A boundary integral method for computing the dynamics of rigid particles in unsteady Stokes and viscoelastic medium

In this paper, we study the dynamics of rigid particles in either viscoelastic flow or unsteady Stokes flow in 3D. We start with the Cauchy momentum equation in time domain, and apply the Fourier transform to recast the problem in frequency domain as a Brinkman equation with imaginary coefficients. The Fourier space equations are reformulated in terms of boundary integrals. This formulation allows us to compute the velocity of the particles if the force is given (forward problem), and to compute the force when the velocity is given (inverse problem). We then propose a special mapping scheme such that the singularity of the surface integrals can be removed and a higher order quadrature can be established. Using a two-particle system, we demonstrate the order of convergence of our algorithms and validate our numerical results with known analytic or asymptotic solutions. Numerical experiments reveal that our method is formally high-order accurate and able to tackle problems with closely packed particles.