# Simulation of the Marangoni Effect and Phase Change Using the Particle Finite Element Method
## 3 Results
### Melting of a Block of Gallium  
A rectangular block of solid gallium slightly below the melting point is in a closed box. The block is exposed to a hot wall on the left, which is of constant temperature significantly above the melting point. The gallium then progressively melts from left to right. This test case combines a buoyancy-driven convective flow and the modelling of phase change, which affects momentum equation and heat. All of these effects are strongly coupled, leading to an interesting test case. A schematic is provided in Figure 8.  
This test case is based on experiments originally conducted by [PERSON] and [PERSON] in 1986 [35]. The advantage of choosing gallium is that the melting point, \(Tm\) = 302.78 K, is near room temperature, greatly simplifying the experimental setup. Likewise, a pure metal exhibits a sharp melting front, which is easier to measure. The remaining material parameters are \(\rho\) = 6093 kg/m\({}^{3}\), \(\mu\) = 1.81 \(\times\) 10\({}^{-3}\) Pa s, \(c_{p}\) = 381.5 J/kg K, \(\kappa\) = 32.0 W/m K, \(\beta_{V}\) = 1.2 \(\times\) 10\({}^{-4}\) K\({}^{-1}\) and \(\Delta h_{L}\) = 8,016,021 J/kg. The hot wall on the left has \(T_{hot}\) = 311.0 K, and the cold wall on the right has \(T_{cold}\) = 301.3 K, which is also used as initial temperature \(T_{0}\) from the entire domain. Note that the initial temperature is very close to the melting point. The experimental setup is shown in Figure 9. The cavity can be adjusted to have different dimensions, resulting in different aspect ratios \(A=H/W\), where \(H\) = 0.0635 m and \(W\) = 0.0889 m are typically used for the height and width of the cavity, respectively, to arrive at an aspect ratio of \(A=0.714\). This aspect ratio is used in the simulations described below, and other aspect ratios mentioned by [PERSON] and [PERSON] are ignored in this work. Figure 7: Isotherms of non-dimensionalized temperature \(\theta\) = \((T-T_{\text{min}})/\Delta T\). Top (**a**–**c**) shows results by [PERSON] and [PERSON] [34], [PERSON] [33] in the middle (**d**–**f**) and this work at the bottom (**g**–**i**). Left (**a**,**d**,**g**) shows case 1 with pure buoyancy-driven flow. Middle (**b**,**e**,**h**) shows case 2, where the Marangoni effect enhances the underlying buoyancy-driven circulation. Right (**c**,**f**,**i**) shows case 3, where the Marangoni effect counteracts the underlying buoyancy-driven circulation. A normalized temperature, \(\theta=T/\Delta T\), is used. The process of obtaining the position of the melting front in the experiment is not simple, as gallium is not visually transparent. First, melted gallium is poured into the cavity through the opening at the top, which is sealed afterwards. Then, the entire cavity is cooled down to the initial condition, which is a fixed temperature slightly below the melting point. At this stage, the gallium has completely solidified in the cavity. The experiment begins by pushing fluid of a constant given temperature through the side walls, the left one hot and the right one cold; this way, the side walls maintain a constant temperature throughout the entire experiment. As time passes, the gallium melts near the hot wall and convective flow sets in. After a given time, the experiment is interrupted and the liquid quickly evacuated from the cavity, leaving only the remaining solid part of the gallium block behind. From this block, the melting front is traced and photographed. For each time duration, the experiment is repeated from the beginning to finally reconstruct the progression of the melting front over time. This way, the melting front position is obtained  
Figure 8: Schematic of the gallium melting test. The entire domain is just below freezing temperature initially, then unfreezes from left to right, with the melting front indicated by a magenta line. The resulting buoyant convective motion of the liquid gallium, indicated by a thin black arrow, influences the advancement of the melting front, resulting in a curved front such as the one shown here.