Lubrication of a Porous Bearing—Stokes’ Solution

Abstract

Coupling of flows induced by the rotation of an infinite cylinder in an eccentric cylindrical hole in a fluid-saturated porous space is investigated in the context of a coupled boundary-value problem in which the Stokes flow outside porous regions and the Darcy flow inside porous regions are connected by continuity requirements on the pressure and normal component of velocity. The configuration is used to model the effects of a thick porous bearing. The solution simplifies considerably in the Reynolds limit of small clearance, and compact approximations for the pressure distribution and other relevant physical variables are derived. It is shown that transverse pressure gradients in the lubricant which are normally neglected in the Reynolds limit do increase, but not significantly, as a result of bearing flow. It follows that candidate Reynolds’ equations may ordinarily ignore effects of transverse pressure gradients in the lubricant even when the bearing is porous. A principal effect of the porous flow on the coupled motion is a diminution of pressure differences which would develop if all solids were impermeable. Corresponding changes in the shear stress resultant, which is neglected relative to the pressure resultant in the impermeable Reynolds limit, can become dominant because of the diminished pressures which attend porous flow. For large eccentricity ratios, the shear resultant is negative, and the load capacity may fall to zero and even change sign.