## 4 Bearing with Longitudinal Microgrooves
### Modeling Circumferential Crossflows in the Inter-Row Zone  
Let us simulate circular air flows in the inter-row zone and estimate their effect on the load capacity of the air gap in this part, depending on the number of microgrooves, which are located in a circle at the same distance from each other on the lines \(\varphi_{k}=\frac{2\pi(k-1)}{n_{g}}\), \(k=1,2,\ldots,n_{g}\). We will assume that the number of microgrooves \(n_{g}\) is an even number. Due to the symmetry of flows in both directions, we will consider the area as \(G=\{0\leq Z\leq L,0\leq\varphi\leq\pi\}\). We divide the area \(G\) into sections \(G_{k}=\{0\leq Z\leq L,\varphi_{k}\leq\varphi<\varphi_{k+1}\}\), \(k=1,2,\ldots,n_{h}\), where \(n_{h}=0.5n_{g}\) is the number of sections. Consider the boundary conditions for the square of the pressure in each section on the pressurization line and microgrooves  
\[\Pi(\varphi)=P_{k0}^{2}(1+\varepsilon\cos\varphi)^{2}. \tag{67}\]  
For the boundary \(Z=0\), due to the symmetry of the flow, we will use the second condition of the boundary value problem (3).  
We divide each section into \(n\times m\) parts and perform a finite-difference replacement of the differential equation of Problem (3). In this case, we obtain an algebraic boundary value problem for the system of Equation (50) with boundary conditions (46), (49), and (66). We solve the problem using the iteration method, using Formula (56) for the interior points. For the boundary points of the region, we use Formulas (66) and (59). The load capacity of the compressed air gap in the \(G\) region was found by summing the force reactions on the \(G_{k}\) segments  
\[W=\sum_{k=1}^{n_{h}}W_{k}.\]  
In the absence of circumferential flows, \(\Psi(Z,\varphi)=\Pi(\varphi)\), therefore  
\[W=\frac{2P_{k0}}{\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{L}(1+\varepsilon \cos\varphi)\cos\varphi dZd\varphi= P_{k0}L\varepsilon.\]  
Figure 6 shows the dependences of the bearing capacity \(W\) of the compressed air gap in the region \(G\) on its length \(L\) for a different number of microgrooves \(n_{g}\) at \(P_{k0}=3\), \(\varepsilon=0.75\)  
The graph shows that an increase in the number of grooves begins to noticeably affect the bearing capacity immediately. Already at \(n_{g}=4\), in comparison with the graph for \(n_{g}=2\), at which there are no barriers to crossflows, \(W\) increases by 11%. However, a significant jump in the growth of the characteristic is observed at \(\text{ng}=6\). For this number of microgrooves, \(W\) increases immediately by 39%. With a further increase in the number of microgrooves, \(W\) increases in a weaker manner. Thus, at \(n_{g}=8\), the growth is 48%, and at \(n_{g}\geq 12\), circumferential air crossflows practically cease, providing an overall efficiency of increasing the bearing capacity from the use of longitudinal microgrooves by 61%. With other parameters, the efficiency may differ from the given estimates and is in the range of 60-75%.  
The data provided testifies to the high efficiency of the use of longitudinal microgrooves to increase the bearing capacity of compressed air in the row-to-row sector of the bearing surface. If we assume that the number of microgrooves \(n_{g}=12\) is optimal, then the refined estimate of their volume in relation to the volume of the bearing gap will be only 3%. Such an insignificant increase in the volume of compressed gas adjacent to the bearing gap cannot contribute to the deterioration of the dynamics of the structure; therefore, the use of microgrooves to significantly improve the static characteristics of the bearing is quite justified.