A BOUNDARY INTEGRAL METHOD FOR COMPUTING THE FORCES OF MOVING BEADS IN A THREE-DIMENSIONAL LINEAR VISCOELASTIC FLOW

Computing the forces acting on particles in fluids is fundamental to understanding particle dynamics and interactions. In this thesis, we study the dynamics of a two-particle system in a three-dimensional linear viscoelastic flow. Using a correspondence principle between unsteady Stokes flow and viscoelastic flow, we reformulate the problem and derive a boundary integral formulation that solves the Brinkman’s equation in the Fourier domain. We show that computational costs can be reduced by carefully eliminating the double-layer potential, and that a unique solution can be obtained by desingularizing the equation. We develop a highly accurate numerical integration scheme to evaluate the resulting boundary integrals. We solve the backward problem by making use of our numerical integration scheme, variable transformations, generalized minimum residual (GMRES) method, and spherical harmonic interpolations. In particular, spherical harmonic interpolations ensure that this numerical scheme is of high accuracy. Our method also has the advantage of working for both unsteady Stokes and linear viscoelastic flow by appropriately adjusting the oscillation frequency. Our numerical results are in agreement with the exact solution for a single-particle system, as well as the asymptotic solution for large particle separation in the two-particle system. Last, we analyze the numerical results for high oscillation frequencies and small particle separation. Our numerical method is shown to only depend on the frequency parameter and the distance between the particles. We find that for high frequencies, the forces on the particles behave differently for unsteady Stokes and linear viscoelastic flows