# 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
Such recirculation bubbles are, however, naturally rather unstable and sensitive to the main flow field. Good agreement regarding the size and position of cells of slow-moving fluid and reversed flow can be seen. For case 2 in Figure 6b,e,h, the effect of the surface tension enhancing the underlying buoyancy-driven circulation can be seen. The streamlines reach much further into the top right corner before changing direction, and the streamlines there also come closer together. Otherwise, the flow pattern is largely the same as in case 1. The agreement is better between [PERSON] and [PERSON] (b) and [PERSON] (e) than with our results (h). Again, we find slightly different recirculation bubbles in the corners, and the vortex center of the main circulation is more centered in (h) than in (b) and (e). For case 3, in Figure 6c,f, the surface tension can clearly be seen counteracting the buoyancy, as the flow pattern is neatly divided into two distinct vortices. A line of zero vorticity, where the two vortices meet, would occur close to the diagonal between bottom left and top right corner. This flow pattern is found to be sensitive to _Ra_ and _Ma_, such that a small change in input parameters would quickly favor one of the two effects over the other, quickly leading to one of the main vortices to dominate over the other. It is possible that this delicate balance was deliberately chosen by [PERSON] and [PERSON] to emphasize the importance of the Marangoni effect, as they argue that the Marangoni effect has been largely neglected in applications where convective flows in liquid metals are of importance. The agreement of all three results is less good than before, and the size and position of the two distinct vortices is slightly different in each case. Given the delicate balance that causes this flow pattern, our result agrees well with the literature, as the differences are very minor. Both [PERSON] [33] and [PERSON] and [PERSON] [34] also present plots of isotherms using a non-dimensionalized temperature \(\theta\) = \((T-T_{\text{min}})/\Delta T\) with a spacing of 0.1. The agreement between all three publications is good. Comparing [PERSON] and [PERSON]'s result in the top row of Figure 7 with [PERSON]'s in the middle and ours on the bottom row, a similar tendency as in the previous comparison can be observed: qualitatively a good agreement when looking at the overall appearance of the plots; the spacing of the isotherms in most cases is especially well captured, indicating that the important temperature gradients are well-captured. On the other hand, several discrepancies can be found for cases 2 and 3 between literature results in Figure 7b,e and Figure 7c,f and our results Figure 7h,i respectively. In both cases 2 and 3, the isotherms appear more twisted in a clockwise direction in the areas where the convective flow is strongest. Unfortunately, no experimental results that can validate the simulation results from the literature could be found, which leaves some uncertainty regarding any of the results presented.