# Simulation of the Marangoni Effect and Phase Change Using the Particle Finite Element Method
## 3 Results
### Melting of a Block of Gallium
Figure 9: Setup for the gallium melting experiment by Gau and Viskanta 1986 (Gau and Viskanta 1986 ([PERSON] and [PERSON] 1986); (**a**) top view and (**b**) side view showing heated side wall with constant temperature, adiabatic side walls made of plexiglass (thick black solid lines) and holes for filling and evacuating. for nine different time marks (2, 3, 6, 8, 10, 12.5, 15, 17 and 19 min). It is not disclosed if the experiment was repeated several times and the results averaged for each time mark. It is also not discussed in detail how the experiment could be flawed. Potential issues can arise from:  
* 3D effects that are not represented in the 2D curves of the melting front;
* Shrinkage after filling the cavity with liquid gallium and letting it solidify, leading to potential loss of contact between the gallium and the walls;
* Heat losses or parasitic heat conduction in the walls;
* Possible transient temperature evolution in the \"constant temperature walls\" upon switching on the constant temperature liquid feeding system;
* Possible continuation of the melting while the liquid is evacuated too slowly, leading to sloshing potentially affecting the roughness of the resulting surface;
* Possible errors from tracing and photography, not explained in detail. Another large uncertainty is introduced through the strong anisotropy of the thermal conduction of solid gallium. This issue is discussed in the work [PERSON] in the context of the inverse experiment, where the gallium is initially liquid and then solidified on a constant temperature cold wall, as opposed to the melting of solid gallium on a hot wall, which is the topic of this work. In the solidification experiment, the physical behavior has been found to be highly dependent on the crystallography of the freshly solidified gallium, introducing significant variability into the position of the solidification front. In the melting experiment, the authors explain that the anisotropy of the solid gallium plays a lesser role, as the physical behavior is dominated by the heat transfer in the liquid, which is plausible. However, a quantitative evaluation of the variability of the experiment should have been included. It is also not clear if the presented results were obtained through several repetitions and subsequent averaging. In 1988, [PERSON] et al. [36] attempted to match the experimental results with a Eulerian Finite Volume Method approach that used an extremely coarse mesh by today's standards. The latent heat absorption was accounted for using an iterative nodal enthalpy formulation that also works on coarse, fixed grids. The authors point out that this scheme creates a diffuse phase change region rather than a sharp front. The momentum equations are modified for the solid using the same Carman-Kozeny equation for porous media as in Section 2.5. This combination of nodal enthalpy with an iterative scheme for the phase change and the porosity model for the solid is called the enthalpy-porosity approach in their work. The authors model the experiment as closely as possible, using the same material properties and a fixed rectangular domain with an aspect ratio \(A\) = 0.714. The boundary conditions at all walls are no-slip conditions. The authors provide results on streamlining and temperature contour plots for four time marks (3, 6, 10, 19 min) as well as the melting front position for all nine time marks, as was done for this experiment. More recently, [PERSON] [33] also used a Eulerian Finite Volume Method approach with the same Carman-Kozeny equation to freeze the fluid's motion as in Section 2.5. For the latent heat, [PERSON] uses a separate source term calculated using an iterative scheme similar to the one used by [PERSON] et al. [36]. [PERSON]'s work unfortunately does not state what numerical parameters (such as time step size, mesh size or residual target) were used. It is also not clear which type of boundary conditions were used on the walls. The latest research publications that feature this test case tend to use it as a validation step.