## 4 Bearing with Longitudinal Microgrooves
### Ideal Air Flow Pattern without circumferential Crossflows
#### 4.4.2 Non-Crossflowing Model for Arbitrary Eccentricities  
In the absence of crossflows, the pressure and air flow rate in the lubricating gap, by analogy with (16) and (23), are determined by the formulas  
\[P(Z)=\left\{\begin{array}{cc}\sqrt{(P_{k}^{2}-1)\frac{L_{1}-Z}{L_{1}}+1},& 0\leq Z\leq L_{1},\\ P_{k},&0\leq Z\leq L_{2},\end{array}\right.\]  
Figure 8: Dependences of the compliance \(K_{e}\) of a lightly loaded bearing on the length \(L\) at different values of injection pressure \(P_{s},\lambda_{1}=0.2\).  
Formulas (33) and (34) remain unchanged.  
As before, we will divide the arc \(\varphi\in[0,\pi]\) into \(m\) parts. Let us set \(\varepsilon\), and for \(\varphi_{j}=\frac{\pi j}{m}\)\((j=0,1,\ldots,m)\), calculate \(H\). Next, for each \(j\), we solve the system of the final two nonlinear equations in (35), representing the balance of flow rates in the SECT as  
\[\left\{\begin{array}{l}\frac{H_{j}^{2}}{L_{1}}\bigg{[}\Big{(}P_{k}^{j}\Big{)} ^{2}-1\bigg{]}=A_{k}\operatorname{Brn}(P_{p}^{j},P_{k}^{j}),\\ A_{p}\operatorname{Brn}(P_{s},P_{p}^{j})=A_{k}H_{j}\operatorname{Brn}(P_{p}^{ j},P_{k}^{j})\end{array}\right.\]  
relative to unknown pressures.  
Having evaluated the pressure vector at the outlet of the annular diaphragms \(P_{k}=\big{(}P_{k}^{0},P_{k}^{1},\ldots,P_{k}^{m}\big{)}\), we found the integral of the bearing capacity by the Simpson quadrature formula  
\[W=\frac{2}{\pi}\!\!\int\limits_{0}^{2\pi}\!\!\int\limits_{0}^{L}\left(P-1 \right)\cos\varphi dZd\varphi=\!\frac{2}{3m}\!\!\sum\limits_{j=0}^{m}s_{j} \cos\varphi_{j}\!\!\left[\frac{2L_{1}\big{(}P_{k}^{3}-1\big{)}}{3\big{(}P_{k} ^{2}-1\big{)}}+L_{2}(P_{k}-1)\right]_{j}, \tag{71}\]  
where \(s_{j}\) are the coefficients of the Simpson formula [29].  
Figure 9 shows the graphs of the dependences of the bearing capacity \(W\) on the eccentricity \(\varepsilon\) calculated by Formula (71) for various values of the length \(L\) for \(P_{s}=5\), \(\lambda_{1}=0.2\), \(\chi=0.45\), \(\zeta=0.15\), and \(n_{g}\geq 12\). The graphs are similar to Figure 2.  
Position \(a\) in the figure indicates the maximum value of the bearing capacity \(W\) of a conventional bearing of length \(L=1\), and position \(b\) indicates the maximum value of \(W\) curves for \(L>1\) (these curves are shown in Figure 2). With \(L=1\), the maximum bearing capacity of a bearing with microgrooves is 1.5 times greater than that of a conventional bearing; with \(L=1.5\), it is 1.8 times greater; with \(L=2\), it is 2.3 times greater; and with \(L=4\), it is 4.5 times greater. The same estimates take place for small and moderate values of eccentricity.  
The data obtained show that the use of longitudinal microgrooves in the inter-row zone at \(L\geq 1\) makes it possible to reduce the compliance by a factor of 1.5 and increase the bearing capacity. Moreover, the larger the relative length of the bearing, the better the performance. Therefore, the use of longitudinal microgrooves is an effective way to improve the force characteristics of a radial aerostatic bearing significantly.  
Figure 9: Dependences of the bearing capacity \(W\) of the bearing with microgrooves on the eccentricity \(\varepsilon\) at different values of length \(L\).