Orientation dynamics of a slender particle in simple shear flow of a viscoelastic fluid: Theory and direct numerical simulations

Previous experiments indicate that viscoelasticity changes the initial condition dependent orientation dynamics of a slender particle in a Newtonian Stokes flow (Jeffery orbits) in myriad ways. Previous theories suggest only one type of motion due to viscoelasticity, where the particle spirals towards the vorticity axis for all initial conditions. Using regular perturbation in polymer concentration and a generalized reciprocal theorem, we construct a theory to explain the effect of viscoelasticity on the motion of a large aspect ratio prolate spheroid in an Oldroyd-B viscoelastic fluid. The orientation trajectories from this theory show a better qualitatively match with the previous experimental observations. Depending on the polymer concentration, relaxation time, and the particle aspect ratio, the particle may end in a small periodic orbit close to the vorticity axis or obtain a stable orientation near the flow direction. A particle near the flow-gradient plane may either spiral or drift monotonically along the flow-vorticity plane. We test our theory and investigate higher polymer concentration effects using a novel finite-difference numerical solver written in prolate spheroidal coordinates, which exactly models the particle surface as one of the coordinate surfaces.