## 4 Results and Discussion
### Comparison of the Present Results with the Earlier Numerical and Experimental Results  
The present computational results are compared with the previously published numerical results in Table 1. Various bubble parameters were previously calculated in the range of B = 1-20, B = 0.28-8, and \(B=4\times 10^{-8}\)-50 by [PERSON] [29], [PERSON] [30], and [PERSON] and [PERSON] [31], respectively. However, [PERSON] [30] used the parameter \(\beta\) in place of \(B\), where the relationship between these is given by  
\[B=\sqrt{8/\beta}\quad\text{ and }\quad X=(x/b)\sqrt{\beta/2}, \tag{27}\]  
where \(X\) denotes the dimensionless parameter of the present work, and \((x/b)\) denotes the dimensionless parameters of [PERSON]'s work. The computed parameters in Table 1 agree well with one another, with the exception of [PERSON]'s parameters. This is because [PERSON] employed a semi-graphical technique of limited accuracy. The earlier methods used an approximate analytical solution for tiny bubbles [31] or an interpolation method with a  
Figure 5: Comparison of the conventional (red) and the improved (black) methods for large bubbles. The tolerance of the ODE solver is set to \(10^{-12}\). (**a**) Floating bubble shapes for the following three large bubble sizes: \(B\) = \(1\times 10^{-7}\), \(1\times 10^{-14}\), and \(1\times 10^{-22}\). (**b**) The dimensionless cap radius \(R_{0}\) is plotted with respect to the parameter \(B\). Figure 6: Comparison of the conventional (red) and the improved (black) methods for small bubbles. The tolerance of the ODE solver is set to \(10^{-12}\). (**a**) Floating bubble shapes for the following three small bubble sizes: \(B\) = 1000, 2000, and 3000. (**b**) The dimensionless absolute height \(|H|\) is plotted with respect to the parameter \(B\). pre-calculated Table [30], so the sizes considered were narrow. [PERSON] and [PERSON] [31] used the boundary conditions of \(Z_{a}=0.00001\) and \(\sin\alpha_{\infty}\leq 0.0001\). [PERSON] [30] searched the junction point until both \(\Delta X_{c}\) and \(\Delta Z_{c}\) were less than \(\varepsilon=0.5\times 10^{-5}\). In contrast, the present calculations maintain an accuracy of \(|\sin\alpha_{\infty}|<5\times 10^{-20}\) and \(\varepsilon_{H}<5\times 10^{-8}\), thus providing much more precise results. The experimental results of [PERSON] et al. [37] are compared with the present numerical results for selected bubble parameters in Figure 7a, where the parameters are plotted with respect to the Bond number based on the cap radius, i.e., \(B_{00}=\rho g\tau_{0}^{2}/\sigma\). Here, the present computations are seen to agree well with the experimental results, except for the \(H^{*}\) values. For this parameter, the numerical results obtained using Surface Evolver S/W [42] have shown a similar trend to the present computations, as described in [37]. Meanwhile, the experimental results of [PERSON] et al. [36] are compared with the present numerical results for selected parameters in Figure 7b, where \(Z_{m}\) denotes the height of the bubble top above the undisturbed surface. All parameters are normalized with respect to the equivalent bubble radius \(R_{e}\). The experimental results agree well with the present computation, except that the present computation overestimates the experimental data for \(R_{0}/R_{e}\). Notably, the numerical results of [PERSON] et al. [36] also overestimate \(R_{0}/R_{e}\) by a similar amount.