The attractive feature of the singularity method for steady Stokes flows is that the hydrodynamic forces acting on the particle can be calculated by the total strength of distributed singularities. For unsteady Stokes flows, however we have to derive hydrodynamic forces acting on a solid body in terms of the strengths of both unsteady Stokeslets as well as unsteady potential dipoles if mass and force sources are both taken into consideration. Since the hydrodynamic force formulation results in a Volterra integral equation of the first kind, and the strengths are numerically approximated by means of the Lubich convolution quadrature method (CQM) in this study. As far as the numerical solutions of time-domain integral formulations of the unsteady Stokes equations are concerned, this paper requires only the Laplace-domain instead of the time- domain fundamental solutions of the governing equations. The stability and accuracy of the proposed method are verified through some well selected numerical examples. In total we include two examples presenting the accuracy of Lubich CQM, and another two examples for calculating general hydrody-namic forces of a sphere in oscillating or non-oscillating unsteady Stokes flows. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions instead to limit to the oscillating flow assumption.

A modified image method is presented for obtaining the solutions of the fundamental singularities in the neighborhood of a plane interface between two semi-infinite, immiscible, and incompressible viscous fluids. The fundamental singularities considered are the stokeslet, rotlet, stresslet, stokes-doublet, source, and source-doublet. The Galerkin vector function representation introduced reduces the complexity of the expressions for the solutions. Moreover, the physical meaning of each solution is clearly identified by these new expressions.