The inertial lift on a small particle in a weak-shear parabolic flow

The lateral migration of a small spherical particle translating within a vertical channel flow with large channel Reynolds numbers Rc is investigated. The weak-shear case is studied when the ratio of the slip velocity to maximum velocity of the channel flow, Vs, is finite, while two other dimensionless groups, particle Reynolds number Rs and Rc−1, are asymptotically small. The disturbance flow at large distances from the sphere is governed by Oseen-like equations. The ratio of Oseen length to channel width is e=ls/l=(Rc|Vs|)−1≪1, i.e., the Oseen region is only a small part of the channel while the major portion of the disturbance flow is inviscid. A solution of the governing equations is constructed in terms of two-dimensional Fourier transform of the disturbance field in a plane parallel to the channel walls. The ordinary differential equation for Fourier transform of lateral velocity Γz(k,z) is solved using the method of matched asymptotic expansions based on e. Several domains in (k,z) space are distin...