## 2 Modeling
### The Reynolds Equation and Energy Equation  
The hydraulic oil used in the piston pumps is assumed incompressible. Therefore, the Navier-Stokes equation of incompressible fluid can be obtained:  
\[\partial\rho\mathbf{u}/\partial t+\
abla(\rho\mathbf{u})=-\
abla p+\
abla( \mu\
abla\mathbf{u}) \tag{7}\]  
The following assumptions can be made: (1) The flow is under the steady-state; (2) compared with the viscous force, the inertial force can be neglected; (3) as the film height is small, the pressure does not change in the direction of the film height; and (4) the velocity gradient of fluid in the direction of film height is much greater than that in other directions. As shown in Figure 3, in the \(\hat{z}\)-axis direction, as the upper boundary of the film contacts with the piston surface, the velocity of the upper boundary is the same as that of the piston. The velocity includes the reciprocating velocity in the \(\hat{y}\)-axis direction and the rotation velocity in the \(\hat{x}\)-axis direction. On the other hand, the lower boundary of the film contacts the surface of the bushing. As the bushing is fixed, the velocity of the lower boundary should be 0. The Reynolds equation can be obtained based on the above conditions:  
\[\frac{\partial}{\partial\hat{x}}\left(\frac{\partial p}{\partial\hat{x}}\frac {h_{p}{}^{3}}{\mu}\right)+\frac{\partial}{\partial\hat{y}}\left(\frac{ \partial p}{\partial\hat{y}}\frac{h_{p}{}^{3}}{\mu}\right)=6\left(u_{p}\frac{ \partial h_{p}}{\partial\hat{x}}+v_{p}\frac{\partial h_{p}}{\partial\hat{y}} +2\frac{dh_{p}}{dt}\right) \tag{8}\]  
where \(h_{p}\) denotes the film height; \(u_{p}\) is the rotation velocity (approximate to the revolution velocity of the piston); and \(v_{p}\) denotes the reciprocating velocity. Figure 3: The fluid film in a rectangular coordinate system. According to [PERSON]'s law of viscosity, the shearing stress per unit area of the piston/bushing surface can be defined by:  
\[\left\{ \begin{array}{l}\tau_{pa}=\left(-\mu\frac{\partial v}{\partial\hat{z}} \right)_{z=h_{p}}\\ \tau_{ba}=\left(-\mu\frac{\partial v}{\partial z}\right)_{z=0}\end{array} \right. \tag{9}\]  
where \(\tau_{pa}\) is the axial shearing stress exerted on the piston; \(\tau_{ba}\) denotes the axial shearing stress exerted on the bushing. The viscous friction force exerted on the piston/bushing can be calculated with:  
\[F_{pa}(\varphi)=\iint\tau_{pa}(\varphi)dA \tag{10}\]  
\[F_{ba}(\varphi)=\iint\tau_{ba}(\varphi)dA \tag{11}\]  
where \(\varphi\) represents the position of the piston. After obtaining the film geometry and the pressure distribution, the temperature distribution can be obtained by solving the energy equation. According to the law of conservation of energy, suppose the effects of volume force and thermal radiation are not considered. In that case, the energy equation for the viscous fluid flow can be expressed as [5]:  
\[\rho\frac{D\big{(}c_{p}T\big{)}}{Dt}=\
abla\Big{(}k_{f}\
abla T\Big{)}-\frac{T }{\rho}\frac{\partial T}{\partial\rho}\frac{Dp}{Dt}+\phi \tag{12}\]  
where \(T\) denotes the temperature; \(c_{p}\) denotes the specific heat at constant pressure; \(k_{f}\) is the thermal conductivity coefficient of hydraulic fluid; and \(\phi\) denotes the energy dissipation term. The convection and heat conduction in the direction of the film height can be neglected. The energy equation of the film can be simplified to:  
\[c_{p}\bigg{(}v_{p}\frac{\partial T}{\partial\hat{g}}+u_{p}\frac{\partial T}{ \partial\hat{x}}\bigg{)}=k_{f}\bigg{(}\frac{\partial^{2}T}{\partial\hat{g}^{2 }}+\frac{\partial^{2}T}{\partial\hat{x}^{2}}\bigg{)}+\phi \tag{13}\]  
where the energy dissipation term \(\phi\) can be calculated with [16]:  
\[\phi=\mu\bigg{(}\bigg{(}\frac{\partial v}{\partial\hat{z}}\bigg{)}^{2}+\bigg{(} \frac{\partial u}{\partial\hat{z}}\bigg{)}^{2}\bigg{)} \tag{14}\]  
Both Equations (8) and (13) belong to the partial differential equation, and the expression of film height is complex. Hence, it is challenging to obtain the analytical solution of these two equations.