# 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
The speed of sound \(c_{s}\) reduced during the rupture of the liquid film, allowing a low Mach number to be maintained to keep the flow weakly compressible. Figure 16: _Cont._  
Shortly after the rupture, the liquid rim adsorbed most of the liquid on the liquid film. As observed from \(t=11.17\) to \(t=11.25\), the size of the liquid rim became larger and larger during the sweeping of its path. As time progressed, the liquid rim gradually accelerated, and not all liquid on the film was adsorbed by the rim. Some of the liquid splashed and formed small droplets, as shown from \(t=11.21\) to \(t=11.25\). This process continued until the whole liquid film forming the soap bubble was broken into droplets, as shown in Figure 16h at \(t=11.27\). The movement of the liquid rim after the rupture of the thin liquid film is in good agreement with the results obtained by experiments reported by [PERSON] and [PERSON] [57]. Figure 17 illustrates the profile of the interface concentration of the surfactant \(f\) in the interface region where the volume fraction was \(0.05<\phi<0.95\). As shown in Figure 17, at the upper part of the interface of the bubble, the interface concentration of the surfactant \(f\) decreased due to the flow of the fluid. As a result, the interface concentration of the surfactant \(f\) decreased on the upper side of the liquid film. The surface tension coefficient \(\sigma(f)\) decreased to a lesser degree. At the lower part of the interface of the bubble, adsorption occurred, so the interface concentration of the surfactant increased. The surface tension coefficient \(\sigma(f)\) decreased sharply and became much smaller than in the upper part of the bubble interface. This adsorption-desorption tendency agrees well with the results of previous studies conducted by [PERSON] [58] using an incompressible solver. The progress of adsorption with a surfactant such as Triton X-100 is strong. Even with a small initial bulk concentration \(F_{0}\), the maximum interface concentration \(f_{max}\) gets close to the saturation concentration while the bubble is rising. After the liquid film was generated, the lower surface of the thin liquid film collided with the upper surface of the ascending bubble, so desorption occurred during the ascending movement, and the interface concentration of the surfactant \(f\) became smaller than that of the upper surface of the liquid film. Figure 16: Volume fraction profile focused on the liquid film area during the rupture process of the liquid film in surfactant solution with a Langmuir number of \(La=15\) from \(t=11.13\) to \(t=11.27\) with an interval of \(\Delta t=0.02\). The region presented in red contains heavy fluid, and the region presented in blue contains light fluid. The AMR leaves are shown as light grey lines. On the interfaces of the liquid film, as shown in Figure 17, the interface concentration \(f\) was the lowest on the top part of the interface of the liquid film. The interface concentration \(f\) increased as the distance increased from the center of the liquid film to the left and the right. This distribution is thought to be due to the downward transport of the surfactant by gravity. Marangoni convection occurred in the direction tangential to the interface to restore this interface concentration gradient, and it is thought that a force acts in the direction opposite to the collapse of the liquid film due to gravity and in the direction that stabilizes the liquid film. In addition, as can be seen from Equation (26), the surface tension coefficient decreased when the interface concentration of the surfactant \(f\) approached the saturation concentration. Figure 18 shows the interface concentration profile during the liquid film rupture process from \(t=11.13\) to \(t=11.27\) with an interval of \(\Delta t=0.02\) when the Langmuir number was \(La=15\). It was observed that the upper rim on the left side had a later generation time.