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Axisymmertic Stokes flow images in spherical free surfaces with applications to rising bubbles

Published online by Cambridge University Press:  17 February 2009

J. F. Harper
J. F. Harper
Affiliation:
Department of mathematics, Victoria University of Wellington, Private Bag, Wellington, New Zealand.
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Abstract

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A theorem is derived for the hydrodynanuc image of an axially symmetric slow viscous (Stokes) flow in a sphere which is impermeable and free of shear stress. A second theorem establishes a sense in which such a flow past an arbitrary rigid surface or shear-free sphere becomes, on inversion in an arbitrary sphere with its centre on the axis of symmetry, a flow past the rigid or shear-free inverse of that surface or sphere.

The theorems are used to simplify the proofs of a number of known results for images of point singularities in plane and spherical rigid and free boundaries, and for a pair of bubbles rising steadily in line in a viscous fluid. They also give for the first time accurate numerical solutions for the velocities of each of a larger number of spherical bubbles rising quasi-steadily in line. These enable one to assess the accuracy of simple approximations to those velocities.

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 2 , October 1983 , pp. 217 - 231
Copyright
Copyright © Australian Mathematical Society 1983

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