(255a) The Hydrodynamic Force on a Slender Particle Executing Oscillatory Translation in Unsteady Stokes Flow

Asymptotic approximations are derived for the hydrodynamic force on a rigid, axisymmetric slender particle executing longitudinal or transverse oscillations in unsteady Stokes flow. A slender particle has an aspect ratio ε = a/L ≪ 1, where L is the half-length of the particle, and a is its characteristic cross-sectional width. The frequency of the oscillations is parameterized by the complex quantity λ² = −iL²Ï‰/ν, where ν is kinematic viscosity and ω is particle oscillation frequency, and i = √-1. Asymptotic approximations for the force are obtained in three frequency regimes: (i) low frequency, ε|λ| ≪ 1; (ii) moderate frequency, ε|λ| ∼ O(1); and (iii) high frequency, ε|λ| ≫ 1. Physical interpretations of the force in each regime are made and compared between the longitudinal and transverse oscillation cases. Our asymptotic predictions are in good agreement with the numerically computed frequency-dependent force on a prolate spheroid (ε = 0.1) for longitudinal and transverse oscillations by Lawrence and Weinbaum (J. Fluid Mech, 1988) and Pozrikidis (Phys. Fluids 1989), respectively.