## 4 Bearing with Longitudinal Microgrooves
### Modeling the Effect of the Depth and Width of the Microgroove on Circumferential Crossflows  
Let us perform a one-dimensional simulation of the compressed air flow between two parallel plates with a unit gap, one of which has a depression of width \(L_{D}\) and depth \(D\). The injection (supply) pressure \(P_{s}\) is applied to the gap inlet; the pressure at the inlet to the depression is equal to \(P_{1}\); at the outlet the pressure is \(P_{2}\), and at the exit from the gap, the pressure is \(P_{a}=1\) (Figure 5). The purpose of the study is to determine at what depth and width of the depression at its edges, and therefore throughout its entire length, the pressure practically does not change, which is evidence that there is nearly no resistance to airflow in the depression. This result will make it possible to determine at what depth and width of the microgroove of the bearing under study it can be assumed that the pressure upon it is constant and equal to the pressure at the outlet of the nearest diaphragm [30]. Figure 4: Unwrapping of the supporting surface with microgrooves. Based on (9), with a constant clearance (\(D\) = 0), the pressure can be found by the formula  
\[P(Z)=\sqrt{(P_{s}^{2}-P_{a}^{2})\frac{L-Z}{L}+P_{a}^{2}}. \tag{62}\]  
If the depression is located in the middle of the gap and \(D\) > 0, then the local coordinate systems are  
\[P(Z)=\left\{ \begin{array}{ll}\sqrt{(P_{s}^{2}-P_{1}^{2})\frac{L_{I}-Z}{L_{H}} +P_{1}^{2}},&0\leq Z\leq L_{H},\\ \sqrt{(P_{1}^{2}-P_{2}^{2})\frac{L_{D}-Z}{L_{D}}+P_{2}^{2}},&0\leq Z\leq L_{D},\\ \sqrt{(P_{2}^{2}-P_{a}^{2})\frac{L_{I}-Z}{L_{H}}+P_{a}^{2}},&0\leq Z\leq L_{H},\end{array} \right. \tag{63}\]  
where \(L_{H}=\frac{1}{2}(L-L_{D})\). Based on (62) and (63), we find the pressures \(P_{1}\) and \(P_{2}\) from the balance equations for the flow rates at the edges of the depression  
\[\left\{ \begin{array}{ll}\frac{P_{s}^{2}-P_{1}^{2}}{L_{H}}=(1+D)^{3}\frac{P_ {1}^{2}-P_{2}^{2}}{L_{D}},\\ \frac{P_{2}^{2}-P_{2}^{2}}{L_{H}}=(1+D)^{3}\frac{P_{1}^{2}-P_{2}^{2}}{L_{D}}. \end{array} \right. \tag{64}\]  
Having solved the system of the equation in (63), we obtain  
\[\left\{ \begin{array}{ll}P_{1}=\sqrt{\frac{P_{s}^{2}(B+1)+BP_{a}^{2}}{(B+1)^{ 2}-B^{2}}},\\ P_{2}=\sqrt{\frac{P_{s}^{2}(B+1)+BP_{s}^{2}}{(B+1)^{2}-B^{2}}},\end{array} \right. \tag{65}\]  
where \(B=\frac{L_{H}}{L_{D}}(1+D)^{3}\). The ratio of the pressure difference at the edges of the step, calculated by Formula (62) at \(D\) = 0 and at \(D\) \(\geq\) 0 and calculated by formulas (64) at \(D\) = \(L_{D}\), is  
\[\Delta=\frac{P(L_{H})-P(L_{H}+L_{D})}{P_{1}-P_{2}}\approx(1+D)^{3}. \tag{66}\]  
From Formula (65), it follows that the drop in resistance to airflow through the depression is proportional to the cube of the sum of the gap value and the size of the depression. At \(D\) = 1.15, the resistance drops by an order of magnitude, and, at \(D\) = 3.64, it drops by two orders of magnitude; therefore, we can assume that at \(D\) \(\geq\) 4, the resistance to the airflow in the depression and, consequently, in the microgrooves practically disappears. This allows us to conclude that if the depth of the microgroove is four times or more than the thickness of the lubricating gap, then the pressure in the microgroove can be approximately considered constant and equal to the air pressure at the outlet of the nearest annular diaphragm. Figure 5: Scheme of compressed air flow in a constant gap with a depression. At the same time, it should be borne in mind that the presence of air volumes in the microgrooves can lead to a deterioration in the dynamic characteristics of the bearing. It is known that in order to maintain stability, it is necessary that additional volumes of air in contact with the bearing gap do not exceed its volume [17]. Let us make the necessary calculations. The volumes of the bearing gap and the microgrooves, respectively, are equal \(v_{h}=4\pi v_{0}lh_{0}\) and \(v_{g}=2n_{g}(l-1_{1})\delta^{2}\), where \(\delta\) is the depth and width of the microgrooves.