## 4 A Close Form Solution to the Flow and Dynamic Force Coefficients in a PGB  
In reference to Figure 1, consider journal center motions with small amplitude (\(\delta\varepsilon_{X}\), \(\delta\varepsilon_{Y}\)) and frequency (\(\omega\)) about an equilibrium position (\(\varepsilon_{X0}\), \(\varepsilon_{Y0}\)). The film thickness (\(h\)) and pressure (\(p\)) fields equal the sum of zeroth-order fields (\(h_{0},p_{0}\)) and first-order or perturbed fields (\(h_{\sigma},p_{\sigma}\)) \(\mid\sigma=X,Y\),  
\[h=h_{0}+\delta\varepsilon_{X}\;h_{X}\;\dot{e}^{i\omega t}+\delta\varepsilon_{Y }\;h_{Y}\;\dot{e}^{i\omega t}\to p=p_{0}+\delta\varepsilon_{\sigma}\;p_{ \sigma}\;\dot{e}^{i\omega t},\sigma=X,\;Y \tag{4}\]  
where \(h_{X}=\cos(\theta)\) and \(h_{Y}=\sin(\theta)\). Above, \(\sigma=X\), \(Y\), \(i=\sqrt{-1}\). Substituting Equation (4) into the Reynolds Equation (1) produces the zero-order equation for \(p_{0}\)  
\[\overset{\rightarrow}{\
abla}\cdot\left(\frac{p_{0}h_{0}}{12\mu}\overset{ \rightarrow}{\
abla}p_{0}\right)=\frac{\Omega R}{2}\frac{\partial(p_{0}h_{0}) }{\partial x}+\frac{\kappa}{2\mu\;t_{p}}\left({p_{0}}^{2}-p_{S}^{2}\right) \tag{5}\]  
and the first-order equation  
\[\overset{\rightarrow}{\
abla}\cdot\left(\frac{h_{0}^{3}}{12\mu} \left[p_{\sigma}\overset{\rightarrow}{\
abla}p_{0}+p_{0}\overset{ \rightarrow}{\
abla}p_{\sigma}\right]\right) =\frac{\Omega R}{2}\frac{\partial(p_{0}h_{\sigma}+p_{0}h_{0})}{ \partial x}+i\;\omega\;(p_{0}h_{\sigma}+p_{\sigma}h_{0}) \tag{6}\] \[-\overset{\rightarrow}{\
abla}\cdot\left[\frac{p_{0}h_{0}}{4\mu }h_{\sigma}\overset{\rightarrow}{\
abla}\cdot p_{0}\right]+\frac{\kappa}{\mu\;t _{p}}p_{0}\;p_{\sigma}\]  
Following San Andres [26], the equations above are integrated in a typical finite element (FE) and assembled over the flow domain to produce sets of algebraic (nonlinear) equations for solution of the pressure fields; and subsequent calculation of the PGB reaction force, drag torque and power, and dynamic force coefficients. Details of the FE model are omitted for brevity. For operation at a centered condition, \(h_{0}=c\), Equation (5) reduces to  
\[\overset{\rightarrow}{\
abla}\cdot\left(\overset{\rightarrow}{\
abla}p_{0} ^{2}\right)=\frac{12\;\kappa}{c^{3}\;t_{p}}\left({p_{0}}^{2}-p_{S}^{2}\right) \tag{7}\]That dispenses with the effects of gas viscosity (\(\mu\)) and shaft speed (\(\Omega\)). Let \(\psi_{0}=\left(p_{0}^{2}-p_{S}^{2}\right)\), then Equation (7) reduces to the simple ordinary differential equation,  
\[\frac{d^{2}}{d\ \overline{z}^{2}}(\psi_{0})-\Lambda_{\kappa}\ \psi_{0}=0. \tag{8}\]  
Above, \(\overline{z}=\left(\frac{z}{\overline{R}}\right)\) with \(\gamma^{2}=\Lambda_{\kappa}=\frac{3}{l_{p}\ c}\left(\frac{D}{c}\right)^{2}\)as the feed flow parameter. The solution of Equation (8) with ambient pressure (\(p_{a}\)) at the bearing sides \(\overline{z}=\pm\left(\frac{L}{D}\right)\) gives  
\[\psi_{0(\overline{z})}=\psi_{0}|_{p}=p_{a}\frac{\cosh(\gamma\overline{z})}{ \cosh\left(\gamma\frac{L}{D}\right)}\ \rightarrow\ p_{0(\overline{z})}=\sqrt{p_{S}^{2}+\left(p_{a}^{2}-p_{S}^{2} \right)\frac{\cosh(\gamma\overline{z})}{\cosh\left(\gamma\frac{L}{D}\right)}}. \tag{9}\]  
The mass flow rate (\(\dot{M}\)) supplied to the bearing through the porous layer equals  
\[\dot{M}=\iint\limits_{\Omega}\ \dot{m}\ R\ d\theta\ dz=\gamma\left(\frac{\pi\ c^{ 3}}{6\ \mu}\right)\ \tanh\left(\gamma\frac{L}{D}\right)\ \frac{\left({p_{S}}^{2}-{p_{a}}^{2}\right)}{R_{S}T}. \tag{10}\]  
Note that \(\dot{M}\) is proportional to \(\sqrt{\frac{\kappa\ c^{3}}{l_{p}}}\), the average gas density \(\rho_{*}=\frac{1}{2}\frac{\left(p_{S}+p_{s}\right)}{R_{S}T}\), and the pressure difference \(\left(p_{S}-p_{a}\right)\).