## 7 Results
### Dendritic Growth with the Phase Field Model  
The same dendrite growth problem, as defined in Section 7.4, was solved with the phase field LB model. For comparison purposes, the density ratio was selected as 40 (\(\rho_{1}=1\) and \(\rho_{\mathrm{g}}=0.025\)) even though the phase field model can simulate much higher density ratios. The results are shown in Figure 10, for the same time values of Figure 8.  
Unlike the Shan-Chen models, the dissolution and reappearance of bubbles related to the EOS were not observed in this model. Therefore, the size of the bubbles remained almost constant during the simulation. A small movement was observed for bubbles beyond the dendrite tips while the bubbles between dendrites were trapped and did not move.  
Figure 9: Results of the enhanced model at time (**a**) 0.0013, (**b**) 0.0026, (**c**) 0.00387, and (**d**) 0.0052 s for \(\mathrm{T}=0.7\mathrm{T}_{\mathrm{cr}}\), which corresponds to a density ratio of 40. (yellow: solid, red: liquid, blue: gas).  
Table 4 summarizes the magnitude of the spurious current, the simulated density ratio, and the computational time. The original Shan-Chen model could only handle the density ratio of four while it diverged for the higher density radios. The phase field method was capable of simulating much higher density ratios, but since the enhanced Shan-Chen model results diverged for the higher density ratios, the selected density ratio was 40 for both models. It can be observed that the original Shan-Chen model was the fastest model, but it produced an unacceptable large spurious current, even for a low density ratio. The phase field model was the most precise model, though at the expense of a significantly higher computational cost.  
To compare the magnitude of the spurious current among all models, the velocity profile at t = 0.0013 s was plotted along the centerline of the domain in both the \(x\) and \(y\)-direction as shown in Figure 11.  
The velocity component was formed by the physical velocity and the spurious current. The reason for the existance of any physical velocity was related to the pressure difference throughout the domain, which was modeled in Shan-Chen models by the EOS and through Equation (38) in the phase field method. From the figure, it is observed that the magnitude of the spurious current was not constant in the whole section. The maximum spurious current took place near the bubble/fluid interface in all the models i.e., in the bottom of the domain in Figure 11a,c. As expected, the phase field method generated the least spurious current, in the order of \(5\times 10^{-5}\) m/s.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Maximum Spurious Velocity (mm/s)** & **Density Ratio** & **CPU-Time (s)** \\ \hline Original Shan-Chen model & 5 & 4 & 1527 \\ The enhanced model & 3.5 & 40 & 2351 \\ The phase field model & 0.03 & 40 & 35,861 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison between different parameters among different LB models.  
Figure 10: Results of the phase field model at time (**a**) 0.0013, (**b**) 0.0026, (**c**) 0.00387, and (**d**) 0.0052 s for a density ratio of 40. (Yellow: solid, red: liquid, blue: gas).