## 2 Materials and Methods
### Equations of Motion and Their Solutions
A detailed derivation of Formula (8) is provided in Appendix A. Now, the projections of the pressure force, buoyant force, and gravitational force acting on the piston can be readily determined, taking into account the value \(v\) from (4):  
\[F_{1s}=\left\{-\frac{\rho}{2}\Bigg{[}\frac{1}{n_{2}^{2}}\bigg{(}\frac{R_{c}}{ R_{2}}\bigg{)}^{2}-1\Bigg{]}\left(r_{0}\sin\varphi\varphi\right)^{2}+\rho g_{x}h+ \frac{mg_{x}}{\pi\big{(}R_{1}^{2}-n_{2}R_{2}^{2}\big{)}}\Bigg{]}\pi\Big{(}R_{1} ^{2}-n_{2}R_{2}^{2}\Big{)},\right. \tag{9}\]  
To determine the friction force between the piston and the fluid, consider axisymmetric, straight-line fluid flows in a cylindrical coordinate system (\(r\), \(\theta\), \(x\)), where \(r\) is the radial coordinate, \(\theta\) is the polar angular coordinate, and the axis \(x\) coincides with the flow symmetry axis. Applying the axisymmetry condition ([49], Section 6.2, Formula 6.37) to the continuity equation and subsequently to the viscous fluid motion equation for steady flow \(u_{x}=u_{x}(r)\), we obtain an ordinary differential equation ([49], Section 6.2, Formula 6.40):  
\[-\frac{\partial p}{\partial x}+\eta\frac{1}{r}\frac{d}{dr}\bigg{(}r\frac{gu_{x }}{\partial r}\bigg{)}+\rho g_{x}=0, \tag{10}\]  
When the pressure gradient \(\partial p/\partial x\) is constant, the general solution of Equation (10) ([49], Section 6.2, Formula 6.43) is as follows:  
\[u_{x}=\frac{1}{4\eta}\bigg{(}\frac{\partial p}{\partial x}-\rho g_{x}\bigg{)} r^{2}+c_{1}\ln r+c_{2}. \tag{11}\]  
The constants \(c_{1}\) and \(c_{2}\) are determined from the flow boundary conditions. Next, we find the projection of the friction force exerted on the lateral surface of the piston. Using the boundary conditions on the lateral surface of the piston and the inner surface of the cylinder as follows:  
\[ \begin{array}{l}r=R_{1}:u_{x}=v;\\ r=R_{c}:u_{x}=0\end{array}  \tag{12}\]  
Equation (11) may be rewritten as follows:  
\[u_{x}=\frac{1}{4\eta}\bigg{(}\frac{\partial p}{\partial x}-\rho g_{x}\bigg{)} \left(r^{2}-R_{c}^{2}\right)+\bigg{[}v+\frac{1}{4\eta}\bigg{(}\frac{\partial p }{\partial x}-\rho g_{x}\bigg{)}\left(R_{c}^{2}-R_{1}^{2}\right)\bigg{]}\frac {\ln\frac{r}{K_{c}}}{\ln(R_{1}/R_{c})}. \tag{13}\]  
Under the conditions \(R_{c}-R_{1}<<R_{1}\) and \(h\), neglecting edge effects, the flow of the fluid between the piston and cylinder can be considered laminar. By substituting the gradient \(\partial p/\partial x\) with \(\Delta p/h\) and taking into account Expression (8) for \(\Delta p\) and Expression (4) for \(v\), and substituting (13) into [PERSON]'s law, we find the friction force acting on the lateral surface of the piston as follows:  
\[ \begin{array}{l}F_{2s}\approx-\left\{-\frac{\rho}{2}\left[\frac{1}{n_{2}^{2}} \left(\frac{R_{c}}{R_{2}}\right)^{2}-1\right]\left(r_{0}\sin\varphi\dot{\varphi} \right)^{2}+\frac{m_{gg_{s}}}{\pi\left(R_{1}^{2}-n_{2}R_{2}^{2}\right)}\right\} \left[\pi R_{1}^{2}+\frac{\pi\left(R_{c}^{2}-R_{1}^{2}\right)}{2\ln\left(R_{1} /R_{c}\right)}\right]\\ -\frac{2\pi\eta h}{\ln\left(R_{1}/R_{c}\right)}r_{0}\sin\varphi\dot{\varphi}, \end{array}  \tag{14}\]  
Now, we find the projections of the friction forces acting on the surfaces of the cylindrical holes under conditions of steady laminar flow at low engine shaft speeds, assuming the following relation holds: \(R_{2}<<h\). Edge effects are neglected. In Equation (11), to determine the constants \(c_{1}\) and \(c_{2}\) we use the following boundary conditions of the flow (see example 6.2.4, Figure 6.13 in [49]):  
\[ \begin{array}{l}r=R_{2}:u_{x}=v;\\ r=kR_{2}:u_{x}=0.\end{array}