## 2 Material and Methods
### VOF Model for Droplet Impact on Powder Bed
#### 2.2.1 Governing Equations of Simulation  
The impact and spreading of liquid droplets on the solid substrate is a transient and isothermal process involving droplet liquid phase and air phase, which means there is no  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Reference [30]** & **Reference [12]** & **This Study** \\ \hline Droplet diameter (mm) & 2.4 & 2.0 & 0.1 \\ Impact momentum (kg\(\cdot\)m/s) & 9.15 \(\times\) 10\({}^{-6}\) & 1.2 \(\times\) 10\({}^{-5}\) & 1.08 \(\times\) 10\({}^{-9}\) \\ Weber number & 47.56 & 13.60 & 8.58 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The comparison of the currently defined parameters droplet diameter, impact momentum, and Weber number with typical studies.  
mass or heat transfer between droplet and air. The continuity and momentum equations of the two-dimensional models can be represented as follows [48]:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overrightarrow{v}\right)=0 \tag{4}\]  
\[\rho\frac{d\overrightarrow{v}}{dt}=-\
abla p+\mu\Delta\overrightarrow{v}+ \frac{\mu}{3}\
abla\Big{(}\
abla\cdot\overrightarrow{v}\Big{)}+\rho \overrightarrow{F} \tag{5}\]  
where \(t\) is the time, \(\rho\) is the density, \(p\) is pressure, \(\mu\) is dynamic viscosity, \(\overrightarrow{v}\) is the velocity vector, \(\overrightarrow{F}\) is the body force.  
The interface of droplet and air was tracked and calculated by the volume fraction of the fluid \(\alpha_{i}\):  
\[\alpha_{i}(x,t)=\left\{\begin{array}{ll}0&\quad x\in\text{ {the primary phase}}\\ a&\quad 0<a<1,\quad\quad x\in\text{{the mixed phase}}\\ 1&\quad x\in\text{{the secondary phase}}\end{array}\right. \tag{6}\]  
And the conservation equation for the VOF function is given by,  
\[\frac{\partial\alpha_{i}}{\partial t}+\
abla\cdot\left(\overrightarrow{v} \alpha_{i}\right)=0 \tag{7}\]  
where \(\overrightarrow{v}\) is the velocity, \(\alpha_{i}\) is the volume fraction of the \(i\) phase.  
The surface tension is a force that acts only on the surface, in which case balance needs to be maintained. It serves to balance the radially inward intermolecular attraction and the radially outward pressure gradient [37]. The continuous surface force (CSF) model proposed by [PERSON] et al. [49] treats the surface tension into a continuous three-dimensional effect rather than a boundary condition of the interface. The surface tension effect is simulated by adding a source term \(\overrightarrow{F}\) to the momentum Equation (5)  
\[\overrightarrow{F}=\sigma_{ij}\frac{\rho k_{i}\
abla\alpha_{i}}{\frac{1}{2} \big{(}\rho_{i}+\rho_{j}\big{)}} \tag{8}\]  
where \(\sigma_{ij}\) is the surface tension, \(\rho\) is the density of the mixing unit in the two-phase flow, \(\alpha_{i}\) is the volume fraction of the \(i\) phase in the unit, \(\alpha_{j}\) is the volume ratio of the \(j\) phase in the unit, \(\rho_{i}\) is the density of the \(i\) phase, \(\rho_{j}\) is the \(j\) phase density, \(k_{i}\) is the surface curvature of the \(i\) phase. For the air-liquid flow model,  
\[\rho=\alpha_{1}\rho_{1}+\alpha_{2}\rho_{2} \tag{9}\]  
According to the continuum surface force (CSF) model [49], \(k_{i}\) can be expressed as,  
\[k_{i}=\
abla\cdot\hat{n} \tag{10}\]  
\[\hat{n}=\frac{\overrightarrow{n}}{\left|\overrightarrow{n}\right|} \tag{11}\]  
\[\overrightarrow{n}=\
abla a_{i} \tag{12}\]  
where \(\overrightarrow{n}\) and \(\hat{n}\) are the normal vector and unit normal vectors of the liquid-gas interface.  
In addition, when the interface is near a solid wall with a specified contact angle \(\theta_{w}\), \(\hat{n}\) should be modified as follows:  
\[\hat{n}=\hat{n}_{w}\cos\theta_{w}+\hat{t}_{w}\sin\theta_{w} \tag{13}\]  
where \(\hat{n}_{w}\) is the unit normal vector and \(\hat{t}_{w}\) is the unit tangential vector of the wall.  
Obviously, \(k_{i}\) should be amended in connection with \(\theta_{w}\) near the wall.  
Thereby, the moving of liquid-gas two-phase flow interface can be well tracked, and mass is conserved by solving the above equations.