## 2 Flow Force Analysis  
Figure 1 shows the stress on the spool, and the flow force can be expressed as follows:  
\[F_{F}=F_{\tau_{rad}}+F_{s}+F_{d} \tag{1}\]  
where \(F_{F}\) is flow force. \(F_{\tau_{rad}}\) is the viscous force generated by fluid acting on the spool. \(F_{s}\) is the static force generated by fluid acting on the spool. \(F_{d}\) is the dynamic force generated by fluid acting on the spool. Selecting 'right' as the positive direction, \(F_{\tau_{rad}}\), \(F_{s}\) and \(F_{d}\) can be expressed as follows:  
\[F_{\tau_{rad}}=\iint\limits_{A_{rad}}\tau_{rad}dA \tag{2}\]  
\[F_{s}=\iint\limits_{A_{r}}p_{s2}dA-\iint\limits_{A_{l}}p_{s1}dA \tag{3}\]  
\[F_{d}=\iint\limits_{A_{r}}p_{d2}dA-\iint\limits_{A_{l}}p_{d1}dA \tag{4}\]  
where \(\tau_{rad}\) is the shear force generated by fluid acting on the spool stem. \(p_{s1}\) and \(p_{s2}\) are the static pressure generated by fluid acting on the left shoulder surface 1 and the right shoulder surface 3 of the spool. \(p_{d1}\) and \(p_{d2}\) are dynamic pressure generated by the fluid acting on the left shoulder surface 1 and the right shoulder surface 3 of the spool. \(A_{rad}\) is the surface area of the spool stem. \(A_{l}\) and \(A_{r}\) are the areas of the left shoulder surface 1 and the right shoulder surface 3 of the spool. So, the flow force can be expressed as follows:  
\[F_{F}=\iint\limits_{A_{rad}}\tau_{rad}dA+\iint\limits_{A_{r}}p_{s2}dA-\iint \limits_{A_{l}}p_{s1}dA+\iint\limits_{A_{r}}p_{d2}dA-\iint\limits_{A_{l}}p_{ d1}dA \tag{5}\]  
In this paper, the forces of the spool in the steady state are studied, and the transient flow force is ignored; that is, the dynamic forces generated by the fluid acting on the spool are ignored. The flow force of the spool can be expressed as follows:  
\[F_{F}=\iint\limits_{A_{rad}}\tau_{rad}dA+\iint\limits_{A_{r}}p_{s2}dA-\iint \limits_{A_{l}}p_{s1}dA \tag{6}\]  
Figure 1: Stress of the spool. Flow force can also be calculated by momentum theorem. Taking the fluid as the research object, the fluid is subjected to two forces: the force exerted by the spool sleeve and the force exerted by the spool. The force that the spool enacts on the fluid and the force that the fluid enacts on the spool are the action and reaction forces. Therefore, the force acting on the fluid can be expressed as follows:  
\[F_{liquid}=F_{sleeve}-F_{F} \tag{7}\]  
where \(F_{liquid}\) is the force on the fluid. \(F_{sleeve}\) is the force exerted by the sleeve on the fluid. \(F_{sleeve}\) can be expressed as follows:  
\[F_{sleeve}=\iint\limits_{A_{sleeve}}\pi_{sleeve}dA\approx\alpha\mu Lq \tag{8}\]  
where \(\tau_{sleeve}\) is the shear force generated by the spool sleeve acting on the fluid. \(A_{sleeve}\) is the contact area between the sleeve and the fluid. \(\alpha\), \(u\) and \(L\) are the geometry coefficient of runner, fluid viscosity and effective length of the fluid, respectively. \(q\) is flow rate. From the momentum conservation theorem:  
\[F_{liquid}=\frac{d}{dt}\iiint\limits_{C.V.}\rho v_{x}dV+\iint\limits_{A_{i}} \rho v_{x}v\times idA+\iint\limits_{A_{o}}\rho v_{x}v\times(-i)dA \tag{9}\]  
where \(\rho\) is fluid density, \(v\) is fluid velocity, \(v_{x}\) is fluid axial velocity, \(i\) is the left unit normal vector, \(A_{i}\) is the inlet area, \(A_{o}\) is the outlet area. Ignoring the transient flow force, the following results can be obtained as follows:  
\[F_{\tau_{rad}}=\iint\limits_{A_{rad}}\tau_{rad}dA \tag{2}\]0  
where \(v_{1}\) and \(v_{2}\) are the average speeds at the inlet and outlet. \(\theta_{1}\) and \(\theta_{2}\) are the jet angles at the inlet and outlet. Steady-state flow force can be expressed as follows:  
\[F_{\tau_{rad}}=\iint\limits_{A_{rad}}\tau_{rad}dA \tag{2}\]1  
According to the above derivation, the steady-state flow force is in the direction of closing the valve, that is, the negative direction. The steady-state flow force value depends on the forces on the left and right shoulder of the spool and the side of the spool stem.