.70705 \(\times 10^{-2}\) & 1.53636 \(\times 10^{-9}\) \\
01 & 3.04666 & 2.280371 & 60.25804 & 81.40099 & 15.50208 & \(-\)98.88515 \(\times 10^{-2}\) & 2.07786 \(\times 10^{-9}\) \\
03 & 3.97637 & 3.660066 & 12.98547 & 13.08179 & 17.51666 & \(-\)4.846210 \(\times 10^{-2}\) & 4.69268 \(\times 10^{-9}\) \\
002 & 6.017231 & 5.2412926 & 44.02652 & 31.90685 & 18.48925 & \(-\)14.64535 \(\times 10^{-2}\) & 21.46472 \(\times 10^{-9}\) \\ \(2\times 10^{-5}\) & 9.4212791 & 8.505337 & 169.43146 & 794.3151 & 22.79988 & \(-\)79.74283 \(\times 10^{-9}\) & 2.221266 \(\times 10^{-8}\) \\ \(4\times 10^{-8}\) & 13.94893 & 12.82268 & 552.7447 & 1764.149 & 25.96071 & \(-\)2.333582 \(\times 10^{-9}\) & 3.959115 \(\times 10^{-8}\) \\ \(3\times 10^{-12}\) & 20.80268 & 9.49009 & 184.4522 & 3967.061 & 33.31493 & \(-\)30.203582 \(\times 10^{-2}\) & 2.72775 \(\times 10^{-9}\) \\ \(2\times 10^{-8}\) & 30.909496 & 29.332307 & 61.37584 & 8881.418 & 45.50420 & \(-\)2.422823 \(\times 10^{-9}\) & 2.28753 \(\times 10^{-8}\) \\ \(1\times 10^{-9}\) & 42.999807 & 41.1080482 & 171.7797 & 54.76092 & \(-\)8.990982 \(\times 10^{-9}\) & 6.88084 \(\times 10^{-9}\) \\ \(1\times 10^{-9}\) & 67.57016 & 65.27884 & 641.3142 & 4264.47 & 80.35366 & \(-\)1.012833 \(\times 10^{-9}\) & 1.70539 \(\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Cent. The calculated bubble shapes for \(10^{-40}<B<1000\) are presented in Figure 9, where each shape is drawn at equal intervals on a log scale of \(B\). In the interval of \(B=10^{-40}\) to \(10^{-4}\), the bubble exhibits an almost hemispherical shape, and the cavity is virtually flat (Figure 9a). However, in the interval \(10^{-4}<B<0.1\), the shape becomes less hemispherical as the bottom of the bubble continues to descend into the liquid (Figure 9b). In the interval \(0.1<B<10\), the shape continues to change from hemispherical to that of a bisected rugby ball (Figure 9c). Interestingly, the depth \(H\) is found to increase to a maximum value and subsequently decrease within this interval. In the interval \(10<B<100\), the bubbles have acquired a nearly spherical shape and are seen to protrude only slightly above the undisturbed liquid surface (Figure 9d). Finally, in the interval \(100<B<1000\), the bubbles are almost spherical and are almost entirely submerged in the liquid (Figure 9e). Figure 9: The computed bubble shapes for \(10^{-4}<B<1000\), where each shape is drawn at equal intervals on a log scale of \(B\). (**a**) Twenty bubbles for \(B=10^{-40}\)–\(10^{-4}\) (\(Bo=5723\)–\(83.6\)). (**b**) Fifteen bubbles for \(B=10^{-4}\)–\(0.1\) (\(Bo=83.6\)–\(11.8\)).