## 4 Nonlinear Mathematical Model of the Bearing Stationary State
### Bearing Model with Slotted SECT  
Consider the mathematical model (1)-(5) and divide a segment \(\varphi\in[0,\pi]\) into even number \(m\) parts and a segment \(Z\in[0,L]\) into even number \(n\) parts. We transform the Reynolds Equation (1) and write it in the form  
\[U\frac{\partial\Psi}{\partial\varphi}+\frac{\partial^{2}\Psi}{\partial\varphi ^{2}}+\frac{\partial^{2}\Psi}{\partial Z^{2}}=0, \tag{36}\]  
where \(\Psi(Z,\varphi)=P^{2}(Z,\varphi)\), \(U(\varphi)=\frac{3H^{\prime}}{H}=\frac{3\sin\varphi}{1-\cos\varphi}\). Let us write Equation (36) in a finite-difference form [28,29]:  
\[U_{j}\frac{\Psi_{i}^{j+1}-\Psi_{i}^{j-1}}{2\
u_{\varphi}}+\frac{\Psi_{i}^{j+1 }-2\Psi_{i}^{j}+\Psi_{i}^{j-1}}{\
u_{\varphi}^{2}}+\frac{\Psi_{i+1}^{j}-2\Psi _{i}^{j}+\Psi_{i-1}^{j}}{\
u_{z}^{2}}=0, \tag{37}\]  
where \(\
u_{\varphi}=\frac{\pi}{m},\
u_{z}=\frac{L}{n}\) are the steps of integration over the variables \(\varphi\) and \(Z\), and \(j=1\), \(2\),..., \(m-1\); \(i=1,2,\ldots,\)\(k-1\), \(k+1\),..., \(n-1\) are the numbers of the nodal points for these variables (for the injection line \(i=k\), a separate equation is shown below). We write the first boundary condition (3) in the form [29]  
\[\frac{-\Psi_{2}^{j}+4\Psi_{1}^{j}-3\Psi_{0}^{j}}{2\
u_{z}}=0. \tag{38}\]  
The second boundary condition (3) for \(j=0\), \(2\),..., \(m\) is  
\[\Psi_{k}^{j}=\Big{(}P_{k}^{2}\Big{)}^{j}, \tag{39}\]  
where \(k=\Big{[}\frac{nL_{1}}{L}\Big{]}\) is the number of the point on the grid of the longitudinal axis, which corresponds to the location of the feed slot \(Z=L_{1}\). The last condition (4) for \(j=0\), \(1\),..., \(m\) is  
\[\Psi_{n}^{j}=1. \tag{40}\]  
The boundary conditions (5) for \(i=0\), \(1\),..., \(n\) can be written as  
\[\Psi_{i}^{1}=\Psi_{i}^{-1},\Psi_{i}^{m+1}=\Psi_{i}^{m-1}. \tag{41}\]  
We represent (37) in the form  
\[\big{(}b_{0}+a_{0}U_{j}\big{)}\Psi_{i}^{j+1}+c_{0}\Psi_{i}^{j}\big{(}b_{0}-a_{ 0}U_{j}\big{)}\Psi_{i}^{j-1}+\Psi_{i+1}^{j}+\Psi_{i-1}^{j}=0, \tag{42}\]where \(a_{0}=\frac{v_{2}^{2}}{2v_{\varphi}},b_{0}=\left(\frac{v_{2}}{v_{\varphi}}\right)^{ 2},c_{0}=-(1+b_{0})\). We write (41) in vector form  
\[\overline{\Psi}_{i+1}+A_{1}\overline{\Psi}_{i}+\overline{\Psi}_{i-1}=0, \tag{43}\]  
where  
\[\overline{\Psi}_{i}=\left[ \begin{array}{c}\Psi_{i}^{0}\\ \Psi_{i}^{1}\\ \ldots\\ \Psi_{i}^{m-1}\\ \Psi_{i}^{m}\end{array} \right]\]  
is a vector of the length \(m+1\), and  
\[A_{1}=\left[ \begin{array}{cccccc}c_{0}&2b_{0}&0&\ldots&0&0&0\\ (b_{0}-a_{0}U_{1})&c_{0}&(b_{0}+a_{0}U_{1})&\ldots&0&0&0\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ 0&0&0&\ldots&(b_{0}-a_{0}U_{m-1})&c_{0}&(b_{0}+a_{0}U_{m-1})\\ 0&0&0&\ldots&0&2b_{0}&c_{0}\end{array} \right] \tag{44}\]  
is a square tridiagonal matrix of size \((m+1)\times(m+1)\). Condition (38) can be written in vector form  
\[-\overline{\Psi}_{2}+4\overline{\Psi}_{1}-3\overline{\Psi}_{0}=0. \tag{45}\]  
Let us write a vector equation similar to (37) corresponding to the line \(i=k\) of the air supply to the carrier gap. To do this, we use the equations for the balance of flow rates (9), (12) and (13). Applying the well-known formulas of the quadratic order of accuracy to represent the first derivative of a function at the edges of the segment [29], we write the formulas for the flow rate in the carrier gap on the pressurization line  
\[ \begin{array}{c}Q_{h,j}=Q_{h1,j}-Q_{h2,j}=\\ =\frac{H_{h}^{3}}{2v_{2}}\Big{(}\Psi_{k+2}^{j}-4\Psi_{k+1}^{j}+6\Psi_{k}^{j}-4 \Psi_{k-1}^{j}+\Psi_{k-2}^{j}\Big{)}.\end{array}  \tag{46}\]  
For the flow rate Formulas (9), we obtain  
\[\left\{ \begin{array}{l}Q_{k,j}=A_{k}\Big{(}P_{p,j}^{2}-P_{k,j}^{2}\Big{)},\\ Q_{p,j}=A_{p}\Big{(}P_{s}^{2}-P_{p,j}^{2}\Big{)}.\end{array} \right. \tag{47}\]  
Using the third equation of System (7) and excluding \(P_{p,j}^{2}\), we get  
\[Q_{k,j}=A_{kp}\Big{(}P_{s}^{2}-P_{k,j}^{2}\Big{)}, \tag{48}\]  
where \(A_{kp}=\frac{A_{k}A_{p}}{A_{k}+A_{k}}\).