Meandering instability of a viscous thread

Abstract

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 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-8" state, which contains a significant 2omega 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 2omega . 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.

Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities

“…This surprisingly complex behavior can be reproduced in a simple yet well-controlled experiment, where a viscous thread falls onto a moving belt: the patterns laid down by the thread are diverse and include meanders, alternating loops, a W pattern, and coiling ( Fig. 1), as well as various resonant patterns such as double coils and double meanders [1][2][3]. This system has been extensively studied [1][2][3][4][5] but has lacked a simple explanation until now.…”

“…1), as well as various resonant patterns such as double coils and double meanders [1][2][3]. This system has been extensively studied [1][2][3][4][5] but has lacked a simple explanation until now. The resemblance of these patterns to the stitch patterns of a sewing machine led Ref.…”

“…By varying these two parameters, one generates a phase diagram for the FMSM [5]. Herein, we work with the typical parameter values used in the literature [2,3] so that 0.5 ≤ H ≤ 1.4. We artificially omit inertia in our simulations, an assumption which is valid in almost this entire range (specifically, for H ≤ 1.2, see Sec.…”

Coiling of elastic rods on rigid substrates

“…Together, these observations on the meandering patterns combined-square root dependence of the amplitude on the control parameter, a finite onset wavelength, and critical slowing down at the onset of the instability-suggest that the meandering instability in our rod deployment pattern formation process is consistent with a Hopf bifurcation, which marks the transition from a stable to an oscillatory state in many other nonlinear systems. Moreover, it is interesting to note that the meandering instability for a viscous thread falling onto a moving belt has been shown to also arise through a Hopf bifurcation (17,18), pointing to universality features and emphasizing the prominence of geometry in these two systems.…”

On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface

“…Similar problems of the curved jets under the influence of gravity, or centrifugal and Coriolis forces but no moving surface, are described in [5,16,[18][19][20]. Various aspects of the problem addressed here have been studied in [2,12,13,17,21]. In this paper we start with description of the jet model, which is partly solved and transformed into a first order ODE on an unknown domain with additional scalar unknown; see Section 2.…”