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Stokes wall effects for particles moving near cylindrical boundaries

Published online by Cambridge University Press:  20 April 2006

A. Falade  and
H. Brenner
A. Falade
Affiliation:
Department of Mechanical Engineering, University of Lagos, Lagos, Nigeria
H. Brenner
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
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Abstract

An asymptotic scheme is derived for calculating values of the ‘reflected’ Stokeslet-field velocity dyadic V and its gradient ${\boldmath \nabla V}$ back at the Stokeslet location for situations in which this singular point lies in close proximity to the wall of an infinitely long circular cylinder. The asymptotic formulas furnished by this scheme permit calculations of first- and second-order wall effects in the non-dimensional parameter κ = a/R0 (a ≡ characteristic particle radius, R0 ≡ cylinder radius) upon the Stokes resistance of a particle of arbitrary shape, location and orientation when translating and/or rotating near the wall of an otherwise quiescent fluid-filled or fluid-surrounded circular cylinder. This reflection-type calculation is applicable for circumstances in which the inequalities κ [Lt ] 1 and 1 − β [Lt ] 1 are each separately satisfied, while simultaneously K/(1 − β) [Lt ] 1. (Here β = b/R0 is the fractionally eccentric Stokeslet location, or equivalently the centre of reaction of the particle, with b its distance from the tube axis.) The main result of this paper is the development of the pair of asymptotic wall-correction formulas \[ W_{jk}\sim {}_0C_{jk}(1-\beta)^{-1}+{}_1C_{jk}+{}_2C_{jk}(1-\beta)+O(1-\beta)^2 \] and \[ W_{jk,\,l}\sim {}_0D_{jkl}(1-\beta)^{-2}+{}_1D_{jkl}(1-\beta)^{-1}+{}_2D_{jkl}+O(1-\beta) \] to particle resistance, with W ≡ Wjk and ${\boldmath \nabla W}\equiv W_{kl,j}$ respectively the non-dimensional normalized wall-effect dyadic and its gradient at the Stokeslet location β. The numerical, rational fraction, β-independent, nCjk and nDjkl coefficients (n = 0, 1, 2) appearing above are evaluated by solving a recursive sequence of Stokes-flow boundary-value problems in the semi-infinite fluid domain bounded by a plane wall. These simple asymptotic formulas are shown to agree excellently in the range near β = 1 with existing values derived from the exact solution of the original circular-cylinder boundary-value problem, involving tedious infinite-series summations of complicated Bessel-function integrands extended over infinite integration domains. Generalizations of the scheme to particle motion in the space external to a circular cylinder is briefly sketched, as too is the case of cylinders of non-circular cross-section.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 154 , May 1985 , pp. 145 - 162
Copyright
© 1985 Cambridge University Press

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