Derivation of Reynolds Equation in Cylindrical Coordinates Applicable to Pin-on-Disk and CMP

Abstract

Traditional tribology references typically provide the cylindrical (or polar) Reynolds equation, which may not be applicable when entrainment velocities vary with radius and/or angle. However, entrainment velocities are known to vary with angle for some cases of pin-on-disk contact and chemical mechanical polishing (CMP). A form of Reynolds equation is derived in this manuscript from the Navier-Stokes equations without entrainment velocity assumptions. Two case studies, related to pin-on-disk and CMP, are presented and results from the derived form of Reynolds equation are compared with results from the traditionally used form. Pressure distributions obtained from the two forms of Reynolds equation varied greatly in magnitude and in pressure shape. Therefore, a new form of the cylindrical Reynolds equation derived in this manuscript is used when entrainment velocities are known to vary with radius or angle.