# 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
For the other cases we superimpose the Marangoni effect; thus, the Marangoni number \(Ma\) (sometimes denoted as \(Mg\) to avoid confusion with the Mach number) that describes the relative importance of the Marangoni effect compared to the viscosity is defined as  
\[Ma=\frac{\
icefrac{{\partial\gamma}}{{\partial T}}\,\Delta TL}{\alpha\mu}= \frac{\
icefrac{{\partial\gamma}}{{\partial T}}\,\rho c_{p}\Delta TL}{\mu\kappa}. \tag{40}\]  
The Marangoni coefficient is \(\partial\gamma/\partial T=2.0\times 10^{-4}\mathrm{N/m\,K}\) for case 2 and \(\partial\gamma/\partial T=-3.5\times 10^{-4}\mathrm{N/m\,K}\) for case 3. This results in \(Ma=0\), \(Ma=1.47\times 10^{4}\) and \(Ma=-8.4\times 10^{3}\) for cases 1, 2 and 3, respectively. The cavity has an edge length of \(L=0.02\) m and a Temperature difference \(\Delta T=100\) K between cold and hot walls. All results are compared after reaching steady state equilibrium. We use an element size of \(\Delta x=0.001\) m and a time step of \(\Delta t=0.001\) s, \(\Delta t=0.0005\) s and \(\Delta t=0.0005\) s for the three cases, respectively, to accommodate the higher local velocities produced by the Marangoni effect. Further refinement of the time step mesh size does not improve the results. [PERSON] uses a Eulerian FVM approach, and does not state mesh size or time step length. [PERSON] and [PERSON] use a steady state Eulerian FVM approach with a similar mesh size, but with a refinement near the boundaries. Both [PERSON] [33] and [PERSON] and [PERSON] [34] compare streamlines that highlight the flow pattern for each case. As we do not have the stream function (see e.g., [41], p. 14) available in our results, we constructed the streamlines by integrating from a given seed point in space over the velocity vector field. For this approach, the seed points must be chosen by the user, and the resulting streamlines do not necessarily represent the same value of the stream function as given by [PERSON] and [PERSON]. We chose each seed point at a location that is intersected by the streamlines from [PERSON] and [PERSON] in an attempt  
Figure 5: Schematic of square cavity with buoyancy driven and Marangoni effect-driven convective flow. The top surface is free to move in \(x\) but not in \(y\) direction. The thin arrows indicate the isolated direction of the flow from the two different effects. Three cases are shown: Case 1 (brown): no Marangoni effect, only buoyancy driven flow (black arrow). Case 2 (blue): [PERSON] effect has the same orientation as buoyancy driven flow (blue and black arrows superimposed). Case 3 (green): [PERSON] effect has the opposite orientation as buoyancy driven flow (green and black arrows superimposed). to produce the most qualitatively similar representation of the main flow field. For the recirculation bubbles that can appear in the corners, we freely chose a convenient location for the seed points with the aim of best capturing the flow pattern in the corners. Crucially, these different procedures for obtaining the streamlines introduce some uncertainty when comparing streamlines. The comparison between [PERSON] and [PERSON] (top), [PERSON] (middle) and our work (right) can be found in Figure 6.  
Figure 6: Streamlines representing the flow field in the cavity. Results from [PERSON] and [PERSON] [34] at the top (**a–c**), [PERSON] [33] in the middle (**d–f**) and this work at the bottom (**g–i**). Left (**a,d,g**) for case 1 with only buoyancy driven flow. Middle (**b,e,h**) showing case 2 with Marangoni effect reinforcing the buoyancy-driven circulation. Right (**c,f,i**) showing case 3 with Marangoni effect counteracting the buoyancy-driven circulation. Note that [PERSON] and [PERSON] give the maxima and minima of the stream function \(\psi\) or each case. The results show good overall agreement, although a few discrepancies can be found as well. For case 1 in Figure 6a,d,g, the main flow fields are nearly indistinguishable. Small differences can be found with the recirculation bubbles in the corners, and no bubble is found in the bottom left corner at all.