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Unsteady Stokes flow near boundaries: the point-particle approximation and the method of reflections

Published online by Cambridge University Press:  01 March 2018

A. Simha Open the ORCID record for A. Simha [Opens in a new window] ,
J. Mo  and
P. J. Morrison
A. Simha*
Affiliation:
Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Austin, TX 78712, USA
J. Mo
Affiliation:
Department of Physics and Center for Nonlinear Dynamics, University of Texas at Austin, Austin, TX 78712, USA
P. J. Morrison
Affiliation:
Department of Physics and Institute for Fusion Studies, University of Texas at Austin, Austin, TX 78712, USA
*
†Email address for correspondence: [email protected]
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Abstract

Problems of particle dynamics involving unsteady Stokes flows in confined geometries are typically harder to solve than their steady counterparts. Approximation techniques are often the only resort. Felderhof (see e.g. J. Phys. Chem. B, vol. 109 (45), 2005, pp. 21406–21412; J. Fluid Mech., vol. 637, 2009, pp. 285–303) has developed a point-particle approximation framework to solve such problems, especially in the context of Brownian motion. Despite excellent agreement with past experiments, this framework produces unsteady drag coefficients that depend on particle density. This is inconsistent, since the problem can be formulated mathematically without any reference to the particle’s density. We address this inconsistency in our work. Upon implementing our modifications, the framework passes consistency checks that it previously failed. Further, it is not obvious that such an approximation should work for short-time-scale motion. We investigate its validity by deriving it from a general formalism based on integral equations through a series of systematic approximations. We also compare results from the point-particle framework against a calculation performed using the method of reflections, for the specific case of a sphere near a full-slip plane boundary. We find from our analysis that the reasons for the success of the point-particle approximation are subtle and have to do with the nature of the unsteady Oseen tensor. Finally, we provide numerical predictions for Brownian motion near a full-slip and a no-slip plane wall based on the point-particle approximation as used by Felderhof, our modified point-particle approximation and the method of reflections. We show that our modifications to Felderhof’s framework would become significant for systems of metallic nanoparticles in liquids.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 841 , 25 April 2018 , pp. 883 - 924
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Copyright
© 2018 Cambridge University Press 

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