## 7 Results
### Column Dendritic Growth with Original Shan-Chen Model  
The growth of five columnar dendrites in a rectangular domain was simulated using the original Shan-Chen model. The simulation was performed for Al-3 wt% Cu binary alloy with initial undercooling of \(\Delta\mathrm{T}=2.0\) K. All the external boundary conditions were considered as stationary walls for the velocity field and insulated for the temperature field except for the left wall where a temperature gradient of 1200 K.m\({}^{-1}\) was imposed. The boundary condition for the concentration field was assumed as insulated in all directions. The preferential crystallographic orientation was 0\({}^{\circ}\) with respect to the \(x\)-axis (horizontal). The domain size was 91.2 \(\upmu\)m \(\times\) 72 \(\upmu\)m, discretized with 304 \(\times\) 240 cells for solving fluid flow and solute transport equations, and 38 \(\times\) 30 cells for solving the energy equation. Five columnar dendrites and 10 bubbles with a radius between 4 \(\upmu\)m to 6 \(\upmu\)m were initially placed in the domain (10). The contact angle was selected as 150\({}^{\circ}\). Other parameters are shown in Table 3.  
Since the focus of this section is to compare the ability of the original and enhanced Shan-Chen models to simulate dendrite-bubble interaction, the gas component rejection into the liquid phase during solidification was not considered in these simulations. The initial condition is depicted in Figure 7.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Gibbs Thomson Coefficient (G)** & **0.0000024 m K** \\ \hline Domain size & **90 \(\upmu\)m\(\times\)90 \(\upmu\)m** \\ \(\mathbf{\Delta T_{\mathrm{undercouling}}}\) & **2 K** \\ Contact Angle & 150\({}^{\circ}\) \\ Kinematic Viscosity & 5.6566 \(\times\) 10\({}^{-8}\) m\({}^{2}\)/s \\ Solute Diffusivity & 3 \(\times\) 10\({}^{-9}\) m\({}^{2}\)/s \\ Thermal Diffusivity & 2.4210\({}^{-5}\) m\({}^{2}\)/s \\ Liquidus Slope (m\({}_{\mathrm{l}}\)) & \(-\)2.6 (K/wt\%) \\ Partitioning Coefficient & 0.17 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters used for dendritic growth with the original Shan-Chen Model [51; 52]  
Figure 7: Initial condition for simulation of dendrite-bubble interaction (Yellow: solid, red: liquid, blue: gas).  
The interaction between dendrites and bubbles is shown in Figure 8. During the simulation, some bubbles tended to dissolve in the fluid or merge to other bubbles. This process made some bubbles bigger while eliminating the rest. The reason for this phenomenon is that the pressure of the bulk fluid was greater than the corresponding saturated value for the smaller bubbles. Therefore, these bubbles condensed. On the other hand, the pressure of the bulk fluid was lower than the saturated value of larger bubbles, causing the larger bubbles to grow. This is related to the EOS and the fact that no repulsive force was present in the domain. Therefore, during the simulations, some bubbles became smaller or vanished, while others became larger.  
The rate of bubble merging obtained with the original Shan-Chen model is not realistic as evidenced by experimental results. Coalescence of bubbles occurred in the experiments, but in a different time scale compared to what is presented in Figure 8. Based on the simulation results, all the small bubbles in the middle of the domain coalesced after 0.005 s while in the experiment, this phenomenon happened in the order of 10 s [(60; 61)]. This model also generates a high spurious current, especially in the interface. The spurious current was in the order of 5 mm/s, which is very high and can completely mask the actual physical flow.