# Cake Icing and Climbing Ropes: the Physics of Curling Threads

A stream of honey on toast, a line of cake icing, and a cascading climbing rope all produce similar curlicues and coils as they fall. More than merely decorative, these curls are used to make nonwoven fabrics such as felt, yet are something to be avoided when laying down underwater fiber-optic cables.

Many experiments have studied the simple case of a viscous thread falling onto a moving conveyer belt, but until now the physics of how the curls form was a mystery.

 Simulations of viscous thread patterns. Image courtesy of Pierre-Thomas Brun, MIT

When a viscous fluid flows onto a moving surface the patterns range from tight coils if the surface isn’t moving to wavy lines for faster moving surfaces. The resemblance to embroidery gave rise to the name “fluid mechanical sewing machine” for any system of a thread curling onto a moving belt.

The experiments are relatively easy to perform: simply drizzle a syrup at a steady rate onto a slowly-accelerating conveyer belt and watch what happens. This wonderfully soothing video, created by Stephen Morris of the University of Toronto, shows the setup. The belt starts out fast and the syrup falls in a straight line. Gradually the belt slows down and the syrup begins to meander in a waves, then forms alternating loops, and finally a repeating coil.

This specific progression of patterns is based on the speed of the belt and can be seen in a variety of threads, both liquid (syrups and molten plastics) and solid (cables and ropes).

Except for the straight line pattern, the speed of the thread exceeds the belt’s ability to carry it away and the excess thread accumulates in waves and loops.

Pierre-Thomas Brun is an applied mathematician at MIT and has been working on the theory behind the curly patterns for the past few years. In an upcoming article in the Physical Review Letters journal, Brun and his colleagues are the first to describe the mathematics of the fluid mechanical sewing machine and have created a simulation that reproduces all of the experimental patterns.

According to their model, the system has a kind memory of its previous behavior. This memory is usually the key ingredient in complex systems, said Brun. For a mass on a spring, inertia (the resistance to change in motion) provides the memory and causes the oscillating mass to speed through its equilibrium point against the pull of the spring.

For curling threads, Brun deliberately took inertia out of the model but the patterns remained the same, indicating that something else was dictating the curls.

Brun and his colleagues focused on the point where the thread first hits the belt, including the angle of the thread at that point. They found that these two properties determine the exact curl of the thread as it falls onto the belt, which in turn determines the new point and angle of the thread. The memory is provided by the angle of the thread, according to their model.

The team even found a rare pattern which they call a W-pattern. This is similar to the alternating loop pattern, but with loops on the same side and spaced far apart. The W-pattern only appears when the belt is speeding up, and not when it is slowing down.

 Credit: Till Krech via flickr

This hysteresis (a different outcome based on different a path or history) is a new insight into the falling thread system, and could find applications in a range of industries.

The same physics governs flexible fiber-optic cables as they are laid on an ocean bed, and Brun’s new model could help prevent coils. But perhaps most promising application, according to Brun, is in the creation of micro-scale fiber structures and fabrics, including new techniques for 3D printing.

And it’s a good excuse to pile on the honey next time you make toast, just to watch it curl.

By Tamela Maciel, also known as “pendulum”