## 4 The Simulation and Analysis of CBN Grain Movement Behavior
### The Coupled Numerical Modeling of Multiphysical Fields
#### 4.1.2 The Governing Equation  
According to the COMSOL software documentation [35], the governing equations of the temperature field analysis, the flow field analysis, and the ultrasonic field analysis can be seen as follows.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Temperature (K)** & **293** & **373** & **473** & **573** & **673** & **773** & **873** & **1073** & **1173** & **1273** \\ \hline Specific heat capacity (J/kgK) & 472 & 480 & 498 & 560 & 586 & 615 & 700 & 806 & 637 & 602 \\ Heat conductivity coefficient (W/mK) & 47.68 & 43.53 & 40.44 & 38.13 & 36.02 & 34.16 & 21.98 & 36.49 & 25.92 & 24.02 \\ Density (kg/m\({}^{3}\)) & 7840 & 7816 & 7785 & 7752 & 7718 & 7683 & 7646 & 7569 & 7529 & 7508 \\ Solidus (K) & & & & & & 1495 & & & & \\ Liquidus (K) & & & & & & 1495 & & & & \\ \hline \end{tabular}
\end{table}
Table 6: The parameter of 45 steel.  
Figure 6: The thermophysical parameters of Cu-Sn19-Ti10 alloy: (**a**) specific heat capacity; (**b**) heat conductivity coefficient; (**c**) density; (**d**) The solid and liquid phase transformation.  
The temperature field  
The heat transfer law determines the temperature variation in laser cladding melt pool. The governing equation of the heat transfer in the fluids interface can be expressed by Equations (2) and (3).  
\[\rho C_{p}\bigg{(}\frac{\partial T}{\partial t}+\mathbf{u}\cdot\
abla T\bigg{)}+ \
abla\cdot(\mathbf{q}+\mathbf{q}_{r})=\alpha_{p}T\bigg{(}\frac{\partial p}{ \partial t}+\mathbf{u}\cdot\
abla p\bigg{)}+\tau:\
abla\mathbf{u}+Q \tag{2}\]  
\[\alpha_{p}=-\frac{1}{\rho}\frac{\partial\rho}{\partial T} \tag{3}\]  
where \(\rho\) denotes the density, \(C_{p}\) denotes the specific heat capacity at constant pressure, \(T\) denotes the absolute temperature, \(\mathbf{u}\) denotes the velocity vector, \(\mathbf{q}\) denotes the heat flux by conduction, \(\mathbf{q}_{r}\) denotes the heat flux by radiation, \(\alpha_{p}\) denotes the coefficient of thermal expansion, \(p\) denotes the pressure, \(\tau\) denotes the viscous stress tensor, and \(Q\) contains heat sources other than viscous dissipation.  
The flow field  
From the modeling assumptions, the metal fluid flow in the melt pool during the laser cladding process is considered as an incompressible laminar flow, so the governing equation can be represented as shown in Equation (4).  
\[\rho\frac{\partial\mathbf{u}}{\partial t}+\rho(\mathbf{u}\cdot\
abla)\mathbf{ u}=\
abla\cdot\Big{[}-p\mathbf{I}+\mu\Big{(}\
abla\mathbf{u}+(\
abla\mathbf{u})^{T} \Big{)}\Big{]}+\mathbf{F} \tag{4}\]  
where \(\rho\) denotes the density, \(\mathbf{u}\) denotes the velocity vector, \(p\) denotes pressure, \(\mathbf{I}\) denotes the identity matrix, \(\mu\) denotes the dynamic viscosity, and \(\mathbf{F}\) denotes the volume force vector.  
The ultrasonic field  
The governing equation for the acoustic analysis in the melt pool can be expressed by Equation (5):  
\[\begin{array}{l}\
abla\cdot\Big{[}-\frac{1}{\rho}(\
abla p-\mathbf{q}_{d}) \Big{]}-\frac{\omega^{2}p}{\rho c^{2}}=Q_{m}\\ \
abla\cdot\Big{[}-\frac{1}{\rho}(\
abla p-\mathbf{q}_{d})\Big{]}-\frac{\omega ^{2}p}{\rho c^{2}}=Q_{m}\end{array} \tag{5}\]  
where \(\rho\) denotes the total density, \(p\) denotes the total pressure, \(\mathbf{q}_{d}\) denotes the dipole source, where \(\omega=2\pi f\) denotes the angular frequency (\(f\) denotes the frequency), \(c\) denotes the isentropic speed of sound, and \(Q_{m}\) denotes the monopole source.