## 2 Mathematical Model  
Figure 2 shows the calculation domain of the unified PAW model, including welding nozzle, gas region and workpiece. Argon enters the nozzle through the inlet and ionizes to be thermal plasma due to the electric voltage between the two electrodes. There is a shielding gas around the welding nozzle, where the argon gas is also used. The plasma arc impacts on a workpiece, and it soon fuses  
part of the workpiece and produces a weld pool. At the same time, the arc causes a huge stagnation pressure on the workpiece and flows out through the upper outlet boundary. To calculate a mobile process, the workpiece is set with a welding speed while the welding torch maintains a still stance. Some assumptions are made to construct the governing equations: the plasma arc is in local thermo-dynamic equilibrium (LTE) and optically thin to radiation so that arc properties can be regarded as a function of temperature; both arc and liquid metal are in laminar flow; thermal properties of the solidus and liquidus metal can be regarded as a function of temperature; metal vapor is ignored as it has little effect on weld pool; the calculation area is symmetrical and half of the welding area is taken for calculation. Mass conservation equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)+\frac{ \partial}{\partial y}(\rho v)+\frac{\partial}{\partial z}(\rho w)=0 \tag{1}\]  
Momentum conservation equations:  
\[ \begin{array}{l}\frac{\partial(\rho u)}{\partial t}+u\frac{\partial}{ \partial x}(\rho u)+v\frac{\partial}{\partial y}(\rho u)+w\frac{\partial}{ \partial z}(\rho u)=\\ -\frac{\partial p}{\partial x}+\mu\Big{(}\frac{\partial^{2}u}{\partial x}+ \frac{\partial^{2}u}{\partial y}+\frac{\partial^{2}u}{\partial z}\Big{)}+j_{y }B_{z}-j_{z}B_{y}+S_{x}\\ \\ \frac{\partial(\rho v)}{\partial t}+u\frac{\partial}{\partial x}(\rho v)+v \frac{\partial}{\partial y}(\rho v)+w\frac{\partial}{\partial z}(\rho v)=\\ -\frac{\partial p}{\partial y}+\mu\Big{(}\frac{\partial^{2}v}{\partial x}+ \frac{\partial^{2}v}{\partial y}+\frac{\partial^{2}v}{\partial z}\Big{)}+j_{ z}B_{x}-j_{x}B_{z}+S_{y}\\ \\ \frac{\partial(\rho w)}{\partial t}+u\frac{\partial}{\partial x}(\rho w)+v \frac{\partial}{\partial y}(\rho w)+w\frac{\partial}{\partial z}(\rho w)=\\ -\frac{\partial p}{\partial z}+\mu\Big{(}\frac{\partial^{2}w}{\partial x}+ \frac{\partial^{2}w}{\partial y}+\frac{\partial^{2}w}{\partial z}\Big{)}+j_{ x}B_{y}-j_{y}B_{x}-\rho g+S_{z}\\ \end{array}  \tag{2}\]  
Energy conservation equation:  
\[ \begin{array}{l}\frac{\partial(\rho H)}{\partial t}+u\frac{\partial(\rho H)} {\partial x}+v\frac{\partial(\rho H)}{\partial y}+w\frac{\partial(\rho H)}{ \partial z}\\ =\frac{\partial}{\partial x}\Big{(}\frac{k}{\rho}\frac{dH}{\partial x}\Big{)}+ \frac{\partial}{\partial y}\Big{(}\frac{k}{\rho}\frac{dH}{\partial y}\Big{)}+ \frac{\partial}{\partial z}\Big{(}\frac{k}{\rho}\frac{dH}{\partial z}\Big{)} +S_{e}\\ \end{array}  \tag{3}\]  
Current conservation equation:  
\[\
abla^{2}\phi=0 \tag{4}\]  
Figure 2: Calculation domain of the unified plasma arc welding (PAW) model. [PERSON]'s law:  
\[\mathbf{J}=-\sigma\
abla\phi \tag{7}\]  
Magnetic potential vector equations:  
\[\
abla^{2}\mathbf{A}=-\mu_{0}\mathbf{J} \tag{8}\]  
\[\mathbf{B}=\
abla\times\mathbf{A} \tag{9}\]  
where \(t\) is time; \(\rho\) is density; \(P\) is pressure; \(u\), \(v\), \(w\), \(j_{x}\), \(j_{y}\), \(j_{z}\), and \(B_{x}\), \(B_{y}\), \(B_{z}\) are velocity, current density, and magnetic induction intensity in \(x\), \(y\), \(z\) direction, respectively; \(\mathbf{A}\) is magnetic potential vector; \(\mu\) is dynamic viscosity; \(T\) is temperature; \(k\) is thermal conductivity; \(\sigma\) is electrical conductivity; \(c_{p}\) is specific heat capacity; \(H\) is total enthalpy; \(S_{x}\), \(S_{y}\), \(S_{z}\) are source term of momentum equations; \(S_{e}\) is source term of energy equation; \(g\) is gravity acceleration; \(\phi\) is electric potential; \(\mu_{0}\) is permeability of vacuum.