# An AMR-Based Liquid Film Simulation with Surfactant Transport Using PLIC-HF Method
## 5 Numerical Results
### Liquid Film Generation and Rupture by a Single Bubble Rising to the Interface
By setting the initial interface concentration of the bubble surface to the equilibrium state, adsorption could easily proceed, and the interface concentration quickly approached the saturation concentration [56]. Therefore, according to Equation (26), the gradient of the surface tension coefficient became steep, the Marangoni effect worked strongly, and the shape of the bubble smoothed. The AMR method was used in the simulations. The initial tree node was set as \(4\times 8\), so the initial resolution was \(16\times 32\). The maximum AMR refinement level was set as \(6\). Thus, the finest resolution was \(1024\times 2048\), with the minimum resolution being \(\Delta h=9.7656\times 10^{-4}\). An NVIDIA V100 GPU mounted on the Flow Type II supercomputer of the Information Technology Center of Nagoya University and a NVIDIA P100 GPU mounted on the TSUBAME 3.0 supercomputer of The Global Scientific Information and Computing Center of Tokyo Institute of Technology were used in the simulation. The volume fraction profile during the rising of the bubble and the generation of the liquid film is shown in Figure 15. When the bubble started to rise, the bottom of the bubble became flat. At around \(t=2.0\), the upper part of the interface of the bubble approached \begin{table}
\begin{tabular}{c c c c} \hline
**Surfactant** & \(\mathbf{k_{\text{ad}}}\left[\mathbf{m^{3}/(mol\cdot s)}\right]\) & \(\mathbf{\beta}[\mathbf{mol/m^{3}}]\) & \(\mathbf{f_{\text{lim}}}\left[\mathbf{mol/m^{2}}\right]\) & \(\mathbf{k_{\text{de}}}\left[\mathbf{s^{-1}}\right]\) \\ \hline Triton X100 & 50.0 & \(6.6\times 10^{-4}\) & \(2.9\times 10^{-6}\) & 0.033 \\ \hline \end{tabular}
\end{table} 
Table 3: Physical properties as a soluble surfactant of Triton X-100. the horizontal interface between the heavy fluid and light fluid. As the bubble continued to rise, the liquid layer over the bubble became thinner. After \(t=5.0\), with the drainage of the upper liquid layer, a thin liquid film was generated. The thin liquid film was kept stable until around \(t=11.0\). Since the present simulation used the PLIC-HF method, the influence of spurious current on the liquid film was greatly suppressed. Under the influence of gravity, drainage continued and the thickness of the thin liquid film gradually decreased. When the thickness of the thin liquid film reached a lower limit, rupture occurred on the liquid film at \(t=11.11\). Figure 16 shows the detailed volume fraction profile and the behavior of the liquid rim during the rupture of the liquid film from \(t=11.13\) to \(t=11.27\) with an interval of \(\Delta t=0.02\). As illustrated by Figure 16, when the rupture occurs, a moving liquid rim is generated at that place. The generation of the liquid rim on the left is shown by Figure 16a at \(t=11.13\) and the generation of the liquid rim on the right is shown by Figure 16c at \(t=11.17\). In the present two-dimensional simulation, the rim moved as droplets on the edges of the liquid film. The moving liquid rim swept the liquid film in its path at a higher  
Figure 15: Volume fraction profile for the bubble rising to an interface starting from the contaminated state and forming liquid film in surfactant solution with a Langmuir number of \(La=15\), from \(t=1.0\) to \(t=12.0\). The region presented in red contains heavy fluid and the region presented in blue contains light fluid. AMR leaves are shown as light gray lines. (**a**) Volume fraction profile for the bubble rising to an interface starting from the contaminated state and forming a liquid film in surfactant solution, \(La=15\), from \(t=1.0\) to \(t=6.0\). (**b**) Volume fraction profile for the bubble rising to an interface starting from the contaminated state and forming liquid film in surfactant solution, \(La=15\), from \(t=7.0\) to \(t=12.0\). speed than the typical velocity. This process can be observed from \(t=11.17\) to \(t=11.25\). The typical velocity of the simulation was \(U=0.5\), and the maximum velocity of the liquid rim was \(U_{max}=3.6\), more than 7 times the typical velocity.