## 7 Results
### The Phase Separation Problem with the Enhanced Model  
The phase separation between the liquid and vapor phases of an initial mixture of both phases was studied in this section. The domain was initially at rest with an average density plus a random variation in in all nodes. All boundary conditions were periodic. The system was unstable based on EOS, and phase separation occurred. The final shape of bubbles or drops was circular since the free energy of the domain tends to minimize, and a circle has the minimum surface area compared to other shapes.  
The problem was simulated by the enhanced model in six different cases. In each case, the initial density of the mixture was different, and a random variation in the order of 0.01 was added to the density. The temperature was selected as \(\mathrm{T}=0.9\mathrm{T}_{\mathrm{cr}}\) which corresponds to a density ratio of 6.  
Based on the value of \(\mathrm{G}_{1}\) and \(\mathrm{G}_{2}\) in Table 1, different scenarios of phase separation occurred as shown in Figure 6. For larger \(\mathrm{G}_{2}\), corresponding to a bigger repulsive force, the coalescence of the drops was prevented in some of the cases. These results are in agreement with Ref. [44], which simulated the same problem. One difference with the simulations of Ref. [44] is that they used the original Shan-Chen EOS while the more realistic C-S EOS was used in this work. However, since the density ratio was almost the same in this study as the one in Ref. [44] (six in this study and five in Ref. [44]), the results can be compared directly. The results show that the midrange-repulsive force can be used to prevent the coalescences of drops, but not bubbles. In the case of bubbles, this method only delays the merging process, but, eventually, just one bubble remains in the domain. For case (a) and (b), one bubble was present in the domain. However, for other cases, the formation of drops was observed. The formation of bubble or drops was related to the initial density. If the initial density is larger than a critical value, after some iterations, only one bubble will result in the whole domain.  
As mentioned before, the enhanced model reduced the spurious current, but even this reduced amount of spurious current was too large for modeling phenomena, such as Marangoni convection, in this scale. The maximum amount of spurious current was observed in the interface between the bubbles and liquid, which is the place where the Marangoni force is exerted. The magnitude of the spurious current is presented in Table 2, showing approximately one order of magnitude reduction of the spurious current in the enhanced Shan-Chen model with respect to the original one, but still a large value at \(5\times 10^{-3}\). On the other hand, when this same problem was simulated with the phase field lattice Boltzmann method, the generated spurious current was in the order of \(10^{-6}\), which is acceptable for most cases of buoyancy and Marangoni flows [31].  
The total computational time of the simulation was also a critical parameter. The simulation time was non-dimensionalized with the time needed to solve the original Shan-Chen model. In Table 2,  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{G_{1}}\) & \(\mathbf{G_{2}}\) & \(\rho_{initial}\) \\ \hline Case1 & \(-1.4\) & 1.00 & 0.20 \\ Case2 & \(-1.4\) & 1.00 & 0.170 \\ Case3 & \(-1.4\) & 1.00 & 0.130 \\ Case4 & \(-1.4\) & 0.95 & 0.130 \\ Case5 & \(-1.4\) & 0.90 & 0.130 \\ Case6 & \(-1.4\) & 1.00 & 0.117 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters used in the simulation.  
\begin{table}
\begin{tabular}{c c c} \hline  & **Spurious Current** & **Normalized Total CPU Time** \\ \hline Original Shan-Chen & 0.03 & 1 \\ The enhanced model & 0.005 & 2.49 \\ \hline \end{tabular}
\end{table}
Table 2: The spurious current magnitude and normalized total time in the original Shan-Chen and the enhanced model.  
it is observed that the model enhancements make it about 250% more expensive in CPU time than the original Shan-Chen model.