## 2 Thermohydrodynamic Lubrication Model
The average Reynolds equation is used for control, as shown in Equation (15):  
\[\frac{\partial}{\partial x}\left(\frac{\varphi_{x}h^{3}\partial\overline{P}}{ \partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\varphi_{y}h^{3} \partial\overline{P}}{\partial y}\right)=\frac{6 Ul\sigma\eta\partial\overline{ h_{T}}}{\partial x}+\frac{6 Ul\sigma\eta\partial\varphi_{s}}{\partial x}+12 \rho\varphi_{c}\frac{\partial\overline{h_{T}}}{\partial t} \tag{15}\]  
where \(h\) is the oil film thickness, \(\overline{P}\) is the average oil film pressure, \(\overline{h_{T}}\) is the average clearance (The gap between microconvex bodies at the microscopic scale is different, and \"the average clearance\" is the average gap between microconvex bodies. ), \(\varphi_{x},\varphi_{y}\) is the pressure flow factor, \(\varphi_{s}\) is the shear force factor, \(\rho\) is the density of the lubricant, \(U\) is the sliding velocity of the piston ring relative to the cylinder liner, \(\sigma\) is the combined roughness of the two contact surfaces:  
\[\sigma=\sqrt{\sigma_{l}^{2}+\sigma_{r}^{2}} \tag{16}\]  
where \(\sigma_{l}\) is the standard deviation of the combined roughness height of the cylinder liner, \(\sigma_{r}\) is the standard deviation of the combined roughness height of the piston ring, \(\varphi_{c}\) is the contact factor, which can be defined by the empirical formula as:  
\[\varphi_{c}=\left\{ \begin{array}{ll}\exp\bigl{(}-0.6912+0.782H-0.304H^{2}+0.0401H^{3}\bigr{)}&,&0\leq H<4\\ 1&,&H\geq 4\end{array} \right. \tag{17}\]  
The pressure flow coefficient is given by the specific formula in Equation (18):  
\[\left\{ \begin{array}{ll}\varphi_{x}=1.0-Ce^{-rH},&\gamma\leq 1\\ \varphi_{x}=1.0+Ce^{-r},&\gamma>1\\ \varphi_{y}(H,\gamma)=\varphi_{x}(H,1/\gamma)\end{array} \right. \tag{18}\]  
where \(C\) and \(r\) are constants, and the film thickness ratio is defined as follows, \(\gamma\) is the surface direction parameter, reflecting the relationship between the length ratio of the micro-convex body on a rough surface and the relative direction of motion:  
\[H=\frac{h}{\sigma} \tag{19}\]  
The shear flow coefficient is given by Equation (20):  
\[\varphi_{s}=\Bigl{(}\frac{\sigma_{r}}{\sigma}\Bigr{)}^{2}\Phi_{s}(H,\gamma_{ r})-\Bigl{(}\frac{\sigma_{l}}{\sigma}\Bigr{)}^{2}\Phi_{s}(H,\gamma_{l}) \tag{20}\]  
The calculation formula is given by Equation (21):  
\[\Phi_{s}=\left\{ \begin{array}{ll}A_{1}H^{a}e^{-bH+cH^{2}}&,&H\leq 5\\ A_{2}e^{-bH}&,&H>5\end{array} \right. \tag{21}\]  
where \(A_{1},\ A_{2},\ a,\ b,\ c\) are constants. They change with the surface orientation parameter \(\gamma\). After determining the oil film thickness, a super-relaxation iterative method is used to solve for the oil film pressure on the piston ring-cylinder liner contact surface. The oil film pressure across the entire piston ring surface is then obtained by integration, as shown in Equation (22):  
\[F_{z}=\iint\limits_{A}pdxdy \tag{22}\]  
In the calculation, it is assumed that the model operates under rich-oil lubrication, with the starting point of the oil film being on the piston ring side near the top of the cylinder liner and the outlet being on the side of the piston ring farthest from the top of the cylinder liner. Therefore, the oil film pressures at the inlet and outlet are equal to the gas pressures at the two sides of the piston ring, as shown in the Figure 3. The boundary conditions are as follows, in Equations (23) and (24):  
\[P_{x=0}=p_{1}(t)\ ;\ P_{x=b}=p_{2}(t) \tag{23}\]  
\[P_{y}=0=p_{0}(t)\ ;\ P_{y=y_{1}}=p_{0}(t) \tag{24}\]  
where \(p_{0}(t)\) is the gas pressure at the piston ring inlet, \(p_{1}(t)\,\ p_{2}(t)\) represent the oil film inlet and outlet pressures at time \(t\), respectively, with the inlet pressure being the engine combustion pressure and the outlet pressure being the gas pressure between the first and second rings.