A numerical investigation of the fluid mechanical sewing machine

TL;DR

This paper numerically simulates the fluid mechanical sewing machine experiment, successfully reproducing most observed patterns and proposing a new classification based on Fourier spectra and frequency locking.

Contribution

It introduces a Discrete Viscous Threads simulation method that captures complex pattern formation and offers a new spectral classification of the patterns.

Findings

Successfully reproduces nine out of ten patterns observed in experiments.

Shows patterns are linked to frequency ratios of steady coiling.

Provides a new spectral classification based on Fourier analysis.

Abstract

A thin thread of viscous fluid falling onto a moving belt generates a surprising variety of patterns depending on the belt speed, fall height, flow rate, and fluid properties. Here we simulate this experiment numerically using the Discrete Viscous Threads method that can predict the non-steady dynamics of thin viscous filaments, capturing the combined effects of inertia, stretching, bending and twisting. Our simulations successfully reproduce nine out of ten different patterns previously seen in the laboratory, and agree closely with the experimental phase diagram of Morris et al.\ (2008). We propose a new classification of the patterns based on the Fourier spectra of the longitudinal and transverse motion of the point of contact of the thread with the belt. These frequencies appear to be locked in most cases to simple ratios of the frequency $\Omega_c$ of steady coiling obtained in the limit of zero belt speed. In particular the intriguing `alternating loops' pattern is produced by combining the first five multiples of $\Omega_c/3$.