# Novel Numerical Analysis of a Self-Circulating Bearing
## 3 Results and Discussion  
The validity of the present computational solution was verified and is illustrated in Figure 4 across a range of eccentricity ratios. Verification was completed using air as the working fluid to prevent error from cavitation not included in this model. As in [PERSON]'s work [20], the ideal gas law was used with [PERSON]'s law governing the injection of air into the bearing from the porous medium. Excellent agreement was shown between the present work and that of [PERSON] using the geometry, characteristics, and fluid of the previous work [24]. The variance in results stems from the different axial boundary conditions used in the simulation. The current work used a Dirichlet boundary approach condition in which the edges of the bearing were equal to atmospheric pressure [19], while the referenced work had set the axial direction pressure gradient equal to zero. The previous work limited the feeding parameters that were explored. The current work expanded this range and revealed trends that were not previously investigated. For the remainder of the results, a finite length porous bearing with L/R = 1, c/r = 0.001, and h/D = 0.1 was analyzed at moderate speed (5000 rpm) and room temperature, 72 \({}^{\circ}\)F. Properties of VG-32 oil were used for the lubricant with a Gumbel cavitation model. The range of permeabilities resulted in a feeding parameter spanning 4.55 to 455, where each change in permeability resulted in an order of magnitude change in the feeding parameter (Equation (14)). The supply pressure surrounding the porous bearing had a constant pressure of \(4\times 10^{3}\) Pa. The density and viscosity were held constant throughout the analysis. For validation and results, the coupled solution converged to a pressure difference of less than 0.1% error (Equation (9)). Figure 5 demonstrates the improved accuracy of a three-dimensional representation of the pressure within the bushing, compared to a simplification of assuming a linear distribution within the porous bushing. The nodes represent the axial position of the pressure distribution, with node 2 being the closest to the axial edge of the bearing and node 10 being approximately 15% towards the bearing centerline. The use of the numerical derivative improved the representation of the porous bushing for approximately 30% of the bearing's length when both ends were accounted for. Figure 6 demonstrates the minimal effect over the range for bushing thickness. In contrast, the range of permeabilities has a significant effect on the injection velocity. This sets the stage for further discussions on the variation of these parameters and their effects  
Figure 4: Comparison of present and published results [24]. Figure 5: Pressure distribution within porous bushing (H = 0.2r). on pressure distribution within the fluid film, which drives bearing load capacity and stability. The lowest permeability would not be ideal for high-load applications, as there is minimal fluid circulation (excluding side leakage), making the fluid more susceptible to cavitation and instability. The lowest permeability demonstrates the least injection velocity and thus will behave most like a solid bearing. As demonstrated by Figure 7, the effect of changing permeability on the bearing load capacity becomes significant as the bearing eccentricity increases. With the lowest permeability, which would provide the most resistance to fluid traveling through the bushing, an eccentricity ratio of 0.9 was an example of the numerical instability caused by the rapid transfer of fluid between the bearing clearance and reservoir due to the large eccentricity. With high eccentricity, the bearing and bushing have a large offset which causes a large pressure differential between the converging and diverging sections. This pressure differential is what drives the fluid motion between the bearing clearance and reservoir; when the exchange occurs at too large of a rate, then numerical convergence cannot be achieved.