# Study on the Performance of Liquid Metal Lubricated V-Groove Bearing Considering Turbulence
## 3 Numerical Model and Simulation
### Equation Solving  
#### 3.2.1 The Discretization of Equations  
In general, the calculation in CCS can directly carry out differential processing of the partial derivative of the Reynolds equation. However, the equations calculated in OCS have cross partial derivatives, and the direct differential processing will make pressure values at non nodes. Therefore, the five-point difference method was not applicable here, and the local integration method was chosen to deal with this problem.  
When the bearing is meshed along the direction of the V-groove and its boundary, there will appear continuous and discontinuous areas of oil film. As an example, the process of the local integration method can be explained by taking the node falling at the continuous area. The integral area is shown in Figure 4, recorded as marked area A.  
Both sides of Equation (6) were conducted on a surface integral in the region shown below in Figure 5:  
\[\iint\limits_{\Delta}\left[\frac{\partial}{\partial\varphi}\left(\frac{H^{3}}{ k_{x}}\frac{\partial P}{\partial\varphi}\right)+\left(\frac{d}{b}\right)^{2} \frac{\partial}{\partial\lambda}\left(\frac{H^{3}}{k_{x}}\frac{\partial P}{ \partial\lambda}\right)\right]\text{d}\varphi\text{d}\lambda=\frac{1}{4}\iint \limits_{\Delta}\left(\frac{\partial H}{\partial\varphi}\right)\text{d} \varphi\text{d}\lambda \tag{9}\]  
Figure 4: Diagram of continuous oil film integration area A.  
Using the Green formula, the surface integral was changed into a curve integral as follows:  
\[\int\limits_{\mathrm{L}}\frac{H^{3}}{k_{\mathrm{x}}}\frac{\partial P}{\partial \varphi}\mathrm{d}\lambda-\left(\frac{d}{b}\right)^{2}\int\limits_{\mathrm{L}} \frac{H^{3}}{k_{\mathrm{z}}}\frac{\partial P}{\partial\lambda}\mathrm{d}\varphi= \frac{1}{4}\int\limits_{L}H\mathrm{d}\lambda \tag{10}\]  
Taking Equations (4) and (5) into Equation (10), the integration was carried out according to the curve direction shown in Figure 5, and the five-point difference formula in oblique coordinate was obtained:  
\[P_{i,j}=\frac{A_{i,j}P_{i+1,j}+B_{i,j}P_{i-1,j}+C_{i,j}P_{i,j+1}+D_{i,j}P_{i,j- 1}-F_{i,j}}{E_{i,j}} \tag{11}\]  
where  
\[A_{i,j}=\frac{\Delta\xi}{\Delta\eta}\sin\frac{\alpha}{2}\frac{H^{3}_{i+0.5,j} }{k_{\mathrm{x}}}-\left(\frac{d}{b}\right)^{2}\left(\frac{-H^{3}_{i,j-0.5}+H^{ 3}_{i,j+0.5}}{2k_{\mathrm{Z}}\tan\left(\frac{\alpha}{2}\right)}-\frac{\Delta \xi}{\Delta\eta}\frac{\cos\left(\frac{\alpha}{2}\right)}{\tan\left(\frac{ \alpha}{2}\right)}\frac{H^{3}_{i+0.5,j}}{k_{\mathrm{z}}}\right) \tag{12}\]  
\[B_{i,j}=\frac{\Delta\xi}{\Delta\eta}\sin\frac{\alpha}{2}\frac{H^{3}_{i-0.5,j} }{k_{\mathrm{x}}}-\left(\frac{d}{b}\right)^{2}\left(\frac{H^{3}_{i,j-0.5}+H^{ 3}_{i,j+0.5}}{2k_{\mathrm{Z}}\tan\left(\frac{\alpha}{2}\right)}-\frac{\Delta \xi}{\Delta\eta}\frac{\cos\left(\frac{\alpha}{2}\right)}{\tan\left(\frac{ \alpha}{2}\right)}\frac{H^{3}_{i-0.5,j}}{k_{\mathrm{z}}}\right) \tag{13}\]  
\[C_{i,j}=-\left(\frac{d}{b}\right)^{2}\left(-\frac{\Delta\eta}{\Delta\xi}\frac {1}{\sin\left(\frac{\alpha}{2}\right)}\frac{H^{3}_{i,j+0.5}}{k_{\mathrm{z}}}- \cos\frac{\alpha}{2}\frac{-H^{3}_{i+0.5,j}+H^{3}_{i-0.5,j}}{2k_{\mathrm{z}} \sin\left(\frac{\alpha}{2}\right)}\right) \tag{14}\]  
\[D_{i,j}=-\left(\frac{d}{b}\right)^{2}\left(-\frac{\Delta\eta}{\Delta\xi}\frac {1}{\sin\left(\frac{\alpha}{2}\right)}\frac{H^{3}_{i,j-0.5}}{k_{\mathrm{z}}} -\cos\frac{\alpha}{2}\frac{H^{3}_{i+0.5,j}-H^{3}_{i-0.5,j}}{2k_{\mathrm{z}} \sin\left(\frac{\alpha}{2}\right)}\right) \tag{15}\]  
\[E_{i,j}=A_{i,j}+B_{i,j}+C_{i,j}+D_{i,j} \tag{16}\]  
\[F_{i,j}=\frac{1}{4}\sin\left(\frac{\alpha}{2}\right)\Delta\xi\left(H_{i+0.5,j} -H_{i-0.5,j}\right) \tag{17}\]  
For the discontinuous areas of oil film (i.e., the nodes falling at the junction of ridge and groove), it is only necessary to locally integrate the left and right parts of Figure 4, respectively. For the dynamic Reynolds equation, the small perturbation method is directly used to deal with this as the discrete form is similar to that of the static equation.