Free-surface cusps associated with flow at low Reynolds number

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.