## 3 Analysis of Flow Force Characteristics of Slide Valves
### Effects of Viscous Forces and Vortex Dissipation  
It is worth noting that the steady flow force includes viscous forces. To verify that the fluid momentum at the non-throttling valve port is the primary factor causing the differences in flow force between the two throttling modes, it is essential to determine the contribution of viscous forces to the total fluid force. Figure 11 shows the variation in viscous forces with valve opening for both throttling modes. The results indicate that the viscous forces on the valve body wall in the inlet throttling mode are greater than those in the outlet throttling mode, but remain very small. Therefore, viscous forces are not the main cause of the asymmetry in steady flow force characteristics. In addition, the inlet throttling mode of the slide valve contains a significant number of vortices; the presence of these vortices has a substantial impact on the jet angle of the fluid within the valve chamber. It is therefore necessary to discuss the relationship between  
Figure 11: Variation curve of viscous force with valve opening. Figure 10: Inlet throttling slide valve jet angle and jet velocity. Figure 9: Flow velocity clouds of the outlet throttling slide valve at different valve openings \(x_{o}\), from left to right: valve openings \(x_{o}\) are 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, and 1.6, respectively. the fluid vortices and the steady flow force characteristic curves. Considering that vortex dissipation is an irreversible process associated with an increase in entropy, this paper employs entropy production theory to analyze the hydraulic losses caused by the vortices in the valve. The flow inside the slide valve is turbulent, and the entropy production rate of fluid particles consists of two parts:  
\[S_{D}^{\bullet\prime\prime}=S_{D}^{\bullet\prime\prime}+S_{D^{\prime\prime}}^{ \bullet\prime\prime} \tag{10}\]  
\[S_{D}^{\bullet\prime}=\frac{\mu}{T}\Bigg{\{}2\Bigg{[}\Bigg{(}\frac{\partial \overline{u}_{1}}{\partial x_{1}}\Bigg{)}^{2}+\Bigg{(}\frac{\partial\overline{u }_{2}}{\partial x_{2}}\Bigg{)}^{2}+\Bigg{(}\frac{\partial\overline{u}_{3}}{ \partial x_{3}}\Bigg{)}^{2}\Bigg{]}+\Bigg{(}\frac{\partial\overline{u}_{2}}{ \partial x_{1}}+\frac{\partial\overline{u}_{1}}{\partial x_{2}}\Bigg{)}^{2}+ \Bigg{(}\frac{\partial\overline{u}_{3}}{\partial x_{1}}+\frac{\partial \overline{u}_{1}}{\partial x_{3}}\Bigg{)}^{2}+\bigg{(}\frac{\partial\overline{ u}_{2}}{\partial x_{3}}+\frac{\partial\overline{u}_{3}}{\partial x_{2}}\Bigg{)}^{2} \Bigg{\}} \tag{11}\]  
\[S_{D^{\prime}}^{\bullet\prime\prime}=\frac{\rho\varepsilon}{T} \tag{12}\]  
where \(S_{D}^{\bullet\prime\prime}\) represents the entropy production caused by turbulent time-averaged velocity; \(S_{D^{\prime}}^{\bullet\prime\prime}\) denotes the entropy production due to turbulent pulsation velocity; \(\mu\) is the effective dynamic viscosity of the flowing medium, in Pa-s; \(T\) is the temperature of the flow field, in \(K\); \(\overline{u}\) is the Reynolds average velocity, in m/s; \(\rho\) is the density of the fluid, in kg/m\({}^{3}\); and \(\varepsilon\) is the turbulent kinetic energy dissipation rate, in m\({}^{2}\)/s\({}^{3}\). Figure 12 illustrates the entropy production distribution within the sliding valve for the two throttling modes. The distinction between the two throttling modes lies in the position and intensity of the vortices. In the inlet throttling slide valve, the fluid accelerates through the throttling orifice, rapidly impacting the spool and subsequently transitioning to axial flow under the constraints of the wall, with vortices primarily forming within the control volume. In contrast, for the outlet throttling slide valve, the fluid exits the valve after passing through the throttling orifice, with vortices predominantly appearing outside the control volume. To investigate the impact of vortices on the asymmetry of flow force performance in the two throttling modes, modifications were made to the flow channel of the slide valve.