The meandering instability of a viscous thread

The meandering instability
of a viscous thread

Physical Review E, 77, 066218 (2008).

Stephen W. Morris [1], Jonathan H. P. Dawes [2],
Neil M. Ribe [3], and John R. Lister [2]

[1] Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario, Canada M5S 1A7.

[2] DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA

[3] Dynamique des Fluides Géologiques, IPGP et Université de Paris-7, CNRS, Tour 14, 2, place Jussieu, 75005 Paris, France

A viscous thread falling from a nozzle onto a surface exhibits the famous rope-coiling effect, in which the thread buckles to form loops. If the surface is replaced by a belt moving with speed U, the rotational symmetry of the buckling instability is broken and a wealth of interesting states are observed [See S. Chiu-Webster and J. R. Lister, J. Fluid Mech., 569, 89 (2006)]. We experimentally studied this "fluid mechanical sewing machine" in a new, more precise apparatus. As U is reduced, the steady catenary thread bifurcates into a meandering state in which the thread displacements are only transverse to the motion of the belt. We measured the amplitude and frequency omega of the meandering close to the bifurcation. For smaller U, single-frequency meandering bifurcates to a two-frequency "figure eight" state, which contains a significant 2 omega component and parallel as well as transverse displacements. This eventually reverts to single-frequency coiling at still smaller U. More complex, highly hysteretic states with additional frequencies are observed for larger nozzle heights. We propose to understand this zoology in terms of the generic amplitude equations appropriate for resonant interactions between two oscillatory modes with frequencies omega and 2 omega. The form of the amplitude equations captures both the axisymmetry of the U=0 coiling state and the symmetry-breaking effects induced by the moving belt.

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