^{1}, John R. Lister

^{2}and Sunny Chiu-Webster

^{2}

### Abstract

A thin thread of viscous fluid that falls on a moving belt acts like a fluid-mechanical “sewing machine,” exhibiting a rich variety of “stitch” patterns including meanders, translated coiling, slanted loops, braiding, figures-of-eight, W-patterns, side kicks, and period-doubled patterns. Using a numerical linear stability analysis, we determine the critical belt speed and oscillation frequency of the first bifurcation, at which a steady dragged viscous thread becomes unstable to transverse oscillations or “meandering.” The predictions of the stability analysis agree closely with the experimental measurements of Chiu-Webster and Lister [J. Fluid Mech.569, 89 (2006)]. Moreover, the critical belt speed and onset frequency for meandering are nearly identical to the contact-point migration speed and angular frequency, respectively, of steady coiling of a viscous thread on a stationary surface, implying a remarkable degree of dynamical similarity between the two phenomena.

We thank C. Clanet and S. Morris for discussions, S. Morris for comments on an earlier draft of the manuscript, and two anonymous referees for constructive reviews. This work was funded in part by the DyETI program of INSU (France). This work is IPGP Contribution No. 2186.

I. INTRODUCTION

II. MATHEMATICAL FORMULATION

A. Governing equations

B. Unperturbed steady state

C. Perturbation equations

III. STABILITY CALCULATIONS

A. Method

B. Results

IV. DISCUSSION

### Key Topics

- Viscosity
- 13.0
- Boundary value problems
- 11.0
- Bending
- 5.0
- Eigenvalues
- 5.0
- Frequency analyzers
- 5.0

## Figures

Geometry of a dragged viscous thread. Fluid with kinematic viscosity , density , and surface tension coefficient is ejected downward at a constant volumetric rate through a hole of diameter ; the fluid thread falls a distance onto a horizontal belt moving with uniform speed . Liquid rope coiling on a stationary surface corresponds to . The origin of the Cartesian coordinates is at the center of the hole (the axes are displaced to the right for clarity).

Geometry of a dragged viscous thread. Fluid with kinematic viscosity , density , and surface tension coefficient is ejected downward at a constant volumetric rate through a hole of diameter ; the fluid thread falls a distance onto a horizontal belt moving with uniform speed . Liquid rope coiling on a stationary surface corresponds to . The origin of the Cartesian coordinates is at the center of the hole (the axes are displaced to the right for clarity).

Selected “stitch” patterns produced by a fluid-mechanical “sewing machine” (Fig. 1) with and different values of the volume flux , the viscosity , the fall height , and the belt speed . In (d), and ; in all other panels, and . (a) Steady dragged thread, , ; (b) meanders, , ; (c) slanted loops, , ; (d) braiding, , ; (e) figures-of-eight, , ; (f) W-pattern, , ; (g) translated coiling, , . The belt moves from left to right, and the smallest division of the scale at the bottom of each image is . The value of the viscosity is approximate due to temperature changes during the experiment.

Selected “stitch” patterns produced by a fluid-mechanical “sewing machine” (Fig. 1) with and different values of the volume flux , the viscosity , the fall height , and the belt speed . In (d), and ; in all other panels, and . (a) Steady dragged thread, , ; (b) meanders, , ; (c) slanted loops, , ; (d) braiding, , ; (e) figures-of-eight, , ; (f) W-pattern, , ; (g) translated coiling, , . The belt moves from left to right, and the smallest division of the scale at the bottom of each image is . The value of the viscosity is approximate due to temperature changes during the experiment.

Shapes of a steady dragged thread as a function of the belt speed , as predicted numerically by solving Eqs. (3) and (4) (black lines) and as observed experimentally (gray images) in experiment 1 with . (a) , (b) , (c) , and (d) .

Shapes of a steady dragged thread as a function of the belt speed , as predicted numerically by solving Eqs. (3) and (4) (black lines) and as observed experimentally (gray images) in experiment 1 with . (a) , (b) , (c) , and (d) .

Dragout distance as a function of belt speed for experiment 9 and . Circles: experimental measurements. Solid line: numerical prediction obtained using the full theory presented herein (with both stretching and bending terms). Dashed line: numerical prediction using the pure stretching theory of CWL (Ref. 16).

Dragout distance as a function of belt speed for experiment 9 and . Circles: experimental measurements. Solid line: numerical prediction obtained using the full theory presented herein (with both stretching and bending terms). Dashed line: numerical prediction using the pure stretching theory of CWL (Ref. 16).

