Sedimentation and dispersion of non-neutrally buoyant Brownian particles in cellular circulatory flows simulating local fluid agitation

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Abstract
Spatially periodic fluid ‘‘vortical’’ velocity fields are utilized to model the role of microscale convection upon the sedimentation and dispersion of isolated spherical particles. In particular, generalized Taylor dispersion theory is employed to calculate the mean Stokes sedimentation velocity vector and dispersivity dyadic for a finite-size spherical Brownian particle settling in a laminar, spatially periodic fluid velocity field with zero mean flow (but nonzero ‘‘circulation’’). Surprisingly, such local cellular vortexlike motions are shown to have no effect whatsoever upon the long-time mean sedimentation velocity of the sphere, which is identical to its steady-state Stokes-law value occurring in a quiescent fluid. This conclusion is, inter alia, independent of the sphere size a relative to the spatial period l of the cellular flow (provided that a/l≪1), as well as of the ratio of the particle settling velocity to the magnitude of the cellular fluid velocity or circulation. On the other hand, local convection does affect particle dispersion relative to the mean particle velocity in these systems, consistent with recent calculations for a particular cellular velocity field (Mofatt [Rep. Prog. Phys. 46, 621 (1983)], Sagues and Horsthemke [Phys. Rev. A 34, 4136 (1986)], Rosenbluth et al. [Phys. Fluids 30, 2636 (1987)]). We obtain a general expression for the dispersion coefficient for sedimenting, finite-size spheres in the limit of low Peclét number; this general result is consistent with the specific formula derived by the latter authors for the particular cellular ‘‘Taylor’’ flow analyzed by them, applicable to the case of a nonsedimenting, point-size tracer particle.

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