## Appendix A  
To determine the nature of the pressure distribution \(p_{r}\) in the radial gap, the Bernoulli equation is written for radii \(r\) and \(r_{1}\)[6] as follows:  
\[\frac{p_{r}}{\rho_{am}}+\frac{V_{r}^{2}}{2}=\frac{p_{am}}{\rho_{am}}+\frac{V_{ 1}^{2}}{2}+E_{r-r1}, \tag{11}\]  
where \(V_{r}\) and \(V_{1}\) are the air flow rates at radii \(r\) and \(r_{1}\), respectively, and \(E_{r-r1}\) is the loss of specific energy of the flow at the interval from \(r\) to \(r_{1}\). The elemental losses of a specific energy of \(dE\) flow on the elemental segment \(dr\) are found by the known dependence [76]:  
\[dE=\lambda_{m}\frac{V_{r}^{2}}{2}\frac{dr}{D_{g}}, \tag{12}\]  
where \(\lambda_{m}\) is the coefficient of the viscous friction of the air to the active surface of the grip and the object of manipulation, which depends on the Reynolds number and surface roughness, and \(D_{g}=2h_{r}\) is the hydraulic flow diameter at the radius \(r\). The coefficient of viscous friction \(\lambda\) for the turbulent air movement mode is conveniently calculated by empirical formula [76]:  
\[\lambda=0.11K_{n}\bigg{(}\frac{\Delta}{D_{g}}+\frac{68}{\text{Re}}\bigg{)}^{0.25}, \tag{13}\]  
where \(K_{n}\) is a coefficient that takes into account the non-circular cross-section of the flow (for an annular section of \(K_{n}=1.1\)) [48]; \(\Delta\) is the equivalent absolute roughness of the surfaces of the flow-forming air, which can be taken as equal to the roughness along Ra of the surface of the manipulated object; and Re is the Reynolds number. The Reynolds number Re for streams of any form is determined with [53]:  
\[\text{Re}=\frac{\rho VD_{g}}{\mu_{d}}, \tag{14}\]  
where \(\mu_{d}=1.71\cdot 10^{-5}+4.94\cdot 10^{-8}t\) is the coefficient of the dynamic viscosity of air (\(\text{kg}/(\text{s}\cdot\text{m})\)) and \(t\) is the ambient temperature (\({}^{\circ}\)C). Given the mass flow of air in the radial annular interval \(G=V_{r}\rho_{a}2\pi h\), we obtain:  
\[\text{Re}=\frac{G}{\pi\mu_{d}r},\ \lambda=0.121\bigg{(}\frac{\Delta}{2h}+\frac{6 8\pi\mu_{d}r}{G}\bigg{)}^{0.25}. \tag{15}\]  
According to the velocity flow continuity equation,  
\[V_{r}=\frac{G}{2\pi\rho_{a}rh},\quad V_{1}=\frac{G}{2\pi\rho_{a}r(h+\delta)}. \tag{16}\]  
We will find the total losses of the specific energy of the flow by integrating Equation (12) and considering (13) and (14):  
\[E_{r-r1}=\frac{1}{16\pi^{2}}\frac{G^{2}}{\rho_{a}^{2}}\int\limits_{r}^{r_{1} }\frac{0.121\Big{(}\frac{\Delta}{2h}+\frac{68\pi\mu_{d}r}{G}\Big{)}^{0.25}}{h^ {3}r^{2}}dr. \tag{17}\]Substituting the result in (A1), after the transformations, the following is obtained:  
\[p_{r}=p_{a}-\frac{G^{2}}{8\pi^{2}\rho_{a}}\left[\frac{1}{h^{2}r^{2}}-\frac{1}{ \left(h+\delta\right)^{2}r_{1}^{2}}-\frac{1}{2}\int\limits_{r}^{r_{1}}\frac{0.12 1\left(\frac{\Delta}{2h}+\frac{68\pi\mu_{g}r}{G}\right)^{0.25}}{h^{3}r^{2}}dr \right]. \tag{10}\]  
In order to determine absolute pressure \(p_{1}\) at the end of the gripping, we must conduct a gas-dynamic analysis of the process of the air flow outflow from the annular nozzle and its expansion in the radial interval \(h\) to the radius \(r_{1}\). To do this, we compose the equation of pulses of forces in projections on the \(x\)-axis for the selected element \(d\phi\) of the radial gap \(h\):  
\[ \begin{array}{c}dGV_{0}\cos\phi+p_{1}hr_{0}d\phi\cos\phi+2h\int\limits_{r_{0} }^{r_{1}}pdr\sin\frac{d\phi}{2}\cos\phi+\\ +2h\int\limits_{r_{0}}^{r_{1}}p_{r}dr\sin\frac{4\phi}{2}\cos\phi=dGV_{1}\cos \phi+p_{r1}hr_{1}d\phi\cos\phi+dF_{f}.\end{array}  \tag{11}\]  
where \(dG=V_{0}\rho_{0}lnr_{0}d\phi\) is the elementary mass flow of air through the annular nozzle and \(dF_{f}\) is the elemental force of the viscous friction of air flow in the end surfaces of the grip and the object. The differential of the second order from the frictional force \(d^{2}F_{f}\)is defined as the double product of the friction stress \(\tau_{r}\) on the elementary area \(dS=dr\cdot r\cdot d\phi\).