Frequency structure of the nonlinear instability of a dragged viscous thread

A thread of viscous fluid falling onto a moving surface exhibits a spectacular variety of types of motion as the surface speed and nozzle height are varied. For modest nozzle heights, four clear regimes are observed. For large surface speed, the thread is dragged into a stretched centenary configuration which is confined to a plane. As the surface speed is lowered, the thread exhibits a supercritical bifurcation to a meandering state. At very low surface speeds, the state resembles the usual coiling motion of a viscous thread falling on a stationary surface. In between the meandering and coiling regimes, a window containing a novel multifrequency state, previously called “figures of eight,” is found. Using an improved visualization technique and a fully automated apparatus, we made detailed measurements of the longitudinal and transverse motion of the thread in all these states. We found that the multifrequency state is characterized by a complex pattern of motion whose main frequencies are locked in a 3:2 ratio. This state appears and disappears with finite amplitude at sharp bifurcations without measurable hysteresis.

Figure 1

A schematic of the experimental apparatus, showing side and top views. The belt moved at speed $U$ in the direction of the arrow. The camera simultaneously collected $x$ and $y$ views by means of a 45${}^{\circ }$ mirror.

Figure 2

The two simultaneous views of the thread, as seen by the camera. The belt moves to the left with speed $U$. The right-hand image is the view of the thread transverse to the belt motion, seen in the mirror, while the left-hand one is the longitudinal view, seen directly. The horizontal line indicates the portion of the image used to determine the $x$ and $y$ positions of the thread. The thread is shown in the catenary state just above the meandering transition.

Figure 3

The state diagram, obtained by automated variation of $H$ and $U$, with automated state classification based on the Fourier spectra of the $x$ and $y$ motions. The window of the alternating loop state (small circles) is bounded on all sides by the meandering (triangles) and coiling (stars) states. The onset of fluid pendulum behavior occurs for $H\ge 6$ cm, just beyond the right-hand edge of this plot.

Figure 4

The Fourier spectra of the $x$ (solid) and $y$ (dashed) motions for the (a) meandering ($U=2.23$ cm/s), (b) one side looping ($U=2.19$ cm/s), (c) alternating loop ($U=1.60$ cm/s), and (d) coiling ($U=1.00$ cm/s) states. Each panel shows the power spectra, with a reconstruction of the pattern traced by the tip of the thread on the belt above and an inset showing the corresponding Lissajous figure. For all these data, $H=5.00$ cm. For the Lissajous figures, the upstream belt direction is at the top.

Figure 5

The frequencies of the main modes of the $x$ (circles) and $y$ (squares) motions of the thread as a function of the belt speed $U$, for $H=4.45$ cm. The solid symbols trace the $U$ evolution of the mode corresponding to the main Hopf frequency of the meandering state. The $x$ and $y$ symbols overlap in the coiling state.

Figure 6

The ratio of the dominant frequencies of the main $x$ and $y$ modes vs $U$, for $H=4.45$ cm, using the same data as Fig. 5. This figure shows clearly the 3:2 nature of the alternating loop state.

Figure 7

The amplitudes of the largest $x$ (open circles) and $y$ (solid squares) Fourier modes as a function of $U$, for $H=4.45$ cm, using the same data as the previous two figures.