## 3 Results
### Data Analysis  
The purpose of this section is to illustrate the effects of the parameters \(d\), \(L\), \(D\)_and_\(p_{2}\) on the values of mass flow rate (\(\dot{m}\)) and total force on the releasing mass (\(F_{M}\)). For a fixed geometry, decreasing outlet pressure (\(p_{2}\)) increments \(\dot{m}\) and \(F_{M}\) (see Figure 15). The incrementation of \(F_{M}\) can be ascribed to both pressure and viscosity components. Indeed, decreasing the outlet pressure, a lower pressure is expected also at the base of the releasing mass (near the outflow). Moreover the viscous component increases due to the higher shear stress acting on the releasing mass lateral surface. Figure 14: Velocity streamlines and pressure contours in the gripper terminal region at different outlet pressures (\(d\) = 0.77 mm, \(L\) = 32.2 mm, \(D\) = 3.6 mm). Figure 15: Dependence on imposed negative pressure: axial pressure force (**top left**), viscous force (**bottom left**) and mass flow rate (**right**). For a fixed outlet pressure, increasing the lateral hole diameter (\(d\)) provides higher mass flow rate (\(\dot{m}\)) and lifting force (\(F_{M}\)) on the releasing mass. The value of \(L\) slightly affects mass flow rate, while it influences the viscous part of the lifting force. It is interesting to note how the lifting force (\(F_{M}\)) is strongly related to \(D\) value (see Figure 16,b). A larger annulus area reduces shear stress on the lateral surface, and consequentially, viscous force (\(F_{V}\)) acting on the releasing mass wall decreases. Similarly, pressure force (\(F_{P}\)) increases as \(D\) decreases. Since the pressure force acting on the releasing mass is related to the pressure difference acting on its two bases, (\(p_{xx}\) and \(p_{dx}\)) a possible explanation for \(F_{P}\) trend can be found out by looking at the pressure drop in the different regions of the gripper. As can be seen from Figure 17, the pressure in the front chamber is approximately constant and equal to the pressure acting on the upstream releasing mass base (\(p_{xx}\)). Considering Figure 17 where average static pressure on cross-sections versus dimensionless axial direction (\(z/L_{tot}\)) is plotted, the pressure difference acting on the releasing mass (\(p_{xx}-p_{dx}\)) can be written as:  
\[p_{xx}-p_{dx}=\Delta p_{j}+\Delta p_{AN} \tag{10}\]  
where \(\Delta p_{j}=p_{xx}-p(z_{3})\) is the pressure difference between the front chamber and the end of the jet region, and \(\Delta p_{AN}\) is the pressure difference in the annular region. Figure 16: Dependence on gripper body diameter (\(D\)): (**a**) axial pressure force; (**b**) axial viscous force. Figure 17: Average static pressure in the axial direction. The pressure drop (\(\Delta p_{j}\)) is controlled by the lateral impinging jet: when the distance between the releasing mass surface and the lateral inlet is smaller, as happens for smaller values of \(D\), the distribution of the mean static pressure on the impinging surface, which typically has a Gaussian shape, exhibits higher peak pressure. As a result, higher pressure drop is expected between the front chamber and the end of the jet region. Despite the mass flow rate decreasing for decreasing \(D\), the pressure drop in the annulus region increases, as shown in Figure 18.  
This behavior can be justified as follows. Assuming that the flow in annulus can be approximated as one-dimensional (average average axial velocity \(u_{z}\) is dominant), the momentum conservation equation reads:  
\[\dot{m}\ du_{z}=-A_{AN}dp-\tau_{w}Pdz \tag{11}\]  
where \(A_{AN}=\pi\frac{D^{2}-{D_{1}}^{2}}{4}\) is the cross-sectional area, \(P=\pi(D+D_{1})\) is the wetted perimeter and \(\tau_{w}\) is the wall shear stress in the annulus region.