# Novel Numerical Analysis of a Self-Circulating Bearing
## 2 Methods  
A self-circulating bearing utilizes the natural pressure variation within the bearing clearance to drive fluid through the porous bushing and into the passive reservoir. This process is reversed in the divergent portion of the bearing, where fluid flows into the bearing clearance (Figure 1). This recirculation prevents the need for an external pumping system and serves to prevent vaporous cavitation as the fluid flows through the natural heat sink of the porous bushing. The focus of the results presented here is to emphasize the improvements achieved through a closed-form solution for the three-dimensional bushing. While there are many studies that could be performed with the model, the trends for this type of bearing, computational time saved, and the improved accuracy for the pressure distribution within the bushing will be the focus. The fluid film in the journal bearing, as shown in Figure 1, was modeled using the Reynolds equation [22]. \[\frac{\partial}{\partial\theta}\left(\frac{\rho h^{3}}{12\mu}\frac{\partial \mathrm{P_{b}}}{\partial\theta}\right)+\frac{\partial}{\partial z}\left(\frac {\rho h^{3}}{12\mu}\frac{\partial\mathrm{P_{b}}}{\partial z}\right)=\frac{ \partial(\rho h)}{\partial t}+\frac{\partial}{\partial\theta}\left(\frac{ \rho hU}{2}\right)+V_{inj} \tag{1}\]  
The bearing eccentricity ratio--the bearing eccentricity divided by the clearance of the bearing--describes the displacement of the journal's axis of rotation relative to the bearing's center. This controls the fluid film thickness, \(h\), as expressed in the following:  
\[\varepsilon=\mathrm{e/c} \tag{2}\]  
\[h=\mathrm{c}\left(1+\varepsilon\,\cos\theta\right) \tag{3}\]  
The permeability of the porous bushing, \(K\), and the pressure difference within the porous bushing radially were used in [PERSON]'s law to represent the injection velocity from the porous bushing into the bearing clearance. \[V_{inj}=-K\frac{dP}{dr} \tag{4}\]  
The working form of the Reynolds equation used Equation (4) to define the injection velocity. \[\frac{\partial}{\partial\theta}\left(\frac{\rho h^{3}}{12\mu}\frac{\partial \mathrm{P_{b}}}{\partial\theta}\right)+\frac{\partial}{\partial z}\left(\frac {\rho h^{3}}{12\mu}\frac{\partial\mathrm{P_{b}}}{\partial z}\right)=\frac{ \partial(\rho h)}{\partial t}+\frac{\partial}{\partial\theta}\left(\frac{ \rho hU}{2}\right)+\rho\frac{dP}{dr}\frac{K}{\mu_{f}} \tag{5}\]  
The pressure gradient in the porous bearing at the fluid-bushing interface was computed using a second order forward derivative, with \(R_{i}\leq r\leq R_{o}\), as in Figure 2b. \[\frac{dP}{dr}=\frac{-3P(1,\theta,z)+4P(2,\theta,z)-P(3,\theta,z)}{2\Delta r} \tag{6}\]  
where \(P(1,\theta,z)\) is the pressure at axial and circumferential locations on the porous medium-fluid film boundary; referenced as the bearing pressure, \(P_{b}\), below. This contrasts the numerical simplifications of the previous works that assumed the porous medium had a linear pressure distribution, where the pressure was calculated as the linear slope between the inner and outer radius of the porous medium [23]. The calculation of the pressure slope at each circumferential location improved the accuracy of the injection velocity (Equation (4)). The grid used within the porous bushing was verified using a convergence study on denser grids in the radial direction. Periodic boundary conditions were used in the circumferential direction while the axial ends of the bearing were exposed to atmospheric pressure. The boundary between the porous media and the fluid film was a no-slip boundary condition, with the pressures in the fluid film equal to the pressure in the porous media at the boundary.