If you like honey on your toast at breakfast, you are ready to perform one of the simplest and most beautiful experiments in the physics of fluids. Plunge a spoon into the honey jar, take it out and then hold it vertically, several centimeters above the toast. The thin stream of falling honey does not approach the toast directly but instead builds up a whirling helical structure. In the late 1950s the resemblance to a pile of coiled rope led the first investigators of this phenomenon, George Barnes and Richard Woodcock, to call it the liquid rope-coil effect.
The three of us had long been fascinated by this effect but never found the opportunity to study it until 10 years ago, when Ribe and Bonn discovered their shared interest by chance at a scientific workshop in Paris. At the time, Bonn had a collaboration with the Institute for Advanced Studies in Basic Sciences in Zanjan, Iran, so we invited Habibi and several others—including, at different times, Ramin Golestanian, Maniya Maleki, Yasser Rahmani and Seyed Hossein Hosseini—to complete the team.
Together we developed a controlled version of the breakfast-table experiment, using silicone oils rather than honey because they come in a broad range of viscosities. Viscosity is a measure of how thick a fluid is—how much it resists flowing because of internal friction. With our apparatus, we vary the flow conditions (such as the rate at which the fluid streams downward and the height from which it drops) and see how they affect the coiling frequency (how fast the descending column of fluid wraps around).
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When we began, we expected coiling to be an all-or-nothing affair that either happened or did not depending on the experimental conditions. We were therefore totally unprepared for the wealth of unexpected behavior that we found. For instance, for a slow flow rate, we found that the farther the fluid fell, the slower it coiled. Yet for higher flow rates, we found just the opposite: as the fall height increased, the frequency also increased rapidly. Moreover, when the fall height was fixed at a certain value, the coiling rope jumped back and forth in a seemingly random way between two states with different frequencies.
In parallel with the experiments, we developed a mathematical model to identify the basic principles at work. The starting point was Newton's laws of motion written in a form appropriate for a slender liquid rope whose length is much greater than its diameter. Two main types of forces act on any piece of such a rope: the downward pull of gravity and the internal viscous, or frictional, forces. The rope can deform in three distinct ways—stretching, bending and twisting—and each of these has an associated viscous force that opposes it. The shape of the rope depends on the relative magnitudes of all these forces as well as the inertia of the fluid (that is, mass times acceleration). The surface tension force, important for many other fluid flows, turned out to have only a minor effect here.
Solving the equations proved to be challenging. In most textbook problems of physics, the boundaries of the system are specified, and the student's task is to determine what is going on inside them. In contrast, liquid rope coiling is what physicists call a “free boundary” problem, in which the shape of the boundary is part of the problem we are trying to solve. With care, we were able to show that coiling in highly viscous fluids can occur in four distinct modes, each involving a different force balance.
Spiral Bubble Waves
Having mapped out the general types of coiling, we imagined that we had a fairly complete picture—but we were wrong. Further experiments, conducted in an exploratory way with no preconceived ideas, revealed remarkable new phenomena.
The first was the appearance of beautiful spiral waves of air bubbles in the thin layer of fluid that spreads away from the coiling rope. They form when the successive loops of the rope are slightly offset from one another, trapping small air pockets. We still do not understand, however, why the spirals have precisely the shape they do or why they only occur for narrow ranges of fluid viscosity, flow rate and fall height.
We also played with silicone oils of much lower viscosity. These fluids coiled more quickly—up to 2,000 times per second—so we needed high-speed cameras to record their behavior. The fluids could coil and even fold in much more complex ways. A given state would persist indefinitely if left undisturbed yet would suddenly switch to another if we gave the apparatus a strong tap with our knuckles.
In all the above experiments, the thin stream of liquid fell onto a stationary surface. But new effects can arise if the surface and the source of the liquid move relative to each other—as observed by Jackson Pollock in his action painting and by manufacturers of textiles using molten polymer threads. Our colleagues Keith Moffatt, Sunny Chiu-Webster and John Lister, all then at the University of Cambridge, experimented with these situations using what amounts to a fluid sewing machine, which extrudes a single thread of viscous fluid onto a horizontal belt moving at a constant speed. At high speeds, the dragged thread leaves a straight trace on the belt. But as the belt slows down, more complex, unsteady patterns emerge, such as meandering, alternating loops, double coiling and even a W shape.
We still have a long way to go before we understand liquid ropes fully. A top priority is to understand the physical mechanism behind spiral bubble waves. Why does the center of the coil start to move in a separate orbit? Another goal is to model the complex secondary coiling that occurs in lower-viscosity fluids. We also plan to extend our exploration to more exotic systems, including complex fluids with nonstandard responses to applied forces, as well as electrically charged fluids coiling at microscales and nanoscales in an electric field. Judging by our past experience, many more surprises are in store for us.