Free Fall of a Sphere in a Partially Lubricated Cylinder

Abstract

The entrainment of lubricant at the entrance of a lubrication zone, such as that of a partially starved slider bearing, is analyzed in a closed system using the method of matched asymptotic expansions. A sphere falling together with a small lens of lubricant in a closely fitting tube is shown to fall under gravity at a speed V = (Mg − Fc)[(RC − RS)/RC]/(16π2μRC), where M denotes the total mass of the system, sphere plus lubricant, g the acceleration of gravity, F c the differential contact force, μ the viscosity of the lubricant, and R C and R S the radii of the tube and the sphere, respectively. Potential biological applications and experimental verification are discussed.