High-Frequency Oscillations of a Sphere in a Viscous Fluid near a Rigid Plane

High-frequency oscillations of a rigid sphere in an incompressible viscous fluid moving normal to a rigid plane are considered when the ratio of minimum clearance to sphere radius is small. Asymptotic expansions are constructed that permit an analytical estimate of the force acting on the sphere as a result of its motion. An inner expansion, valid in the neighborhood of the minimum gap, reflects the dominance of viscous effects and fluid inertia. An outer expansion, valid outside the gap, reflects the dominance of fluid inertia with a correction for an oscillating viscous boundary layer. The results are applied to the hydrodynamics of the tapping mode of an atomic force microscope and to the dynamic calibration of its cantilevers.