Lift force models for a spherical particle immersed in linear and quadratic shear flows

The lift force models for a particle in wall-bounded linear shear flow have been extensively investigated; however, the influence of the curvature of the velocity profile ( ${\textit{Sg}}$ ) on the lift force at finite slip Reynolds numbers ( ${\textit{Re}}$ ) remains unexplored. In the present work, direct numerical simulations (DNS) are performed to investigate the lift on a spherical particle in unbounded linear shear flow, single-wall-bounded linear shear flow and Poiseuille flow. Based on our DNS data, we first extend the existing unbounded and single-wall-bounded linear shear-slip-induced lift models to higher non-dimensional shear rates ( $|Sr|=2.5$ ) for $0.1\leqslant Re\leqslant 20$ . Based on the empirical model for Couette flow or the analytical model for unbounded Poiseuille flow, the lift models are then modified to account for the curvature effect of the parabolic velocity profile, which reduce to the linear shear-slip-induced lift models in the high ${\textit{Re}}$ and low ${\textit{Sg}}$ limits. We also modify the rotation-induced lift model of linear shear flow to account for the parabolic shear effect, which causes lift enhancement for the leading particle and lift attenuation for the lagging particle in the Poiseuille flow, compared to the linear shear case. This lift attenuation may give rise to an inverse Magnus force at low slip Reynolds numbers. In addition, the model for the particle free rotation rate for the Poiseuille flow is established by correcting the one for the linear shear flow, providing a more accurate prediction of the torque on the particle.