Inertial migration of a spherical particle in laminar square channel flows from low to high Reynolds numbers

The lateral migration properties of a rigid spherical particle suspended in a pressure-driven flow through channels with square cross-sections were investigated numerically, in the range of Reynolds numbers (Re) from 20 to 1000. The flow field around the particle was computed by the immersed boundary method to calculate the lateral forces exerted on the particle and its trajectories, starting from various initial positions. The numerical simulation showed that eight equilibrium positions of the particle are present at the centres of the channel faces and near the corners of the channel cross-section. The equilibrium positions at the centres of the channel faces are always stable, whereas the equilibrium positions at the corners are unstable until Re exceeds a certain critical value, Re_c. At Re ≈ Re_c, additional equilibrium positions appear on a heteroclinic orbit that joins the channel face and corner equilibrium positions, and the lateral forces along the heteroclinic orbit are very small. As Re increases, the channel face equilibrium positions are shifted towards the channel wall at first, and then shifted away from the channel wall. The channel corner equilibrium positions exhibit a monotonic shift towards the channel corner with increasing Re. Migration behaviours of the particle in the cross-section are also predicted for various values of Re. These numerical results account for the experimental observations of particle distributions in the cross-section of micro and millimetre scale channels, including the characteristic alignment and focusing of the particles, the absence of the corner equilibrium positions at low Re and the progressive shift of the equilibrium positions with Re.