Critical belt speed as a function of fall height , for experiments (a) 5, (b) 7, and (c) 8. Black circles: experimental measurements. Solid lines: prediction of the linear stability analysis described in the text. Dotted lines (nearly indistinguishable from the solid ones): axial velocity at the bottom of a thread coiling on a stationary surface with the same experimental parameters. Dashed lines: critical belt speed at which the pure stretching solution of CWL (Ref. 16) ceases to exist.

Critical belt speed as a function of fall height , for experiments (a) 5, (b) 7, and (c) 8. Black circles: experimental measurements. Solid lines: prediction of the linear stability analysis described in the text. Dotted lines (nearly indistinguishable from the solid ones): axial velocity at the bottom of a thread coiling on a stationary surface with the same experimental parameters. Dashed lines: critical belt speed at which the pure stretching solution of CWL (Ref. 16) ceases to exist.

Numerically calculated, neutrally stable shapes of a steady dragged thread for the parameters of experiment 5 and . The solid, long-dashed, and short-dashed lines indicate the position of the thread’s axis, and correspond to critical belt speeds , , and , respectively. The horizontal scale is exaggerated by a factor of 15 relative to the vertical scale.

Numerically calculated, neutrally stable shapes of a steady dragged thread for the parameters of experiment 5 and . The solid, long-dashed, and short-dashed lines indicate the position of the thread’s axis, and correspond to critical belt speeds , , and , respectively. The horizontal scale is exaggerated by a factor of 15 relative to the vertical scale.

Angular frequency of oscillation at the critical belt speed as a function of fall height , for experiments (a) 5, (b) 7, and (c) 8. Black circles: experimental measurements. Solid lines: frequency calculated via the linear stability analysis described in the text. Dashed lines: angular frequency of steady coiling on a motionless surface for the same experimental parameters.

Angular frequency of oscillation at the critical belt speed as a function of fall height , for experiments (a) 5, (b) 7, and (c) 8. Black circles: experimental measurements. Solid lines: frequency calculated via the linear stability analysis described in the text. Dashed lines: angular frequency of steady coiling on a motionless surface for the same experimental parameters.

Comparison of numerically predicted shapes of viscous threads falling onto moving (a,c) and stationary (b,d) surfaces. The fall height is , and the values of , , , , and are those of experiment 5 in Table I. The vertical and horizontal scales are equal in each view. (a) Critical (marginally stable) steady shape of a thread dragged with velocity . (b) Steady (in the corotating frame) shape of a thread coiling with angular frequency on a stationary surface. Horizontal arrows indicate the points below which the thread is in compression . (c,d) Magnified views of the lower portions of (a) and (b), respectively, viewed parallel to the axial unit vector at the contact point. The size of the grid squares is in (a) and (b) and in (c) and (d). The black dots in (b) and (d) mark the point on the surface directly below the ejection hole.

Comparison of numerically predicted shapes of viscous threads falling onto moving (a,c) and stationary (b,d) surfaces. The fall height is , and the values of , , , , and are those of experiment 5 in Table I. The vertical and horizontal scales are equal in each view. (a) Critical (marginally stable) steady shape of a thread dragged with velocity . (b) Steady (in the corotating frame) shape of a thread coiling with angular frequency on a stationary surface. Horizontal arrows indicate the points below which the thread is in compression . (c,d) Magnified views of the lower portions of (a) and (b), respectively, viewed parallel to the axial unit vector at the contact point. The size of the grid squares is in (a) and (b) and in (c) and (d). The black dots in (b) and (d) mark the point on the surface directly below the ejection hole.

(a) Inclination of the thread axis from the horizontal as a function of arclength along the dragged thread shown in Fig. 8(a) (solid) and the coiling thread shown in Fig. 8(b) (dashed). (b) Same as (a), but for the axial velocity scaled by the ejection velocity . (b) Same as (a), but for the total rate of viscous dissipation due to bending plus twisting, scaled by .

(a) Inclination of the thread axis from the horizontal as a function of arclength along the dragged thread shown in Fig. 8(a) (solid) and the coiling thread shown in Fig. 8(b) (dashed). (b) Same as (a), but for the axial velocity scaled by the ejection velocity . (b) Same as (a), but for the total rate of viscous dissipation due to bending plus twisting, scaled by .

## Tables

Parameters of the laboratory experiments referred to in this paper, numbered as in Table I of CWL (Ref. 16) is the density, is the volume flux, and is the diameter of the hole from which the fluid is ejected. The two values of the kinematic viscosity are for the maximum and minimum temperatures, respectively, during the experiment. The coefficient of surface tension is .

Parameters of the laboratory experiments referred to in this paper, numbered as in Table I of CWL (Ref. 16) is the density, is the volume flux, and is the diameter of the hole from which the fluid is ejected. The two values of the kinematic viscosity are for the maximum and minimum temperatures, respectively, during the experiment. The coefficient of surface tension is .

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