## 4 The Original Shan-Chen Model for Multiphase Flow  
The original Shan-Chen model [34] introduces a cohesion force term, \(\mathrm{F}(\mathbf{x},\mathbf{t})\), responsible for phase separation. This cohesion force satisfies the non-ideal equation of state. Based on the pressure difference, the phase separation between different phases takes place:  
\[\mathrm{F}(\mathbf{x},\mathbf{t})=-\mathrm{G}_{1}\psi(\mathbf{x},\mathbf{t}) \sum_{\alpha}\omega_{\alpha}\psi(\mathbf{x}+\mathbf{e}_{\alpha}\Delta\mathbf{ t},\mathbf{t})\mathbf{e}_{\alpha} \tag{10}\]  
The parameter, \(\psi\), is called the effective mass and is related to the equation of state. Here, \(\mathrm{G}_{1}\) controls the force exerted at each node by surrounding particles, with a positive (negative) value leading to a repulsive (attractive) force between particles. In this model, the phase separation occurs when \(\mathrm{G}_{1}\) is higher than a critical value. Using this scheme, only eight neighbor nodes are considered for calculating the cohesion force in the D\({}_{2}\)Q\({}_{9}\) lattice.  
[PERSON] and [PERSON] [34] proposed the following equation for the effective mass:  
\[\psi(\rho)=\psi_{0}\bigg{(}-\exp\bigg{(}\frac{-\rho_{0}}{\rho}\bigg{)}\bigg{)} \tag{11}\]  
Other researchers [7] have suggested different equations for effective mass as: \(\psi(\rho)\!=\!\rho_{0}(1-\exp\bigg{(}\frac{-\rho_{0}}{\rho}\bigg{)}\bigg{)}\), or even as \(\psi(\rho)=\rho_{0}\). In this article, Equation (11) was used for calculating effective mass, where \(\psi_{0}\) and \(\rho_{0}\) are constant. The pressure term in this model has an extra term compared to the ideal gas equation of state. The pressure term is defined as:  
\[\mathrm{p}=\mathrm{c}_{\mathrm{s}}^{2}\rho+\frac{\mathrm{c}_{\mathrm{s}}^{2} \mathrm{G}_{1}}{2}\psi^{2}(\rho) \tag{12}\]  
To model the contact angle, the scheme developed by [PERSON] et al. [55] was employed. They suggested that adhesion force can be implemented in a similar way to cohesion force as follows:  
\[\mathrm{F}(\mathbf{x},\mathbf{t})=-\mathrm{G}_{1}\psi(\mathbf{x},\mathbf{t}) \sum_{\alpha}\omega_{\alpha}\psi(\rho_{\mathrm{w}})\mathbf{s}(\mathbf{x}+ \mathbf{e}_{\alpha}\Delta\mathbf{t},\mathbf{t})\mathbf{e}_{\alpha} \tag{13}\]  
Here, \(\mathbf{s}(\mathbf{x}+\mathbf{e}_{\alpha}\Delta\mathbf{t},\mathbf{t})\) is an indication parameter and has the value of 0 or 1 for fluid or solid nodes, respectively. \(\Psi(\rho_{\mathrm{w}})\) is the effective mass at the wall; by changing \(\psi(\rho_{\mathrm{w}})\), different contact angles can be achieved.  
Depending on the node type (solid or liquid), Equation (10) or (13) was used to simulate the interaction between nodes.  
The fluid velocity is calculated as:  
\[\rho\mathbf{u}_{\alpha}=\sum_{\alpha}\mathrm{f}_{\mathrm{f}}\mathrm{e}_{ \mathrm{i}\alpha} \tag{14}\]  
As mentioned in the introduction, the original Shan-Chen model generates a high spurious current that can mask other important phenomena. Another issue with this model is the unrealistic rate of coalescence between bubbles (drops) that ultimately results in a single bubble (drop) remaining in the whole domain. These drawbacks motivated us to look for a more robust Shan-Chen model for dendritic solidification simulations.