## 2 Mathematical Modeling
### Lubrication Model
#### 2.3.2 Pressure Distribution in the Liquid Film  
To calculate the pressure distribution, a coordinate system Kntz is established (See Figure 7). K is the meshing point on the screw groove flank. Axis \(n\) is vertical to the screw groove flank. Axis \(\tau\) is along the screw groove flank. Axis \(z\) is along the tooth length direction. Kntz is a moving system since the coordinate origin K and the directions of axes \(n\), \(\tau\), z vary along the meshing process. The velocity \(U\) of the liquid flowing through the gap is composed of 3 components \(u\), \(v\), \(w\). Velocities \(u\), \(v\), \(w\) are along the directions of \(n\), \(\tau\), \(z\) respectively. Usually, the Reynolds equation is used to solve the hydrodynamic lubrication problems. The inertia term is ignored during the derivation process of this equation. However, the inertia term cannot be always neglected, even in laminar flow [39]. Especially for the water flooded SSC, the inertia term must be considered since the water viscosity is very small. Therefore, the Reynolds equation is not applicable for this problem. The liquid film pressure model is deduced based on the N-S equations for Newtonian fluid:  
\[\rho\frac{D\overset{\rightarrow}{U}}{Dt}=-\
abla\mathbf{p}+\
abla\left( \lambda V\overset{\rightarrow}{U}\right)+\
abla\left(2\mu\overset{\rightarrow }{S}\right)+\rho\overset{\rightarrow}{f} \tag{8}\]  
Figure 8: The hydrophilicity of the materials: (**a**) the screw surface (**b**) the gate-rotor surface. where \(\rho\) is the liquid density; \(t\) is the time; \(p\) is the liquid film pressure; \(\mu\) is the dynamic viscosity; \(\lambda\) is the second viscosity coefficient; \(S\) is the stress tensor; \(f\) is the body force. To obtain the governing equation, several assumptions are made and Equation (8) is simplified accordingly [20]:  
1) The flow is viewed steady state under a certain gate-rotor rotating angle \(\varphi_{\text{gr}}\). Hence, the item \(\partial\)/\(\partial\)t is neglected. Since water is incompressible, divergence of velocity equals 0. Because liquid injection amount is very large and the heat transfer time is very short, the temperature rise of liquid is small (in a 6 m\({}^{3}\)-min\({}^{-1}\) prototype, the tested temperature rise often does not exceed 10K) and coefficients \(\lambda\) and \(\mu\) are viewed as constants. Body force \(f\), such as gravity, is ignored. Then, Equation (8) is simplified to the following equation:  
\[\left\{ \begin{array}{l}\rho\bigg{(}u\frac{\partial u}{\partial n}+v\frac{ \partial u}{\partial z}+w\frac{\partial u}{\partial z}\bigg{)}=-\frac{\partial p }{\partial n}+\mu\bigg{(}\frac{\partial^{2}u}{\partial n^{2}}+\frac{\partial ^{2}u}{\partial z^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\bigg{)}\\ \rho\bigg{(}u\frac{\partial v}{\partial n}+v\frac{\partial v}{\partial z}+w \frac{\partial v}{\partial z}\bigg{)}=-\frac{\partial p}{\partial\tau}+\mu \bigg{(}\frac{\partial^{2}v}{\partial n^{2}}+\frac{\partial^{2}v}{\partial z^ {2}}+\frac{\partial^{2}v}{\partial z^{2}}\bigg{)}\\ \rho\bigg{(}u\frac{\partial w}{\partial n}+v\frac{\partial w}{\partial\tau}+w \frac{\partial w}{\partial z}\bigg{)}=-\frac{\partial p}{\partial z}+\mu \bigg{(}\frac{\partial^{2}w}{\partial n^{2}}+\frac{\partial^{2}w}{\partial\tau ^{2}}+\frac{\partial^{2}w}{\partial z^{2}}\bigg{)}\end{array} \right. \tag{9}\]  
2) Since water film thickness is usually only dozens of microns, the velocity component \(u\) along the liquid film thickness direction \(n\) is ignored and the gradient \(\partial^{2}\)/\(\partial\)z\({}^{2}\) and \(\partial^{2}\)/\(\partial\)\(\tau^{2}\) along the other two directions are neglected. Further, the differential equation is integrated in liquid film thickness direction and Equation (9) is deduced to:  
\[\left\{