# Simulation of the Marangoni Effect and Phase Change Using the Particle Finite Element Method
## 3 Results
### Buoyancy-Driven Convection in a Square Cavity  
This simple 2D test case contains a square cavity in which a fluid encounters a hot wall on one side and a cold wall on the opposite side, leading to natural convection, as illustrated in Figure 2. For any choice of material parameters and experimental setup, the resulting flow and the balance between heat conduction and heat convection can be compared using dimensionless numbers, which use the thermal diffusivity, \(\alpha\):  
\[\alpha=\frac{\kappa}{\rho c_{p}} \tag{37}\]  
It has the dimension m\({}^{2}\)/s and characterizes the rate of conductive heat transfer through a medium. The first relevant dimensionless number in this case is the Prandtl number \(Pr\), which reads  
\[Pr=\frac{\
u}{\alpha}=\frac{c_{p}\mu}{\kappa}, \tag{38}\]  
where \(\
u\) is the kinematic viscosity. \(Pr\) describes the ratio between momentum diffusion and thermal diffusion. The second relevant dimensionless number is the Rayleigh number, \(Ra\), which reads:  
\[Ra=\frac{g\beta\
u\Delta TL^{3}}{\alpha\
u}=\frac{\rho^{2}g\beta\
u c_{p} \Delta TL^{3}}{\mu\kappa}, \tag{39}\]  
Figure 2: Schematic of cavity with buoyancy-driven convective flow. Thin black arrow indicates the expected sense of rotation of the convective flow. Velocities are extracted from the dashed centerlines. where \(\Delta T\) is the characteristic temperature difference and \(L\) is the characteristic length scale of the problem; \(Ra\) describes the ratio between convective heat transfer and conductive heat transfer. Following [PERSON] et al. (2018), we use an edge length \(L=1\) m and a temperature difference \(\Delta T=1\) K, where the left side is arbitrarily chosen by the authors to be \"hot\" at 20.5 K and the right side to be cold at 19.5 K. The authors have taken these settings from [PERSON] et al. (2018), whom they use as their reference. The dimensionless numbers are given as \(\text{Pr}=1\) and \(Ra=10^{6}\). To achieve these, we chose \(\rho=1\text{kg}/\text{m}^{3}\), \(\mu=10^{-4}\) Pa s, \(c_{p}=1\text{J}/\text{kg}\) K, \(\kappa=10^{-4}\text{W}/\text{m}\) K, \(\beta_{V}=10^{-3}\) K\({}^{-1}\), \(g=-10\text{m}/\text{s}^{2}\), although the other authors do not state their exact choice and their mesh specifics are also not mentioned. We use a uniform triangular mesh with an average edge length of \(h=0.0125\) m, and the time step is set to \(\Delta t=0.5\) s.  
When using the same \(Pr\) and \(Ra\), the steady state solutions are expected to match. [PERSON] et al. (2018) reach the steady state solution after a long simulation time of 396 s. In Figure 3a, good agreement of the temperature distribution can be observed. The color maps may not match exactly, as the color distribution over the range of temperatures is not provided by [PERSON] et al., but the temperature fields are obviously very similar. Next, the transient evolution of the temperature field before reaching the steady state is compared in Figure 4; good agreement during the whole simulation time can be observed. The gradual mixing of cold and hot fluid, along with the developing convective flow are clearly visible and no major differences can be found. It is interesting to note that a different combination of input parameters can produce the same \(Pr\) and \(Ra\), and therefore the same steady state solution, whereas the evolution over time may be at a different rate. This can be explained by the fact that the input parameters are non-dimensional, while the time is dimensional. The use of a non-dimensionalized time, such as the one discussed in the following section, would be more appropriate. [PERSON] and [PERSON] (2018), the reference used by [PERSON] et al., only features the steady state results; thus, no dimensionless time is required. Also note that [PERSON] and [PERSON] define \(Ra\) using \(K\) (instead of the usual \(\alpha\)) as the symbol for the thermal diffusivity, which led [PERSON] et al. to accidentally define \(Ra\) with a \(\kappa\), denoting thermal conductivity, in the place of thermal diffusivity, \(K\).