## 2 The Governing Equations for a PGB and Derived Operating Parameters  
The performance of PCBs is determined by their geometry (radial clearance \(c\), length \(L\), and diameter \(D=2R\)), the gas physical properties (density \(\rho\) and viscosity \(\mu\)), the magnitudes of supply pressure (\(p_{S}\)) and ambient pressure (\(p_{a}\)), the porous material permeability coefficient (\(\kappa\)) and the liner radial thickness (\(t_{p}\)), and the operating conditions of shaft angular speed (\(\Omega\) and precessional frequency (\(\omega\)).  
Figure 1 displays a schematic diagram of a cylindrical PGB with the journal spinning with speed \(\Omega\). The film thickness is \(h=(c+c_{X}\cos\theta+c_{Y}\sin\theta)\), where (\(c_{X}\), \(c_{Y}\)) are the components of the journal static eccentricity. Consider an ideal gas with density \(\rho=p/(R_{S}T)\), whose temperature \(T\) is constant. Then, the film pressure (\(p\)) is governed by ([PERSON], 1979) [5]. A detailed investigation on the influence of non-zero slip flow condition and tangential flow within the porous material is out of scope in the current analysis.  
\[\overrightarrow{\
abla}\cdot\left(\frac{p\ h^{3}}{12\mu}\overrightarrow{ \
abla}p\right)=\frac{\Omega R}{2}\frac{\partial(ph)}{\partial x}+\frac{ \partial(ph)}{\partial t}+\frac{\kappa}{2\mu\ t_{p}}(p^{2}-p_{S}^{2}) \tag{1}\]  
where \(\overrightarrow{\
abla}\) is the vector gradient operator, \(\mu\) is the gas viscosity, and \(p_{S}\) is the pressure of the pressurized gas supplied through the porous layer. The field \(p\) is periodic around the bearing circumference,\(p_{(\theta,\ z,t)}=p_{(\theta+2\pi,z,t)}\), and on the axial ends of the bearing \(p_{(\theta,\ \frac{1}{2}L,t)}=p_{(\theta,-\ \frac{1}{2}L,t)}=p_{a}\).  
Note the supplied mass flow rate through the porous material follows [PERSON]'s law [6; 7; 8]:  
\[\dot{m}=\frac{\rho}{\mu}\frac{\kappa}{t_{p}}(p_{S}-p)\ \sim\frac{1}{2}\frac{1}{\mu\ R_{S}T}\frac{ \kappa}{t_{p}}\left(p_{S}^{2}-p^{2}\right) \tag{2}\]  
Flow unsteadiness, circumferential flow through the porous media, and inertial effects due to the volume in the porous material are not particularly important in modern commercial products such as carbon graphite [3; 9; 10].  
The PGB geometry, porous material properties, and operating conditions combine to produce three fundamental (dimensionless) parameters,  
\[\Lambda_{\kappa}=\frac{12\kappa}{t_{p}\ c}\left(\frac{R}{c}\right)^{2};\ \Lambda_{\Omega}=6\frac{\mu\ \Omega}{p_{S}}\left(\frac{R}{c}\right)^{2};\ \Lambda_{\omega}=12\frac{\mu\ \omega}{p_{S}}\left(\frac{R}{c}\right)^{2}, \tag{3}\]  
Figure 1: Schematic diagram of a cylindrical journal and bearing with porous material layer (not to scale). Coordinate system noted for reference.  
known as the flow feeding number, the speed number, and the frequency number, respectively. Clearly \(p_{S}>p_{a}\) for the pressurized gas to flow through both the porous layer and then through the film clearance to exit at the bearing sides. Most PGBs have a large slenderness ratio, \(L/D>1\), hence giving enough surface area for the pressure field to produce a significant bearing load.  
The \(\Lambda_{\Omega}\) aerodynamic effects due to shaft rotation dominate over aerostatic effects (\(-p_{S}\)). Similarly, squeeze film effects (high frequency) dominate for \(\Lambda_{\omega}>1\). Incidentally, the porosity parameter \(\Lambda_{x}\) relates the flow conductance across the porous layer (\(\kappa/\!\!\!\!/_{p}\)) to the conductance through the film, which is proportional to (\(c^{3}/R^{2}\)).