## 2 Mathematical Modeling
### Lubrication Model  
#### 2.3.1 The Favorable Conditions to Form Hydrodynamic Lubrication  
1) Wedge  
There is a wedge in the meshing pair, which is surrounded by the dashed lines in the clearance (the quadrilateral AKK'A'), as shown in Figure 7. Since the wedge is a part of the gap, its included angle varies along with \(\varphi_{\rm gr}\) and \(l\). Moreover, due to the meshing point sliding on the tooth flank during the meshing process, the wedge length changes except SSLT.  
2) Relative velocity between the two meshing surfaces  
The liquid is not only a Poiseuille flow but also a Couette flow because the relative move of the meshing pair. The relative velocity \(v_{\rm r}\) between gate-rotor and screw is given by:  
\[\begin{array}{ll}\overrightarrow{v_{\rm r}}&=\overrightarrow{v_{\rm gr}}- \overrightarrow{v_{\rm sc}}\\ &=\overrightarrow{\omega_{gr}}\cdot r_{gr,K}-\overrightarrow{\omega_{\rm sc }}\cdot r_{sc,K}\end{array} \tag{7}\]  
where \(v_{\rm gr}\) is the velocity of the of the gate-rotor tooth at the meshing point 'K'; \(v_{\rm sc}\) is the velocity of the of the screw at the meshing point K. \(\omega_{\rm gr}\) and \(\omega_{\rm sc}\) are the angular velocity of the gate-rotor and the screw, respectively; \(r_{\rm gr,K}\) and \(r_{\rm sc,K}\) are the radii from the meshing point K to the gate-rotor axis and screw axis, respectively.  
3) Flow direction  
The liquid flows from the high pressure side to the low pressure side. The bigger opening of the wedge is exactly located at the high pressure side. Therefore, the inlet is bigger than the outlet of the wedge.  
4) The wedge is filled with liquid  
In general, the liquid injection quantity is very huge in SSCs. For example, the injection flow in a SSC with discharge capacity of 6 m\({}^{3}\)-min\({}^{-1}\) is about 60 L-min\({}^{-1}\). Under the effect of differential pressure and relative velocity, a large amount of liquid is brought to the inlet of the wedge. Since the wedge width is only dozens of micrometers, the liquid accumulated at the inlet can hardly be drained out through the clearance. The gas is almost impossible to cross the clearance.  
Furthermore, references [33; 34; 35] study the leakage characteristics of SSCs. During calculation of the leakage properties of the gap between the tooth flank and screw thread flank, the flow in the clearance is supposed to be single phase of liquid. The simulating volume efficiencies are consistent with the experimental data. Therefore, the deduction that the wedge is filled with liquid is reasonable.  
5) The hydrophilicity of the material For the water lubricated friction pair, the hydrophilicity of the material is beneficial for the formation of the hydrodynamic water film [36]. In most water flooded SSCs sold in the market, screws are made of tin bronze and gate-rotors are made of PEEK. A metallic oxide film which is polar always emerged on the surface of copper alloy [37]. Therefore, the screw surface can be viewed as hydrophilic. The PEEK material has a wetting angle range of 20-83\({}^{\circ}\), which is smaller than 90\({}^{\circ}\). Therefore, it should be classified to hydrophilic material [38].  
In this paper, an observational experimental has been conducted to verify whether the materials are hydrophilic. Several drops of deionized water with different size were dropped to the surfaces of the screw and the gate-rotor, as shown in Figure 8. It can be seen that the wetting angle of the droplet on the bronze surface is slightly larger than that on the PEEK surface. However, the maximum wetting angle on the screw is still smaller than 60\({}^{\circ}\), which proves that the meshing pair surfaces have better hydrophilicity and are helpful to establish hydrodynamic lubrication.