## 4 A Close Form Solution to the Flow and Dynamic Force Coefficients in a PGB
If the bearing clearance (\(c\)) is very large (as when there is no shaft inserted in the bearing), then \(p=p_{a}\) over the flow domain, and the supplied flow rate is the largest and equal to  
\[\dot{M}_{\max}\limits_{c\rightarrow\infty}=\frac{1}{2}\frac{\kappa}{\mu\ t_{ p}}\frac{\left(p_{S}^{2}-p_{a}^{2}\right)}{R_{S}T}\left(\pi\ D\ L\right) \tag{11}\]  
\[\frac{\dot{M}}{\dot{M}_{\max}\limits_{C_{r}\rightarrow\infty}}=\frac{\tanh \left(\gamma\frac{L}{D}\right)}{\left(\gamma\frac{L}{D}\right)}. \tag{12}\]  
The ratio of flows is only a function of the bearing geometry and the permeability coefficient (\(\kappa\)), since \(\left(\gamma\frac{L}{D}\right)=\sqrt{\frac{3\kappa}{l_{p}c}}\frac{L}{c}\). Equation (11), known as the flow coefficient for PGBs [6; 7], serves to estimate the permeability coefficient (\(\kappa\)) from measurements of the supplied flow (\(\dot{M}_{\max}\)) for various supply pressures, as will be shown later. Incidentally, the shear drag torque (\(T_{o}\)) and power loss (\(P_{loss}\)) for the laminar flow in a centered journal bearing equal  
\[\overset{\rightarrow}{\
abla}\cdot\left(\frac{p_{0}h_{0}}{12\mu}\overset{ \rightarrow}{\
abla}p_{0}\right)=\frac{\Omega R}{2}\frac{\partial(p_{0}h_{0}) }{\partial x}+\frac{\kappa}{2\mu\;t_{p}}\left({p_{0}}^{2}-p_{S}^{2}\right) \tag{5}\]0  
Note that for an air film, the drag friction coefficient \(f=T_{o}/(W^{*}\ R)<<1\). As the analysis considers small amplitude motions about the centered condition (\(e=0\)), then from \(p=p_{0}+\ (\delta e_{X}\ p_{X}+\ \delta e_{Y}p_{Y})e^{i\ \omega\ t}\), deduce  
\[\overset{\rightarrow}{\
abla}\cdot\left(\frac{p_{0}h_{0}}{12\mu}\overset{ \rightarrow}{\
abla}p_{0}\right)=\frac{\Omega R}{2}\frac{\partial(p_{0}h_{0}) }{\partial x}+\frac{\kappa}{2\mu\;t_{p}}\left({p_{0}}^{2}-p_{S}^{2}\right) \tag{5}\]1  
Hence, \(\psi_{X}=2p_{0}p_{X};\ \psi_{Y}=2p_{0}p_{Y}\). Presently, for small amplitude motions about \(e=0\), let  
\[\overset{\rightarrow}{\
abla}\cdot\left(\frac{p_{0}h_{0}}{12\mu}\overset{ \rightarrow}{\
abla}p_{0}\right)=\frac{\Omega R}{2}\frac{\partial(p_{0}h_{0}) }{\partial x}+\frac{\kappa}{2\mu\;t_{p}}\left({p_{0}}^{2}-p_{S}^{2}\right) \tag{5}\]2Then, write the first order Equation (6) for the perturbed pressure fields as  
\[\begin{array}{l}\frac{d^{2}}{d\,\overline{z}^{2}}\left[ \begin{array}{c}\psi_{X _{c}}\\ \psi_{X_{s}}\end{array} \right]-\mathbf{A}\left[ \begin{array}{c}\psi_{X_{c}}\\ \psi_{X_{s}}\end{array} \right]=\mathbf{C}_{1}\cosh(\gamma\overline{z})+\mathbf{ C}_{2};\\ \frac{d^{2}}{d\,\overline{z}^{2}}\left[ \begin{array}{c}\psi_{Y_{c}}\\ -\psi_{Y_{s}}\end{array} \right]-\mathbf{A}^{T}\left[ \begin{array}{c}\psi_{Y_ {c}}\\ -\psi_{Y_{s}}\end{array} \right]=\mathbf{C}_{1}\cosh(\gamma\overline{z})+\mathbf{ C}_{2}\\ \\ \mbox{where }\mathbf{A}=\left[ \begin{array}{cc}(1+\gamma^{2})+i \overline{\Lambda}_{\omega}&\overline{\Lambda}_{\Omega}\\ -\overline{\Lambda}_{\Omega}&(1+\gamma^{2})+i\overline{\Lambda}_{\omega}\end{array}  \right],\\ \mathbf{C}_{1}=\left[ \begin{array}{cc}i\,2\overline{\Lambda}_{\omega}-3\, \gamma^{2}\\ -2\overline{\Lambda}_{\Omega}\end{array} \right]\frac{\psi_{\Xi}}{c}\frac{1}{ \cosh\left(\gamma\,\frac{1}{D}\right)}\mbox{ and }\mathbf{C}_{2}=\left[ \begin{array}{c}i\,2\overline{\Lambda}_{\omega}\\ -2\overline{\Lambda}_{\Omega}\end{array} \right]\frac{p_{\Xi}^{2}}{c}\\ \\ \mbox{with }\psi_{as}=\left(p_{a}^{2}-p_{S}^{2}\right),\overline{\Lambda}_{ \Omega}=\frac{12\mu\,\Omega}{2\overline{p}_{0}}\left(\frac{R^{2}}{c^{2}}\right) ;\ \overline{\Lambda}_{\omega}=\frac{24\mu\,\omega}{2\overline{p}_{0}} \left(\frac{R^{2}}{c^{2}}\right) \tag{18}\]  
using average pressure \(\overline{p}_{0}=\frac{1}{L}J_{0}^{L}\,p_{0}\,dz\). The boundary conditions for the perturbed fields are homogeneous, i.e.,\(\psi_{X_{\left(\Xi=\pm\frac{1}{D}\right)}}=\psi_{Y_{\left(\Xi=\pm\frac{1}{D}\right)}}=0\). Then,  
\[\psi_{Xc}=\psi_{Y_{s}}=\,\,\psi_{X_{s}}=\psi_{Yc}=0\mbox{ at }\overline{z}=\pm\frac{L}{D}\;. \tag{19}\]  
Note that Equation (16) reveals that \(\psi_{Xc}=\psi_{Y_{s}},\,\,\psi_{X_{s}}=-\psi_{Yc}\).