The inertial lift on a spherical particle settling in a horizontal viscous flow through a vertical slot

The inertial lift force exerted on a small rigid sphere settling due to gravity in a horizontal channel flow between vertical walls is investigated. The method of matched asymptotic expansions is used to obtain solutions for the disturbance flow on the length scales of the particle radius and the channel width (inner and outer regions, respectively). The channel Reynolds number is finite, while the particle Reynolds numbers that are based on the slip velocity and the mean shear rate are small. The inner flow is described by the linear Stokes equations. The outer problem is governed by the linear Oseen-like equations with the particle effect approximated by a point force. The outer equations are solved numerically using the two-dimensional Fourier transform of the disturbance velocity field. The lift coefficient is evaluated as a function of governing dimensionless parameters: the particle coordinate across the channel, the channel Reynolds number, and the slip parameter. The particle always migrates away from the walls, with an equilibrium position being on the channel centerline. Close to the walls, the lift coefficient is the same regardless of the slip velocity and the channel Reynolds number. At large channel Reynolds numbers, a local maximum of the migration velocity forms near the channel centerline due to a combined effect of the slip, the linear shear, and the curvature of the undisturbed velocity profile. The results obtained are extended to the case when the drag on a particle has components both parallel and perpendicular to the undisturbed flow. One of primary applications of the results is modeling of the cross-flow migration of settling particles during particle transport in a hydraulic fracture.