# Comprehensive Investigations on Fluid Flow and Cavitation Characteristics in Rotating Disk System
## 2 Numerical Analysis
### Reynolds Equation and Its Dimensionlessification  
#### 2.2.1 Reynolds Equation  
The existing literature contains a large amount of cavitation inception data with water as the medium, our research is to analyze the shear cavitation phenomenon of the flow field between a pair of rotating friction disks, and the simplified model is shown in Figure 2.  
Because the working medium is chosen to be a viscous hydraulic fluid, the calculation is simplified assuming that the fluid is a Newtonian fluid and the viscosity is constant. The following assumptions are made:  
1. No relative slippage on the contact surface of fluid oil and friction disk;
2. The disks are always parallel to each other without bias grinding, the oil film thickness is always greater than 0, and there is no convex peak contact;
3. Flow field flow is laminar flow;
4. The effect of volume force and inertia terms are neglected and only the effect of centrifugal terms is considered;
5. The axial movement of oil is neglected.  
Based on the above assumptions the Reynolds equation for the flow of oil between frictional pair can be obtained [38; 39]:  
\[\frac{\partial}{\partial r}\left(rh^{3}\frac{\partial p}{\partial r}\right)+ \frac{1}{r}\frac{\partial}{\partial\theta}\left(h^{3}\frac{\partial p}{ \partial\theta}\right)=6\mu\omega r\frac{\partial(h)}{\partial\theta}+\frac{3 }{10}\rho_{L}\omega^{2}\frac{\partial\left(r^{2}h^{3}\right)}{\partial r} \tag{6}\]  
Figure 2: Simplified model for friction pair (\(\omega_{1}\) is rotation speed of friction plate; \(\omega_{2}\) is rotation speed of separator plate; \(r_{1}\) is inner radius; \(r_{2}\) is outer radius; \(h\) is oil film thickness; \(q\) and nearby arrows are directions of oil flow).  
where \(p\) is pressure, Pa; \(\mu\) is oil dynamic viscosity, Pa\(\cdot\)s; \(h\) is oil film thickness, m; \(\omega\) is angular velocity, rad\(/\)s. To make Equation (6) dimensionless, introduce dimensionless parameters as described in Equation (7):  
\[P=\frac{p}{p_{0}},\ \Phi=\frac{\theta}{2\pi},\ H=\frac{h}{h_{0}},\ R=\frac{r}{r_{ 2}},\ K_{1}=\frac{3\mu\omega r_{2}^{2}}{\pi p_{0}h_{0}^{2}},\ K_{2}=\frac{\pi \rho_{L}\omega h_{0}^{2}}{10\mu} \tag{7}\]  
The Reynolds equation was made dimensionless by combining Equation (6), as:  
\[\frac{\partial}{\partial R}\bigg{(}RH^{3}\frac{\partial P}{\partial R}\bigg{)} +\frac{1}{4\pi^{2}R}\frac{\partial}{\partial\Phi}\bigg{(}H^{3}\frac{\partial P }{\partial\Phi}\bigg{)}=K_{1}\bigg{[}R\frac{\partial(H)}{\partial\Phi}+K_{2} \frac{\partial(R^{2}H^{3})}{\partial R}\bigg{]} \tag{8}\]  
Given the periodicity in oil film, the sector-shaped oil film with angle \(\alpha\) is selected for the solution, and the dimensionless boundary condition of Equation (8) is:  
\[\left\{\begin{array}{c}P\big{|}_{R=\tau_{1}/r_{2}}=1,\ P|_{R=1}=0\\ P|_{\Phi=0}=P|_{\Phi=\alpha/2\pi},\ \frac{dP}{d\Phi}\big{|}_{\Phi=0}=\frac{dP}{d \Phi}|_{\Phi=\alpha/2\pi}\end{array}\right. \tag{9}\]  
The friction disk rotates fast, and the velocity in the flow field can be viewed as the synthesis of radial and circumferential velocities, and the dimensionless velocity components are as follows [40]:  
\[\left\{\begin{array}{c}u^{*}=u/\frac{p_{0}h_{0}^{2}}{8\pi r_{0}}=-H^{2} \Big{(}\frac{\partial P}{\partial R}-K_{1}K_{2}R\Big{)}\\ v^{*}=v/\frac{p_{0}h_{0}^{2}}{10\pi r\mu_{0}}=-\frac{H^{2}}{R}\frac{\partial P }{\partial\Phi}+4\pi^{2}K_{1}R\end{array}\right. \tag{10}\]