Note on the memory force on a slightly eccentric fluid spheroid in unsteady creeping flows

Abstract

The transient Stokes equations are solved inside and outside a slightly eccentric fluid spheroid, which oscillates about its zero-mean position in an unbounded viscous medium. By using techniques of regular perturbations, the internal and the external flow fields are determined, and a first approximation of the hydrodynamic force is derived. In addition to the classical perturbed forces: the quasisteady drag, the added-mass force and the history force, the unsteady drag contains an additional history term due mainly to the eccentricity of the particle. For viscous spheroids, its kernel is found to be different from that of the history force acting on spherical droplets. So, we examine here the effects of the eccentricity and the viscosity ratio on its magnitude, its limits for solid and gaseous spheroids, and its asymptotic behaviors at low and high frequencies.