Frequency structure of the nonlinear instability of a dragged viscous thread

A thread of viscous fluid falling onto a moving belt exhibits a spectacular variety of modes of motion as the belt speed and nozzle height are varied [1]. For modest nozzle heights, four clear regimes are observed. For large belt speed, the thread is dragged into a stretched centenary configuration which is confined to a plane. As the belt speed is lowered, this exhibits a supercritical Hopf bifurcation to a meandering mode [2]. At very low belt speeds, the motion resembles the usual coiling motion of a viscous thread falling on a stationary surface. In between the meandering and coiling regimes, a window of novel multifrequency motion, previously called ``figures of eight" is found. We examined the longitudinal and transverse motion of the thread in all these states, using an automated apparatus that allows a detailed exploration of the parameter space. We found that the multifrequency window is characterized by a complex pattern of motion whose main frequencies are locked in a 3:2 ratio. This motion appears and disappears with finite amplitude at sharp bifurcations, without measurable hysteresis. \\[0pt] [1] S. Chiu-Webster and J. R. Lister, J. Fluid Mech., 569, 89 (2006).\\[0pt] [2] S. W. Morris, J. H. P. Dawes, N. M. Ribe and J. R. Lister, Phys. Rev. E, 77, 066218 (2008).