## 2 Mathematical Model
In arc region, there are only electromagnetic force and gravity, so the momentum source terms in Equations (2)-(4) are all zero. In consideration of Joule heat, electron transfer energy and radiative heat loss in the arc, the energy source term is expressed as  
\[S_{e}=\frac{{j_{x}}^{2}+{j_{y}}^{2}+{j_{z}}^{2}}{\sigma}+\frac{5k_{b}}{2e}(j_ {x}\frac{\partial T}{\partial x}+j_{y}\frac{\partial T}{\partial y}+j_{z} \frac{\partial T}{\partial z})-S_{r} \tag{10}\]  
where \(k_{b}\) is the Boltzmann constant; \(e\) is electronic charge; \(S_{r}\) is radiant heat loss. In the workpiece region, the Enthalpy-porous technique [31] is applied to deal with the melting and solidification problem. It is considered that there is a solidus line and a liquidus line when the workpiece is fused. The zone between the two-phase line is considered full of porous medium composed of both liquid phase and solidus phase. The liquidus volume fraction is expressed as  
\[f_{l}=\left\{ \begin{array}{ll}0&T\leq T_{s}\\ (T-T_{s})/\left(T_{l}-T_{s}\right)&T_{s}<T<T_{l}\\ 1&T\geq T_{l}\end{array} \right. \tag{11}\]  
where \(T_{s}\) is solidus temperature; \(T_{l}\) is liquidus temperature. Darcy damping force takes effect in the porous zone, and a buoyancy force also appears due to temperature differences in the whole weld pool, so the momentum source terms are expressed as  
\[S_{x}=-\frac{\mu_{l}}{K}u,\quad S_{y}=-\frac{\mu_{l}}{K}v,\quad S_{z}=-\frac{ \mu_{l}}{K}w+\rho g\beta(T-T_{m}). \tag{12}\]  
\[K=\frac{f_{l}^{3}}{A_{0}(1-f_{l})^{2}} \tag{13}\]  
where \(\mu_{l}\) is dynamic viscosity of liquid mental; \(\beta\) is thermal expansion coefficient; \(T_{m}\) is melting temperature; \(K\) is permeability; \(A_{0}\) is a morphology coefficient related to the porous zone. Based on the Enthalpy-porous technique, total enthalpy \(H\) is expressed as  
\[H=h+f_{l}L=\int_{T_{0}}^{T}c_{p}dT+f_{l}L \tag{14}\]  
where \(h\) is sensible enthalpy; \(L\) is latent heat. Special treatment is needed at the interface of plasma arc and workpiece because a thin non-LTE sheath appears there, and the physical mechanism is too complex and still unknown. The \"LTE-diffusion approximation\" method [32] is generally applied to deal with the heat transfer problem from arc to workpiece. It neglects the real sheath but sets mesh size at the interface approximate to the diffusion length of electrons. In this way, a heat source term, which includes heat conduction, electron condensation and radiative heat loss, is simply added to the interface. As heat conduction is calculated through the coupled interface, then the other terms should be included in the heat source. \[S=\left|\dot{j}\right|\varphi_{w}-\varepsilon\alpha T^{4} \tag{15}\]  
where \(j\) is current density; \(\varphi_{w}\) is work function of workpiece; \(\varepsilon\) is emissivity; \(\alpha\) is the Stefan-Boltzmann constant. Figure 3 shows all the driving forces in weld pool. Plasma arc pressure \(p_{arc}\) impacts vertically on the top surface, and it exhibits a normal distribution with a maximum pressure \(P_{\text{max}}\) at the center and a sharp reduce within a radius \(r_{0}\). The pressure can be obtained according to the governing equations in the arc region. \[p_{arc}=P_{\text{max}}\exp(-3r^{2}/r_{0}^{2}) \tag{16}\]  
When a keyhole appears, the arc fills up the whole keyhole. It is found that the front keyhole wall almost coincides with the solid-liquid interface when the plasma arc is penetrating through the workpiece. Therefore, as a simplification of arc force, all the melted zone is assumed full of arc pressure, and it gradually grows in the whole process. Plasma arc drag force \(\tau_{\text{drag}}\), Marangoni shear \(\tau_{\text{M}}\), electromagnetic force \(F_{\text{m}}\) and buoyancy force \(F_{\text{b}}\) are all taken into account in the model. As shown in Figure 3, plasma arc drag force and Marangoni shear both drive liquid metal outwards, while the electromagnetic force pushes the liquid metal inwards, and the buoyancy force pushes the liquid metal upwards.