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Free-surface cusps associated with flow at low Reynolds number

Published online by Cambridge University Press:  26 April 2006

Jae-Tack Jeong  and
H. K. Moffatt
Jae-Tack Jeong
Affiliation:
Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK Present address: Department of Mechanical Engineering, Kum-Oh National Institute of Technology, 188 Shinpyung Dong, Kumi, Kyung Buk, Republic of Korea, 730-701.
H. K. Moffatt
Affiliation:
Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
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Abstract

When two cylinders are counter-rotated at low Reynolds number about parallel horizontal axes below the free surface of a viscous fluid, the rotation being such as to induce convergence of the flow on the free surface, then above a certain critical angular velocity Ωc, the free surface dips downwards and a cusp forms. This paper provides an analysis of the flow in the neighbourhood of the cusp, via an idealized problem which is solved completely: the cylinders are represented by a vortex dipole and the solution is obtained by complex variable techniques. Surface tension effects are included, but gravity is neglected. The solution is analytic for finite capillary number [Cscr ], but the radius of curvature on the line of symmetry on the free surface is proportional to exp (−32π[Cscr ]) and is extremely small for [Cscr ] [gsim ] 0.25, implying (in a real fluid) the formation of a cusp. The equation of the free surface is cubic in (x, y) with coefficients depending on [Cscr ], and with a cusp singularity when [Cscr ] = ∞.

The influence of gravity is considered through a stability analysis of the free surface subjected to converging uniform strain, and a necessary condition for the development of a finite-amplitude disturbance of the free surface is obtained.

An experiment was carried out using the counter-rotating cylinders as described above, over a range of capillary numbers from zero to 60; the resulting photographs of a cross-section of the free surface are shown in figure 13. For Ω < Ωc, a rounded crest forms in the neighbourhood of the central line of symmetry; for Ω > Ωc, the downward-pointing cusp forms, and its structure shows good agreement with the foregoing theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 1 - 22
Copyright
© 1992 Cambridge University Press

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