Multiple coexisting states of liquid rope coiling

N.M. Ribe*, H.E. Huppert, M.A. Hallworth, M. Habibi, Daniel Bonn

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

57 Citations (Scopus)


A thin 'rope' of viscous fluid falling from a sufficient height onto a surface forms a series of regular coils. Here we investigate theoretically and experimentally a curious feature of this instability: the existence of multiple states with different frequencies at a fixed value of the fall height. Using a numerical model based on asymptotic 'thin rope' theory, we determine curves of coiling frequency Ω vs. fall height as functions of the fluid viscosity v, the diameter d of the injection hole, the volumetric injection rate Q, and the acceleration due to gravity g. In addition to the three coiling modes previously identified (viscous, gravitational and inertial), we find a new multivalued 'inertio-gravitational' mode that occurs at heights intermediate between gravitational and inertial coiling. In the limit when the rope is strongly stretched by gravity and Π1 ≡ (v5/gQ3)1/5≫ 1, inertio-gravititational coiling occurs in the height range O(Π1 -1/6 ≤ H(g/v2)1/3 ≤ O(Π1 -5/48). The frequencies of the individual branches are proportional to (g/H)1/2, and agree closely with the eigenfrequencies of a whirling liquid string with negligible resistance to bending and twisting. The number of coexisting branches scales as Π1 5/32. The predictions of the numerical model are in excellent agreement with laboratory experiments performed by two independent groups using different apparatus and working fluids. The experiments further show that interbranch transitions in the inertio-gravitational regime occur via an intermediate state with a 'figure of eight' geometry that changes the sense of rotation of the coiling.

Original languageEnglish
Pages (from-to)275-297
JournalJournal of Fluid Mechanics
Publication statusPublished - 25 May 2006
Externally publishedYes


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