## 3 Theoretical Basis for the Analysis of Self-Acting Gas Bearings  
The primary task associated with the modeling and calculations of gas bearings is to determine the distribution of gas pressure in the lubrication gap and the resulting reaction force of the gas film against the external load, i.e., the load capacity of the bearing. Due to the low dynamic viscosity coefficient of gases and, therefore, the small amount of energy dissipated in the lubrication gap, the assumption of isothermal flow is usually made in the calculations of gas bearings. This assumption allows the flow to be described by a system of three equations:  
* Navier-Stokes equations;
* Continuity equations;
* Isotherm equations. To solve this system of equations, additional simplifications are introduced, including the following:  
Figure 1: (a) Geometrical dimensions of the analyzed hybrid bearing supplied obliquely. (b) Coordinate system describing the position of the journal in the bearing. * The film thickness \(c\) is assumed to be small relative to the bearing radius \(R\), i.e., \[c/R=\text{O}\Big{(}10^{-3}\Big{)}\ \ll\ 1\] This assumption is typically employed in gas bearing analysis and is reasonable for most cases. However, when the film thickness is not sufficiently small, the results may be less accurate. * The Taylor number, \(Ta\), is less than 40. \[Ta=Re^{2}\sqrt{\frac{c}{R}}\ <\ 40,\] (1) where the Reynolds number is expressed by the following equation: \[Re=\frac{c\omega Re_{a}}{\mu}\] This assumption is also used in practice, but the limit for the Taylor number can vary depending on the specific operating conditions of the bearing. * In the Navier-Stokes equations, inertial forces are neglected. This simplification is used to simplify the equations. In some cases, inertial forces can have a significant impact on the results, particularly at higher rotational speeds. * The flow is assumed to be laminar [20]. In reality, the flow can be laminar, transitional, or turbulent, depending on the operating conditions of the bearing. By assuming laminar flow, we ignore the phenomena associated with turbulent flow, leading to simpler equations. However, for some gas bearings, especially at higher rotational speeds, turbulence effects can be significant. Under the above assumptions, the pressure distribution in a gas film is described by the Reynolds equation:  
\[-\frac{\partial}{R\partial\theta}\left(\frac{\rho h^{3}}{\mu}\frac{\partial p }{R\partial\theta}\right)-\frac{\partial}{\partial z}\left(\frac{\rho h^{3}}{ \mu}\frac{\partial p}{\partial z}\right)+12\frac{\partial}{\partial t}(\rho h )+6\omega\frac{\partial}{\partial\theta}\left(\rho h\right)=\frac{d\dot{m}_{ k}}{Rd\theta dz} \tag{2}\]  
The right-hand side of the equation corresponds to additional sources of mass in the form of gas supplied through the feed holes. In the absence of additional mass sources (\(d\dot{m}_{k}=0\)), Equation (2) takes the following form:  
\[-\frac{\partial}{R\partial\theta}\left(\frac{\rho h^{3}}{\mu}\frac{\partial p }{R\partial\theta}\right)-\frac{\partial}{\partial z}\left(\frac{\rho h^{3}}{ \mu}\frac{\partial p}{\partial z}\right)+12\frac{\partial}{\partial t}(\rho h )+6\omega\frac{\partial}{\partial\theta}\left(\rho h\right)=0 \tag{3}\]  
As an analytical solution to the Reynolds equation is not possible, the pressure distributions in the gas film of the bearings must be determined numerically. Usually, the finite element method (FEM), the finite difference method (FDM), or computational fluid dynamics (CFD) based on the finite volume method are used. Thanks to the finite element method, calculations are possible even in the case of irregular geometries of externally pressurized bearings. The finite difference method (FDM), when applied to bearing calculations, allows pressure distributions, load capacity, stiffness, and volumetric flow rate to be determined.