# Simulation of the Marangoni Effect and Phase Change Using the Particle Finite Element Method
## 3 Results
### Buoyancy and Marangoni-Driven Convection in a Square Cavity  
This 2D test case also uses a square cavity with a buoyancy driven flow, as above, but with additional surface tension acting on the top surface, see Figure 5. Comparisons are done with literature results from [PERSON]'s PhD thesis from 2012 [PERSON] (2012), who himself uses older results from [PERSON] and [PERSON] from 1988 as a reference [PERSON] and [PERSON] (1988). Since all the authors use a Eulerian Finite Volume Method, the top surface is assumed to remain flat, limiting the effect of surface tension to the tangential component, which is referred to as the Marangoni effect. Three test cases are investigated; the first one does not include the Marangoni effect, recovering a test case similar to those in the previous section. The second test case includes a negative Marangoni coefficient \(\partial\gamma/\partial T\), causing the Marangoni effect to push fluid at the surface in the same direction as the circulation caused by the underlying buoyancy-driven convective flow, reinforcing the circulation. The third case uses a positive \(\partial\gamma/\partial T\), leading to an opposite effect where the surface tension pushes against the underlying buoyancy-driven flow, counteracting the circulation at the surface. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data** & **this Work** &
\begin{tabular}{c} **[PERSON] \&** \\ **[PERSON]** \\ \end{tabular} & **[PERSON]** & **[PERSON] et al.** & **[PERSON] et al.** \\ \hline \(\partial_{x,\text{max}}\) & 65.115 & 65.45 & 65.81 & 64.558 & 64.483 \\ \(\hat{y}_{v,\text{max}}\) & 0.854 & 0.85 & 0.852 & 0.851 & 0.845 \\ \(\hat{y}_{v,\text{max}}\) & 225.913 & 213.07 & 214.64 & 221.572 & 218.054 \\ \(\hat{x}_{v,\text{max}}\) & 0.0404 & 0.0474 & 0.0396 & 0.067 & 0.037 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Resulting maximum velocities (\(\hat{v}_{x,\text{max}}\) and \(\hat{v}_{y,\text{max}}\)) on the centerlines (\(\hat{x}=0.5\) and \(\hat{y}=0.5\) respectively) and positions of maxima (\(\hat{y}_{v,\text{max}}\) and \(\hat{x}_{v,\text{max}}\) respectively) along centerlines \(Ra=10^{6}\). All values are non-dimensionalized. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data** & **this Work** &
\begin{tabular}{c} **[PERSON] \&** \\ **[PERSON]** \\ \end{tabular} & **[PERSON]** & **[PERSON] et al.** & **[PERSON] et al.** \\ \hline \(\hat{v}_{x,\text{max}}\) & 15.948 & 16.250 & 16.182 & 16.282 & 15.982 \\ \(\hat{y}_{v,\text{max}}\) & 0.820 & 0.821 & 0.823 & 0.822 & 0.824 \\ \(\hat{y}_{y,\text{max}}\) & 19.551 & 19.541 & 19.509 & 19.547 & 19.378 \\ \(\hat{x}_{v,\text{max}}\) & 0.120 & 0.115 & 0.120 & 0.123 & 0.116 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Resulting maximum velocities (\(\hat{v}_{x,\text{max}}\) and \(\hat{v}_{y,\text{max}}\)) on the centerlines (\(\hat{x}=0.5\) and \(\hat{y}=0.5\) respectively) and positions of maxima (\(\hat{y}_{v,\text{max}}\) and \(\hat{x}_{v,\text{max}}\) respectively) along centerlines \(Ra=10^{4}\). All values are non-dimensionalized. The material data are given in detail by [PERSON] and [PERSON], resembling molten aluminum at 660 \({}^{\circ}\)C: \(\rho\) = 2385 kg/m\({}^{3}\), \(\mu\) = 1.3 \(\times\) 10\({}^{-3}\) Pa s, \(c_{p}\) = 1080 J/kg K, \(\kappa\) = 94.03 W/m K, \(\beta_{V}\) = 1.17 \(\times\) 10\({}^{-4}\) K\({}^{-1}\). This gives us \(Pr\) = 0.0149 and \(Ra\) = 4.6 \(\times\) 10\({}^{4}\). These settings fully describe case 1, where only the buoyancy effect is driving the flow.