## 2 Mathematical Modeling
### Lubrication Model
#### 2.3.3 The Micro Deflecting Motion Driven by the Liquid Film Force  
Supposing \(\delta=0\) at initial time, liquid film force at leading side of the \(ith\) tooth generates a torque \(T_{\mathrm{a,i}}\) which rotates the gate-rotor clockwise. At trailing side, the liquid film torque \(T_{\mathrm{b,i}}\) rotates the gate-rotor anticlockwise. The liquid film torque applied on one tooth flank can be calculated by:  
\[T=\int_{0}^{l_{\mathrm{m}}}\int_{-t_{\mathrm{r}}}^{0}p\Bigg{(}\sqrt{R_{\mathrm{ gr}}^{2}-\left(\frac{b}{2}\right)^{2}}-l\Bigg{)}dtdl+\int_{l_{\mathrm{m}}}^{l_{ \mathrm{r}}}P_{0}l\Bigg{(}\sqrt{R_{\mathrm{gr}}^{2}-\left(\frac{b}{2}\right)^{ 2}}-l\Bigg{)}dtdl \tag{18}\]  
where \(l_{\mathrm{m}}\) is the length of the tooth flank meshing in the screw; \(l_{\mathrm{t}}\) is the tooth thickness; \(b\) is the tooth width; \(l_{\mathrm{t}}\) is the whole tooth length; \(l\) is the tooth length parameter. Supposing only the \(i\)th tooth meshes with the screw, the gate-rotor will deflect anticlockwise under the condition that \(T_{\mathrm{a,i}}\) is smaller than \(T_{\mathrm{b,i}}\). Further, the gap at leading side decreases and \(T_{\mathrm{a,i}}\) increases. Meanwhile, the gap at trailing side increases and \(T_{\mathrm{b,i}}\) decreases until \(T_{\mathrm{a,i}}\) and \(T_{\mathrm{b,i}}\) get balanced. However, if the increased \(T_{\mathrm{a,i}}\) still cannot balance the decreased \(T_{\mathrm{b,i}}\), the gap at leading side will decrease until the first pair of micro convex bodies contact at tooth tip. The state turns to mixed lubrication and the contacting force \(F_{\mathrm{n}}\) starts to act. In case \(T_{\mathrm{b,i}}\) cannot even be balanced by \(T_{\mathrm{a,i}}\) and the torque generated by \(F_{\mathrm{n}}\), the gap at leading side will further decrease and the liquid film thickness reduces to several molecular layer. The state comes to boundary lubrication and \(T_{\mathrm{a,i}}\) is finally balanced by the contact force. Both in mixed and boundary lubrication conditions, abrasion occurs on the working surfaces. However, this paper does not discuss on wear characteristics under mixed or boundary states but focuses on the ability to keep full fluid film lubrication of the MPP. Therefore, when the full fluid film lubrication state turns to mixed lubrication, the fluid film is viewed to be broken and the clearance of the tooth tip \(h_{\text{t}}\) at this moment is defined as the minimum allowable clearance \(h_{\text{min}}\). To determine the \(h_{\text{min}}\) value, the dividing criteria of lubrication state should be introduced first. The lubrication state can be defined by film thickness ratio \(\lambda\). When the mating surfaces are under boundary lubrication condition with preponderant contact on the roughness of contacting surfaces, \(\lambda<1\); when the mating surfaces are under mixed lubrication, \(1\leq\lambda<3\); when the mating surfaces are under fluid film lubrication, \(\lambda\geq 3\)[40]. Therefore, to ensure full fluid film lubrication, the \(\lambda\) value should be 3 and \(h_{\text{min}}\) which is given by [40,41]:  
\[h_{\text{min}}=3\sigma \tag{19}\]  
where \(\sigma\) is the composite surface roughness and is expressed as:  
\[\sigma=\sqrt{\text{R}_{q1}^{2}+\text{R}_{q2}^{2}} \tag{20}\]  
where \(R_{q1}\) and \(R_{q2}\) are the root mean square deviation of roughness of the tooth flank and the screw groove flank respectively. The \(\sigma\) value under different \(R_{q1}\) and \(R_{q2}\) are calculated and listed in Table 1.  
Different processing method will lead to different \(R_{q1}\), \(R_{q2}\) and \(\sigma\). In our lab, the tooth flank is ground by a grinding wheel and \(R_{q1}\) can reach 1.6 \(\upmu\)m, the screw groove flank is milled by a milling cutter and \(R_{q2}\) can reach 3.2 \(\upmu\)m. Therefore, \(\sigma=3.578\)\(\upmu\)m and \(h_{\text{min}}\) equals to 10.7 \(\upmu\)m.  
The clearance of the tooth tip \(h_{\text{t}}\) change with the variation of the deflecting angle \(\delta\). Their relationship is deduced as follows (see Figures 5 and 6):  
\[