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Free-surface cusps associated with flow at low Reynolds number

  • Jae-Tack Jeong (a1) (a2) and H. K. Moffatt (a1)


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.



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Byrd, P. F. & Friedman M. D. 1971 Handbook of Elliptic integrals for Engineers and Scientists, 2nd Edn. Springer.
Dussan V. E. B. & Davis, S. H. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65, 7195.
Gradshteyn, I. S. & Ryzhik I. M. 1980 Table of Integrals, Series and Products (Corrected and Enlarged Edn). Academic.
Griggs D. 1939 A theory of mountain building. Am. J. Sci. 237, 611650.
Joseph D. D., Nelson J., Renardy, M. & Renardy Y. 1991 Two-dimensional cusped interfaces. J. Fluid Mech. 223, 383409.
Lister J. R. 1989 Selective withdrawal from a viscous two-layer system. J. Fluid Mech. 198, 231254.
Muskhelishvili N. I. 1953 Some Basic Problems of the Mathematical Theory of Elasticity, 3rd Edn. P. Noordhoff.
Richardson S. 1968 Two-dimensional bubbles in slow viscous flows. J. Fluid Mech. 33, 476493.
Saffman P. G. 1986 Viscous fingering in Hele-Shaw cells. J. Fluid Mech. 173, 7394.
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Free-surface cusps associated with flow at low Reynolds number

  • Jae-Tack Jeong (a1) (a2) and H. K. Moffatt (a1)


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