## 2 Analysis of Smooth-Walled Radial Bearing Ignoring Cavitation Effects  
### Governing Equations  
In the process of high-speed rotation of hydrostatic bearings, the oil film of journal bearings will form negative pressure under the action of oil wedge. This cavitation phenomenon has been considered by a large number of scholars, and the most common is the half-Sommerfeld method. However, it is more practical to explain this phenomenon through gas-liquid two-phase flow [13; 15]. Therefore, the conservation of momentum and energy are formulated as follows:  
Mass conservation equation:  
\[\
abla\cdot\vec{V}=0 \tag{1}\]  
Momentum equation:  
\[\rho\frac{D\vec{V}}{Dt}=-\
abla p+\rho\vec{g}+\mu\
abla^{2}\vec{V} \tag{2}\]  
We can see that the momentum equation is related to the density, viscosity and the volume velocity of the micro element, extending Equation (2):  
\[\frac{\partial}{\partial x_{j}}\big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{(}\mu\Bigg{(}\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)} \Bigg{)}+\frac{\partial}{\partial x_{j}}\Big{(}-\rho\overline{u_{i}^{\prime} u_{j}^{\prime}}\Big{)} \tag{3}\]  
Energy equation:  
\[\frac{\partial}{\partial t}(\rho E)+\
abla\cdot[\vec{v}(\rho E+p)]=\
abla \cdot\Big{[}k_{eff}\
abla T\Big{]}+S_{h} \tag{4}\]  
The two-dimensional schematic diagram of a radial hydrostatic bearing, shown in Figure 1.  
The main parameters of oil are: density 810 kg/m\({}^{3}\); specific heat capacity \(2000\) J/(kg \(\cdot\) K); thermal conductivity \(0.37\) W/(m \(\cdot\) K); dynamic viscosity of lubricating oil is \(3.85\times 10^{-3}\) Pa \(\cdot\) s (temperature \(T\) = 293 K). Analysis parameters are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Journal** & **Bearing** & **Radius** & & & & **Rotating** \\
**Diameter** & **Length** & **Clearance** & **Eccentricity** & **Lubricant** & **Speed** \\ \hline
20 mm & 20 mm & 0.02 mm & 0.7 &
\begin{tabular}{c} **Mobil Velocite oil** \\ **No. 2** \\ \end{tabular} & 5000 r/min \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation parameters for radial hydrostatic bearing. Figure 1: Schematic diagram of a radial hydrostatic bearing. According to the Reynolds number calculation formula:  
\[Re=\frac{\rho vd}{\mu} \tag{5}\]  
where \(d\) is length scale, and \(v\) is reference velocity. Based on the maximum oil film thickness, let \(d=2h_{\text{max}}\), the Reynolds number calculated as about 75, which is much lower than the range of 1000 to 2000, which can generate turbulent flow. Therefore, this problem can be calculated according to the laminar flow theory, and the obtained pressure distribution in the oil film is shown in Figure 2.  
Based on the maximum oil film thickness, the Reynolds number is calculated as about 75, which is much lower than the range of 1000 to 2000, which can generate a turbulent flow. Therefore, this problem can be calculated according to the laminar flow theory, and the obtained pressure distribution in the oil film is shown in Figure 2.  
In order to verify mesh independence, three different proportions of mesh number are compared with the FEM method. As shown in Table 2.  
It can be seen that without considering the cavitation effect, the minimum pressure may be negative pressure, which is obviously inconsistent with the actual situation. Therefore, the cavitation effect must be considered to obtain reasonable results. At the same time, the number of the mesh has little influence on the calculation results, which proves the independence of the mesh. The process of generating, compressing, expanding and collapsing of tiny bubbles in a liquid is called cavitation. Several scholars have made outstanding contributions to the mathematical theory of cavitation effects.