## Appendix A  
To determine the pressure difference between the fluid above and below the piston, consider the flow tube with cross-sections that coincide with the cylinder's cross-section and the lower boundary of the piston, the cross-sections of the holes, the clearance around the piston, and the upper boundary of the piston. For these cross-sections, the continuity equation is as follows:  
\[S_{c}v=(S_{c}-S_{1})\langle u_{1x}\rangle+n_{2}S_{2}\langle u_{2x}\rangle, \tag{1}\]  
where \(S_{c}=\pi R_{c}^{2}\) is the cross-sectional area of the cylinder, \(R_{c}\) is the radius of the cylinder, \(S_{1}=\pi R_{1}^{2}\) is the cross-sectional area of the piston, \(R_{1}\) is the radius of the piston, \(\langle u_{1x}\rangle\) is the average velocity of the fluid flow in the clearance between the piston and the cylinder,\(n_{2}\) is the number of holes, \(S_{2}=\pi R_{2}^{2}\) is the cross-sectional area of a piston hole, \(R_{2}\) is the radius of a hole, and \(\langle u_{2x}\rangle\) is the average velocity of the fluid flow in the hole cross-section. Assuming that \((S_{c}-S_{1})\langle u_{1x}\rangle\ll n_{2}S_{2}\langle u_{2x}\rangle\), Equation (A1) can be rewritten as follows:  
\[S_{c}v\approx n_{2}S_{2}\langle u_{2x}\rangle.\] (A2)  
From Equation (A2), we find  
\[\langle u_{2x}\rangle=\frac{S_{c}v}{n_{2}S_{2}}=\frac{1}{n_{2}}\bigg{(}\frac{ R_{c}}{R_{2}}\bigg{)}^{2}v.\] (A3)  
[PERSON]'s equation neglecting internal fluid friction is as follows:  
\[p_{1}+\rho gh_{1}+\frac{\rho v^{2}}{2}=p_{2}+\rho g(h_{1}+h)+\frac{\rho\langle u _{2x}\rangle^{2}}{2}+\frac{mg}{S_{1}-nS_{2}}\] (A4)  
in the case of the piston moving upward, and  
\[p_{1}+\rho g(h_{1}+h)+\frac{\rho v^{2}}{2}+\frac{mg}{S_{1}-nS_{2}}=p_{2}+\rho gh _{1}+\frac{\rho\langle u_{2x}\rangle^{2}}{2}\] (A5)  
in the case of the piston moving downward. Here, \(p_{1}\) is the pressure of the fluid on the lower surface of the piston at height \(h_{1}\), \(p_{2}\) is the pressure of the fluid on the upper surface of the piston at height \(h_{1}+h\) (\(h-\)height of the piston), \(\rho\) is the fluid density, \(m\) is the mass of the piston, and \(g\) is the acceleration due to gravity.  
From Equations (A4) and (A5), taking into account (A3), we find the following pressure difference of the fluid between the sections of the flow tube:  
\[\Delta p=p_{2}-p_{1}=-\frac{\rho}{2}\Bigg{[}\frac{1}{n_{2}^{2}}\bigg{(}\frac{ R_{c}}{R_{2}}\bigg{)}^{4}-1\Bigg{]}v^{2}-\rho gh-\frac{mg}{\pi\big{(}R_{1}^{2}- nR_{2}^{2}\big{)}}\] (A6)  
- in the case of fluid flow downward through the holes, and  
\[\Delta p=p_{2}-p_{1}=-\frac{\rho}{2}\Bigg{[}\frac{1}{n_{2}^{2}}\bigg{(}\frac{ R_{c}}{R_{2}}\bigg{)}^{4}-1\Bigg{]}v^{2}+\rho gh+\frac{mg}{\pi\big{(}R_{1}^{2}- nR_{2}^{2}\big{)}}\] (A7)  
- in the case of the fluid flowing upward through the holes.  
Combining Formulas (A6) and (A7), we obtain  
\[\Delta p=p_{2}-p_{1}=-\frac{\rho}{2}\Bigg{[}\frac{1}{n_{2}^{2}}\bigg{(}\frac{ R_{c}}{R_{2}}\bigg{)}^{4}-1\Bigg{]}v^{2}+\rho g_{x}h+\frac{mg_{x}}{\pi\big{(}R_{1}^{2}- nR_{2}^{2}\big{)}}.\] (A8)  
Here, \(g_{x}\) is the projection of gravitational acceleration on the direction of fluid flow through the piston holes and the clearance around the piston, \(g_{x}=-g\) when the fluid flows downward, and \(g_{x}=g\) when the fluid flows upward.