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 R-c is investigated. The weak-shear case is studied when the ratio of the slip velocity to maximum velocity of the channel flow, V-s, is finite, while two other dimensionless groups, particle Reynolds number R-s and R-c(-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 epsilon =l(s)/l=(R(c)parallel toV(s)parallel to)(-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 Gamma (z)(k,z) is solved using the method of matched asymptotic expansions based on epsilon. Several domains in (k,z) space are distinguished that correspond to different regions in physical space: Oseen region, inviscid region, viscous wake, boundary, and critical layers. For the weak-shear case the dominant contribution to the lift gives not velocity disturbances within the Oseen region but the large-scale inviscid disturbances. For such disturbances both effects due to an inviscid interaction with the walls and the curvature of the undisturbed velocity profile should be taken into account. The particle equilibrium positions at the distances from the wall of the order of the channel width arise for both negative and positive slip velocities. (C) 2002 American Institute of Physics.