A singularity method for unsteady linearized flow

Oscillatory viscous flow in the presence of particles is considered. The flow is represented in terms of fundamental solutions to the governing unsteady Stokes equation including the oscillating Stokeslet and dipole. It is shown that the force on a particle in arbitrary oscillatory flow is proportional to the total strength of the distributed oscillating Stokeslets necessary to represent the flow. Faxen’s laws for solid and fluid particles are derived in terms of singularity representations and slender-body theory is discussed. The theoretical results are applied in studies of the translational and rotational oscillations of a solid sphere, the translational oscillation of a spherical drop, and the axial and transverse oscillation of a prolate spheroid. It is demonstrated that a spherical drop convected in a uniform streaming flow or moving under the influence of a body force, maintains its shape, independently of surface tension. The force F on a prolate spheroid executing longitudinal or transverse oscillations is computed in an extended range of frequencies ω. The results verify the accuracy of the correlation F(ω) proposed by Lawrence and Weinbaum [J. Fluid Mech. 189, 463 (1988)].