The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on February 13, 2009
The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(2):131-148; doi:10.1093/qjmam/hbp002
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Nonaxisymmetric stokes flow between concentric cones
( Mathematics Research Institute, School of Engineering, Computing and Mathematics, University of Exeter, Exeter EX4 4QF )
( School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland )
( Mathematics Research Institute, School of Engineering, Computing and Mathematics, University of Exeter, Exeter EX4 4QF )
Corresponding author. 
chris.hills{at}dit.ie
Received 4 July 2008. Revise 28 November 2008. Accepted 3 December 2008.
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Abstract |
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We study the fully three-dimensional Stokes flow within a geometry consisting of two infinite cones with coincident apices. The Stokes approximation is valid near the apex and we consider the dominant flow features as it is approached. The cones are assumed to be stationary and the flow to be driven by an arbitrary far-field disturbance. We express the flow quantities in terms of eigenfunction expansions and allow for the first time for nonaxisymmetric flow regimes through an azimuthal wave number. The eigenvalue problem is solved numerically for successive wave numbers. Both real and complex sequences of eigenvalues are found, their relative dominance affecting the flow features observed. The implications for the presence of eddy-like structures (analogous to those found in other corner geometries) are discussed and we find that these flow features depend not only upon the internal angles of the two cones but also upon the symmetry of the driving mechanism. For an arbitrary disturbance, the dominant flow mode is not axisymmetric but rather is associated with wave number one and, by breaking axisymmetry, eddies can be avoided in this geometry.