# Influence of Cavitation and Shaft Deformation in the Analysis of Lubrication of the Stern Bearing
## 2 Governing Equations of Lubrication Model
### Mixed Lubrication Model Considering Surface Roughness  
The morphological characteristics of the lubricating surface (as shown in Figure 2) affect the status of the oil film, which must be considered in the analysis of hydrodynamic lubrication. According to the rough surface contact theory proposed by [PERSON] and [PERSON] [34], the film thickness ratio (\(H\)) is used as a measure of the lubrication state:  
\[H=h/\sigma \tag{12}\]  
where \(h\) represents the mean square value of the oil film thickness between the surfaces, and \(\sigma\) represents the surface profile height. When the film thickness ratio is less than 4, the bearing is in a mixed lubrication state. With an increase in the film thickness ratio, the contact micro-convex is gradually separated. When the film thickness ratio is greater  
Figure 1: The force diagram on the bearing and the force balance condition.  
than 4, it can be considered that the contact part is completely separated. The mean flow model proposed by [PERSON] and [PERSON]. [35] was introduced to describe the lubrication-governing equation of a shaft-bearing system considering lubrication surface topography characteristics, which can be expressed as:  
\[\frac{\partial}{\partial\theta}\left(\phi_{x}\rho l^{3}\frac{\partial P}{ \partial\theta}\right)+R^{2}\frac{\partial}{\partial z}\left(\phi_{z}\rho l^{3 }\frac{\partial P}{\partial z}\right)=6\mu\rho UR(\frac{\partial\bar{h}}{ \partial\theta}+\sigma\frac{\partial\phi_{s}}{\partial\theta}) \tag{13}\]  
where \(h\) is the nominal oil film thickness, which refers to the oil film thickness between two surfaces without considering the surface roughness; \(h_{T}\) refers to the actual oil film thickness between two surfaces when surface roughness is considered, as shown in Figure 2; and \(\bar{h}_{T}\) is the actual average oil film thickness:  
\[\bar{h}_{T}=E\{h_{T}\} \tag{14}\]  
where \(E\{\}\) represents the mathematical expectation, \(\mu\) is lubricating oil viscosity, \(U\) is the relative linear velocity of the shaft and bearing, \(\phi_{x}\) and \(\phi_{z}\) represent the pressure flow factors, and \(\phi_{s}\) is the shear flow factor. As shown in Figure 2, when the distribution characteristics of surface roughness are determined, \(\bar{h}_{T}\) is a function of \(h\). According to the derivative rule of complex functions, Equation (15) can be obtained:  
\[\frac{\partial\bar{h}_{T}}{\partial\theta}=\frac{\partial\bar{h}_{T}}{ \partial h}\frac{\partial h}{\partial\theta} \tag{15}\]  
The dimensionless parameter proposed by [PERSON] et al. [36] is adopted to define the contact factor (\(\phi_{c}\)):  
\[\phi_{c}=\frac{\partial\bar{h}_{T}}{\partial h} \tag{16}\]  
The asperity heights are assumed to vary randomly with a certain probability distribution. The density function of distribution of roughness is \(\varphi(s)\); then:  
\[\phi_{c}=\int_{-H}^{+\infty}\varphi(s)\text{d}s \tag{17}\]  
Assume that the contact factors are distributed according to the following Gaussian equation:  
\[\varphi(s)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{s^{2}}{2}\right) \tag{18}\]  
Substitute Equation (18) into Equation (17):  
\[\begin{array}{l}\phi_{c}=\int_{-H}^{\infty}\frac{1}{\sqrt{2\pi}}\exp\left(- \frac{s^{2}}{\Sigma}\right)\text{d}s\\ =\int_{-H}^{0}\left[\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{s^{2}}{\Sigma} \right)\right]\text{d}s+\int_{0}^{\infty}\left[\frac{1}{\sqrt{2\pi}}\exp\left( -\frac{s^{2}}{\Sigma}\right)\right]\text{d}s\\ =\frac{1}{2}\Big{[}1+Er\left(f\left(\frac{H}{\sqrt{2}}\right)\right)\Big{]} \text{d}s\end{array} \tag{19}\]  
Figure 2: Thickness of lubricating oil film considering the surface roughness.  
The error function is  
\[Er(f(x))=\int_{\xi}^{x}\frac{2}{\sqrt{\pi}}\mathrm{d}\xi \tag{20}\]  
Equation (19) should not be used directly and can be replaced by the following fitting formula, with the maximum error not exceeding 0.5%.  
\[\phi_{c}=\left\{\begin{array}{c}\exp\left(\begin{array}{c}-0.6912+0.782H\\ -0.304H^{2}+0.0401H^{3}\\ 1\end{array}\right)0\leq H<3\\ H\geq 3\end{array}\right. \tag{21}\]