The cross-stream migration of particles in viscoelastic pressure-driven flows has been widely studied and has an important influence on the rheology of suspensions and the properties of composite materials processed in a molten state. However, it is not clear from the previous literature which factors determine the magnitude and direction of the cross-stream velocity of a freely suspended particle or the cross-stream force on a particle constrained to a fixed cross-stream position. We will present complementary small Deborah number perturbation analysis and finite element simulations for a circular particle (or infinitely long cylinder) in a two-dimensional pressure-driven flow of an Oldroyd B fluid. A neutrally buoyant particle is found to migrate toward the center of the channel for all particle sizes and cross-stream positions. The migration of a small particle throughout most of the channel is determined by the coupled effects of the linear and quadratic variations of the imposed velocity. Closer to the wall and/or for larger particles, the hydrodynamic reflection of the particle with the wall contributes to the migration of the particle toward the center plane. Very close to the wall, the lubrication flow leads to a diverging non-Newtonian cross-stream force on the particle, although the Newtonian normal motion lubrication force maintains a small migration velocity. Finite element simulations confirm the accuracy of the asymptotic analysis for Deborah numbers less than about 0.5 and reveal the modifications of the migration force and fluid velocity with increasing Deborah number and polymer concentration.
A cylinder that is held at a fixed position experiences a cross-stream force directed toward the wall as a result of the coupled effects of the local shear flow and the relative translation of the fluid and cylinder. We compare our results for this case with the experimental study of Dhahir and Walters1. A freely suspended particle experiencing a body force parallel to the bounding walls can migrate toward the wall or toward the center plane depending on the relative value of the settling velocity and the center-line fluid velocity of the undisturbed flow.
1. S.A. Dhahir and K. Walters. On non-Newtonian flow past a cylinder in confined flow. J. Rheol., 33:781, 1989.