# 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  
Thin liquid film generation and rupture by a single bubble freely rising to an interface was simulated by a weakly compressible scheme with a pressure projection equation, the PLIC-HF method, and the Langmuir model. The initial geometrical settings, including the size of the computational domain, the size of the bubble, and the position of the bubble in Section 5.2 were used. A horizontal interface between the heavy and light phases was set at the center of the computational domain where \(y=1.0\), as illustrated by Figure 14.  
Surfactant transport was calculated in the present simulation by using the Langmuir model. The surface tension coefficient \(\sigma\) changes accordingly to the interface concentration \(f\) of the surfactant and causes the Marangoni effect. The physical properties of the surfactant in terms of adsorption and desorption were set in accordance with Triton X-100 [55], as shown in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Number of Cells on** & \
\begin{tabular}{c} **Maximum Number** \\ **of Cells on Finest** \\ **256 \(\times\) 512 AMR Grid** \\ \end{tabular} &
\begin{tabular}{c} **Maximum Number** \\ **of Cells on Finest** \\ **512 \(\times\) 1024 AMR** \\ **Grid** \\ \end{tabular} \\ \hline Case A & 131,072 & 21,728 & 43,328 \\ Case B & 524,288 & 29,696 & 62,432 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of cells on a uniform mesh and maximum number of cells on different AMR grids. Figure 14: A schematic diagram of liquid film generation and rupture simulation by a single bubble freely rising to an interface. The region presented in light blue is occupied by heavy fluid and the region presented in white is occupied by light fluid. The Langmuir number \(La\) is a dimensionless number that expresses the ease of desorption of a surfactant. It is typically defined as  
\[La=\frac{Fk_{\text{ad}}}{k_{\text{de}}} \tag{67}\]  
where \(k_{\text{ad}}\) is the adsorption rate coefficient, and \(k_{\text{de}}\) is the desorption rate coefficient. It is a measure of the overall equilibrium between adsorption and desorption for the surfactant at the liquid surface, and it is typically used in the study of surfactant adsorption and desorption at the surface of a liquid. A high Langmuir number indicates that the adsorption rate of the surfactant is dominant over the desorption rate, meaning that the surfactant tends to adsorb onto the surface of the liquid and remain there. In contrast, a low Langmuir number indicates that the desorption rate is dominant over the adsorption rate, meaning that the surfactant tends to desorb from the surface and enter the bulk of the liquid. The surfactant Triton X-100, whose properties were used in the present research, has a relatively high adsorption rate coefficient to desorption rate coefficient ratio, \(\frac{k_{\text{ad}}}{k_{\text{de}}}=50\). Cases with a Langmuir number of \(La=15\) were simulated. According to Equation (67), with a Langmuir number of \(La=15\), the initial bulk concentration of the surfactant is \(F_{0}=1.0\times 10^{-2}\). The interface concentration on the initial interface \(f_{0}\) was set to the concentration in the equilibrium state, where \(j=0\) in Equations (20) and (21). From Equation (22), when \(j=0\),  
\[j=k_{\text{ad}}F(f_{\text{lim}}-f)-k_{\text{de}}f=0. \tag{68}\]  
Therefore, in the beginning of the simulation when \(t=0\),  
\[f=\frac{k_{\text{ad}}\cdot F\cdot f_{\text{lim}}}{k_{\text{de}}+k_{\text{ad}} \cdot F} \tag{69}\]  
The saturation concentration on the interface was \(f_{lim}=2.9\times 10^{-6}\). Since the Langmuir number was \(La=15\), the equilibrium concentration on the initial interface was \(2.72\times 10^{-6}\), which is 93% of \(f_{lim}\). Physically, it was assumed that the bubbles were contaminated in a stationary state for a long time.