A boundary integral method for motion of particles in unsteady Stokes and linear viscoelastic flows

In this paper, we develop a boundary integral method (BIM) for interacting particles in unsteady Stokes flow or linear viscoelastic (LVE) flow. The idea is to exploit a correspondence principle between them so a BIM can be established in the Fourier domain. Since the unsteady Stokes equation in the frequency domain is analogous to Brinkman equation in the time domain, our method can also be used for flow through porous media. In addition to dimension reduction vested in a boundary integral method, our formulation further reduces the computational cost by eliminating double-layer integrals. To evaluate the single-layer integrals more efficiently, we develop a hybrid numerical integration scheme based on kernel decompositions. The resulting method is third-order accurate. We first compare our numerical results with a known analytic solution for motion of one particle, and then apply the method to motion of two particles. Accurately capturing the hydrodynamic interaction of two particles in purely viscous or viscoelastic flow is of fundamental importance for studying two-particle microrheology and binding kinetics of particles in those flows. We compare our numerical results with an existing asymptotic solution for motion of two particles, which was derived for the case of large separation of the particles, and document its accuracy when the two particles come close to each other.