Pattern formation in coiling of falling viscous threads: Revisiting the geometric model

TL;DR

This study revisits pattern formation of a viscous thread falling onto a moving belt, focusing on moderate heights where non-inertial dynamics prevail. By collecting new experiments and comparing them to the Geometric Model (GM), the authors confirm the basic pattern sequence—translating coiling, alternating loops, W, and meander—and reveal a period-doubled W and rare resonant patterns. While GM captures the general order and many frequency/period trends, it fails to accurately predict the W-region extent and meander dynamics, indicating missing physics or parameter calibration in the curvature law. The work highlights the need for refined modeling and possibly data-driven curvature updates, and it motivates a global bifurcation analysis to better understand transitions and the emergence of disordered and resonant states. Overall, the findings deepen understanding of non-inertial coiling dynamics and point to richer dynamics beyond the simplified geometric picture.

Abstract

The "Fluid Mechanic Sewing Machine" creates periodic patterns through the coiling nature of a viscous fluid falling onto a moving surface. At relatively moderate heights, the reported patterns are translating coiling, alternating loops, W pattern, and meander. A simplified theoretical model based on the geometry and local bending of the contact point can predict these patterns. We experimentally explore the patterns in this region by collecting new data to compare with the model. Our review of the model's bifurcation diagram reveals additional patterns beyond the ones reported, although current experiments have not shown their existence. The W pattern, previously omitted in a regime diagram because of its small region, is now shown explicitly. We report on the consistent appearance of a period-doubled version of the W pattern, as well as rare appearances of resonant patterns, both reported for the first time. Comparing the theoretical model to experimental data, we find that the predicted phase diagram and the meander variation deviate from observations. These deviations hint at an unaccounted dynamics that merits further study.

Figures (14)

• Figure 1: (Color online) Belt patterns appearing at relatively low to moderate height levels. Patterns a-d are catenary (straight), meander, alternating loops and translating coiling, respectively. The arrow on the left indicates the order of increasing $\hat{V}$ with the critical speed $V_{crit}$ indicated by the dotted red line. On the right (e), the less common W pattern is shown. The dashed blue line highlights the heel of the thread.
• Figure 2: (Color online) Schematic of the experimental setup. On the top view of the belt, an example of the meander pattern is reconstructed (red, solid line) using the orbit on the left (blue, dashed line). The orbit is obtained by tracking the motion of the thread seen by pointing the camera to the mirror (demonstrated with the $\hat{x}$ and $\hat{y}$ arrows).
• Figure 3: The meander pattern in the $(\hat{x},\hat{y})$ plane (top left), the time domain (top right), and the frequency domain (bottom). A similar conversion sequence is used for other patterns for classification.
• Figure 4: (Color online) Repeatability of the observed patterns at a fall height of 4.5 cm. Empty markers represent the data collected for an increasing $\hat{V}$ and solid markers are data collected for decreasing $\hat{V}$.
• Figure 5: (Color online) Examples of the basic patterns observed and their frequency spectrum. The orbit and the belt patterns are represented in dashed blue and solid red respectively. The light blue dot represents the thread position at the point this moment in time. Data were obtained at fall height $\hat{H}$ = 4.5 cm for increasing belt speed $\hat{V}$.