Oscillatory dynamics of three-dimensional rigid particles in unsteady Stokes and linear viscoelastic fluids

We present a numerical study of oscillating three-dimensional rigid particles in both linear viscoelastic and viscous Newtonian fluids. Based on the correspondence principle between unsteady Stokes flow and linear viscoelastic fluids, we establish a unified integral formulation in the frequency domain, which is solved using an accurate boundary integral method. For prescribed oscillatory particle motions, we compute the normal and shear stresses on the particle surfaces, as well as the flow fields in their vicinity. Both axisymmetric and non-axisymmetric motions are investigated for a pair of rigid particles. The flow characteristics are examined under various conditions, including different Weissenberg and frequency numbers, particle separation distances, and particle sizes, highlighting the complexity of stress distributions on the particle surfaces. We observe that the shape of the stress profiles remains unchanged in the case of unsteady Stokes flow as the oscillation frequency changes, whereas in the case of viscoelastic flow, the behavior is quite nonlinear. Traces of swirling motions are observed in the velocity fields near the particles. Remarkably, in three-dimensional flows, we find that for a fixed polar angle, the normal and shear stresses on spherical particle surfaces are proportional to the cosine of the azimuthal angle in spherical coordinates.