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Moffatt-type flows in a trihedral cone

Published online by Cambridge University Press:  14 May 2013

Julian F. Scott*
Affiliation:
LMFA, Université de Lyon, Ecole Centrale de Lyon, Ecully 69134, France
*
Email address for correspondence: julian.scott@ec-lyon.fr

Abstract

The three-dimensional analogue of Moffatt eddies is derived for a corner formed by the intersection of three orthogonal planes. The complex exponents of the first few modes are determined and the flows resulting from the primary modes (those which decay least rapidly as the apex is approached and, hence, should dominate the near-apex flow) examined in detail. There are two independent primary modes, one symmetric, the other antisymmetric, with respect to reflection in one of the symmetry planes of the cone. Any linear combination of these modes yields a possible primary flow. Thus, there is not one, but a two-parameter family of such flows. The particle-trajectory equations are integrated numerically to determine the streamlines of primary flows. Three special cases in which the flow is antisymmetric under reflection lead to closed streamlines. However, for all other cases, the streamlines are not closed and quasi-periodic limiting trajectories are approached when the trajectory equations are integrated either forwards or backwards in time. A generic streamline follows the backward-time trajectory in from infinity, undergoes a transient phase in which particle motion is no longer quasi-periodic, before being thrown back out to infinity along the forward-time trajectory.

Type
Papers
Copyright
©2013 Cambridge University Press 

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References

Gomilko, A. M., Malyuga, V. S. & Meleshko, V. V. 2003 On steady Stokes flow in a trihedral rectangular corner. J. Fluid Mech. 476, 159177.Google Scholar
Hall, O., Hills, C. P. & Gilbert, A. D. 2009 Non-axisymmetric Stokes flow between concentric cones. Q. J. Mech. Appl. Maths 62 (2), 131148.Google Scholar
Hills, C. P. & Moffatt, H. K. 2000 Rotary honing: a variant of the Taylor paint-scraper problem. J. Fluid Mech. 418, 119135.Google Scholar
Leriche, E. & Labrosse, G. 2011 Are there localized eddies in the trihedral corners of the Stokes eigenmodes in cubical cavity? Comput. Fluids 43 (1), 98101.CrossRefGoogle Scholar
Liu, C. H. & Joseph, D. D. 1978 Stokes flow in conical trenches. SIAM J. Appl. Maths 34, 286296.Google Scholar
Malhotra, C. P., Weidman, P. D. & Davis, A. M. J. 2005 Nested toroidal vortices between concentric cones. J. Fluid Mech. 522, 117139.Google Scholar
Malyuga, V. S. 2005 Viscous eddies in a circular cone. J. Fluid Mech. 522, 101116.CrossRefGoogle Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 118.Google Scholar
Moffatt, H. K. & Mak, V. 1998 Corner singularities in three-dimensional Stokes flow. In Nonlinear Singularities in Deformation and Flow (ed. Durban, D. & Pearson, J. R. A.), pp. 2126. Kluwer.Google Scholar
Shankar, P. N. 2005 Moffatt eddies in the cone. J. Fluid Mech. 539, 113135.Google Scholar
Shankar, P. N. 2007 Slow Viscous Flows. Imperial College Press.Google Scholar
Wakiya, S. 1976 Axisymmetric flow of a viscous fluid near the vertex of a body. J. Fluid Mech. 78, 737747.Google Scholar