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"H-states'': exact solutions for a rotating hollow vortex

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Title: "H-states'': exact solutions for a rotating hollow vortex
Authors: Crowdy, D
Nelson, R
Krishnamurthy, V
Item Type: Journal Article
Abstract: Exact solutions are found for an N-fold rotationally symmetric, steadily rotating hollow vortex where a continuous real parameter governs its deformation from a circular shape and N≥2 is an integer. The vortex shape is found as part of the solution. Following the designation ‘V-states’ assigned to steadily rotating vortex patches (Deem & Zabusky, Phys. Rev. Lett., vol. 40, 1978, pp. 859–862) we call the analogous rotating hollow vortices ‘H-states’. Unlike V-states where all but the N=2 solution – the Kirchhoff ellipse – must be found numerically, it is shown that all H-state solutions can be written down in closed form. Surface tension is not present on the boundaries of the rotating H-states but the latter are shown to be intimately related to solutions for a non-rotating hollow vortex with surface tension on its boundary (Crowdy, Phys. Fluids, vol. 11, 1999a, pp. 2836–2845). It is also shown how the results here relate to recent work on constant-vorticity water waves (Hur & Wheeler, J. Fluid Mech., vol. 896, 2020, R1) where a connection to classical capillary waves (Crapper, J. Fluid Mech., vol. 2, 1957, pp. 532–540) is made.
Issue Date: 25-Apr-2021
Date of Acceptance: 12-Jan-2021
URI: http://hdl.handle.net/10044/1/86833
DOI: 10.1017/jfm.2021.55
ISSN: 0022-1120
Publisher: Cambridge University Press
Start Page: R5-1
End Page: R5-11
Journal / Book Title: Journal of Fluid Mechanics
Volume: 913
Copyright Statement: © The Author(s), 2021. Published by Cambridge University Press. This paper has been accepted for publication and will appear in a revised form, subsequent to peer-review and/or editorial input by Cambridge University Press.
Keywords: 01 Mathematical Sciences
09 Engineering
Fluids & Plasmas
Publication Status: Published
Online Publication Date: 2021-03-01
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences