The slow motion of a sphere through a viscous fluid towards a plane surface. II - Small gap widths, including inertial effects.

Abstract Singular perturbation techniques are employed to calculate the hydrodynamic force experienced by a sphere moving, at small Reynolds numbers, perpendicular to a solid plane wall bounding a semi-infinite viscous fluid, for the limiting case where the gap width between the sphere and plane tends to zero. Two distinct, but related, analyses of the problem are presented. In the first analysis, the exact bipolar-coordinate expression for the force given independently by M aude [1] and B renner [2] for the quasistatic Stokes flow case is expanded via a novel asymptotic procedure. The second analysis, which is somewhat more general in scope, provides a perturbation solution of the unsteady Navier—Stokes equations for a more general axisymmetric particle than a sphere, taking accounts of the finiteness of the Reynolds number. When the Reynolds number is taken into account, the force on the particle differs, according as it moves towards or away from the wall. Using the techniques developed in the first portion of this paper, formulas are derived for the Stokes couple required to maintain the symmetrical rotation of a dumbbell in an unbounded fluid, as well as the couples required to maintain the symmetrical rotation of a sphere touching a rigid plane wall and touching a planar free surface.