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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on January 27, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2007 60(1):27-48; doi:10.1093/qjmam/hbl024
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Q. Jl Mech. Appl. Math, Vol. 60. No. 1 © The author 2007. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Slow flow between concentric cones

Oskar Hall{dagger}

( Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter EX4 4QF )

Christopher P. Hills

( School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland )

Andrew D. Gilbert{ddagger}

( Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter EX4 4QF )

{dagger} oskar.hall{at}gmail.com

{ddagger} a.d.gilbert{at}exeter.ac.uk

Received 25 July 2006. Revise 28 September 2006.
  Abstract

This paper considers the low-Reynolds-number flow of an incompressible fluid contained in the gap between two coaxial cones with coincident apices and bounded by a spherical lid. The two cones and the lid are allowed to rotate independently about their common axis, generating a swirling motion. The swirl induces a secondary, meridional circulation through inertial effects. For specific configurations complex eigenmodes representing an infinite sequence of eddies, analogous to those found in two-dimensional corner flows and some three-dimensional geometries, form a component of this secondary circulation. When the cones rotate these eigenmodes, arising from the geometry, compete with the forced modes to determine the flow near the apex.

This paper studies the relative dominance of these two effects and maps out regions of parameter space, with attention to how shear and overall rotation can destroy the infinite sequence of eddies that may be present when only the lid is rotated. A qualitative picture of the number of eddies visible in the meridional circulation is obtained as a function of the rotation rates of cones and lid, for various choices of angles. The results are discussed in the context of previous work, including their significance for applications to the mixing of viscous fluids in this geometry.


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