# Numerical Investigation of Air Entrapment Dynamics for High-Speed Thermal Spraying
## 2 Numerical Model  
Since the molten droplet is spherical, an axis-symmetric model was used [15], as shown in Figure 1. A molten metal droplet with radius \(R\) and initial temperature \(T_{\text{d0}}\) impacted on a cold smooth substrate (\(L\times H_{2}\), \(T_{\text{s0}}\)) with an initial velocity \(v\) and air as its surroundings (\(L\times H_{1}\), \(T_{0}=293.15\) K). The substrate surface was set as a wetted wall with a constant contact angle \(\theta\). Surface roughness was neglected here.  
Figure 1: Schematic of the computation domain.  
The fluid was treated as a Newtonian incompressible laminar flow and solved by the Navier-Stokes equation and the continuity equation, as shown in Equation (1),  
\[\rho\frac{\partial\mathbf{v}}{\partial t}+\rho\
abla\cdot(\mathbf{v}\mathbf{v})=-p+\mu\
abla^ {2}\mathbf{v}+\rho g+\mathbf{F}_{\rm vol}-S\mathbf{v},\ \ \rho\
abla\mathbf{v}=0 \tag{1}\]  
\(\rho\) and \(\mu\) are the density and dynamic viscosity of the fluid, \(\mathbf{g}\) is the gravitational acceleration, \(\mathbf{F}_{\rm vol}\) is the volumetric force and mainly refers to the surface tension force at the droplet-air interface. The source term \(S\mathbf{v}\) accounts for the Darcy effect of the mushy zone, which is treated as a porous medium and used to describe the mixed liquid-solid region during solidification (\(T_{\rm m}-\Delta T\leq T\leq T_{\rm m}+\Delta T\)) [22]. This is written as follows  
\[S\mathbf{v}=\frac{C(1-\alpha)^{2}}{\alpha^{3}+\varepsilon}\mathbf{v} \tag{2}\]  
\(C=10^{5}\)-\(10^{9}\) is the mushy zone constant, \(\varepsilon\) is a small number (0.001) to avoid having a zero denominator [23], \(\alpha\) is the liquid volume fraction and calculated based on the enthalpy-porosity method [24]. The liquid fraction is 0 for the solid phase (\(T<T_{\rm m}-\Delta T\)) and 1 (\(T>T_{\rm m}+\Delta T\)) for the liquid phase, as shown in Equation (3). When \(\alpha=0\), this means the material in this cell is completely solidified and the velocity should drop to zero, which is accomplished by applying an extreme large source term to the momentum equation Equation (1).  
\[\begin{array}{c}\alpha=0,\ \ \ (T<T_{m})\\ \alpha=\frac{T-(T_{m}-\Delta T)}{2\Delta T},\ \ \ \ (T_{m}-\Delta T<T<T_{m}+ \Delta T)\\ \alpha=1,\ \ \ \ (\ T>T_{m})\end{array} \tag{3}\]  
The molten droplet-air interface is tracked by the Level-set method [22], and written as  
\[\frac{\partial\phi}{\partial t}+\
abla\cdot(\mathbf{v}\phi)=\gamma\
abla\cdot \left[\varepsilon_{ls}\
abla\phi-\phi(1-\phi)\frac{\
abla\phi}{|\
abla\phi|}\right] \tag{4}\]  
\(\gamma\) is the reinitialization parameter, \(\varepsilon_{ls}\) is the parameter controlling interface thickness, \(\phi\) is the level-set function represented by a smeared Heaviside function here, which is 0 if filled with air, and 1 if filled with molten nickel.  
Energy conservation combined with the enthalpy-transforming model was used to describe the heat transfer, as shown in Equation (5),  
\[\rho C_{p}\frac{\partial T}{\partial t}+\rho C_{p}\mathbf{v}\cdot\
abla T+\
abla \cdot(k\
abla T)-Q=0 \tag{5}\]  
\(C_{p}\) is the heat capacity at constant pressure, \(k\) is the thermal conductivity, \(Q\) is the heat source. The total enthalpy in the droplet consists of sensible heat, which is due to the heat transfer either by heat conduction or convection, and latent heat, which is caused by phase change. Accordingly, a temperature-dependent heat capacity \(C_{p}\) is used for this description, as shown in Equation (6).  
\[C_{p}=\frac{1}{\rho}\left(\alpha\rho_{l}C_{pl}+(1-\alpha)\rho_{s}C_{ps}\right) +L\frac{\partial\alpha}{\partial T} \tag{6}\]  
\(L\) is the latent heat, and the subscripts \"\(s\)\" and \"\(l\)\" represent solid and liquid, respectively. The density and thermal conductivity all evaluated by the average of the solidus phase and the liquidus phase, i.e., \(\rho=\alpha\rho_{1}+(1-\alpha)\rho_{s}\), \(k=ak_{1}+(1-\alpha)\,k_{\rm s}\). Here, \(Q=0\), as there is no external heat source.