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On the slow motion of a sphere parallel to a nearby plane wall

Published online by Cambridge University Press:  28 March 2006

M. E. O'Neill  and
K. Stewartson
M. E. O'Neill
Affiliation:
Department of Mathematics, University College, London
K. Stewartson
Affiliation:
Department of Mathematics, University College, London
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Abstract

A new method using a matched asymptotic expansions technique is presented for obtaining the Stokes flow solution for a rigid sphere of radius a moving uniformly in a direction parallel to a fixed infinite plane wall when the minimum clearance ea between the sphere and the plane is very much less than a. An ‘inner’ solution is constructed valid for the region in the neighbourhood of the nearest points of the sphere and the plane where the velocity gradients and pressure are large; in this region the leading term of the asymptotic expansion of the solution satisfies the equations of lubrication theory. A matching ‘outer’ solution is constructed which is valid in the remainder of the fluid where velocity gradients are moderate but it is possible to assume that ε = 0. The forces and couples acting on the sphere and the plane are shown to be of the form (α0+α1ε) log ε + β0 + O(ε) where α0, α1 and β0 are constants which have been determined explicitly.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 27 , Issue 4 , 20 March 1967 , pp. 705 - 724
Copyright
© 1967 Cambridge University Press

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