A falling stream of water will break into droplets due to the Plateau-Rayleigh instability. Small disturbances can create a wavy perturbation in the falling jet. Under the right conditions, the pressure caused by surface tension will be larger in the narrower regions and smaller in the wider ones. This imbalance will drive flow toward the wider regions and away from the narrower ones, thereby increasing the waviness in the jet. Eventually, the wavy jet breaks into droplets, which enclose the same volume of water with less surface area than the perturbed jet did. The instability is named for Joseph Plateau and Lord Rayleigh, who studied it in the late 19th century and showed that a falling jet of a non-viscous fluid would break into droplets if the wavelength of its disturbance was larger than the jet’s circumference. (Image credit: N. Morberg)
Tag: instability
Sharkskin Instability
Homemade spaghetti noodles exhibit a roughened surface that’s the result of viscoelastic behavior known as the sharkskin instability. It’s usually observed in the industrial extrusion of polymer plastics. In the case of spaghetti, the long, complex polymer molecules necessary for the instability come from the proteins in eggs. The characteristically rough surface of the extruded material is caused by the transition from flow through the die to air. Inside the die, friction from the walls exerts a strong shear force on the outer part of the fluid while the inner portion flows freely. When the material exits the die, the sudden lack of friction on the outer portion of the fluid causes it to accelerate to the same velocity as the middle of the flow. This acceleration stretches the polymers until they snap free of the die; after the strained polymers relax, the material keeps a rough, saw-tooth pattern. In industry, the sharkskin instability can be prevented by regulating temperature or flow speed. In the case of spaghetti, though, Modernist Cuisine suggests the roughness is desirable because it helps trap the pasta sauce. Bon appetit! (Image credit: Modernist Cuisine)
The Dance of the Droplets
Milk and juice vibrating on a speaker can put on a veritable fireworks display of fluid dynamics. Vibrating a fluid can cause small standing waves, called Faraday waves, on the surface of the fluid. Add more energy and the instabilities grow nonlinearly, quickly leading to tiny ligaments and jets of liquid shooting upward. With sufficiently high energy, the jets shoot beyond the point where surface tension can hold the liquid together, resulting in a spray of droplets. (Image credit: vurt runner, source video; h/t to @jchawner)
Reader Question: Rippling Runoff
Reader junolivi asks:
When shallow water (like runoff from melting snow) flows across pavement, it creates small repeated wave-like ripples. What creates that texture and why isn’t it just a steady flow?
This is a great question that’s probably crossed the mind of anyone who’s seen water running down the gutter of a street after a storm. The short answer is that this gravity-driven flow is becoming unstable.
Fluid dynamicists often like to characterize flows into two main types: laminar and turbulent. Most flows in nature are turbulent, like the wild swirls you see behind cars driving in the rain. But there are laminar flows in nature as well. Often flows that begin as laminar will become turbulent. This happens because those laminar flows are unstable to disturbances.
The classic example of stability is a ball on a hill. If the ball is at the top of the hill and you disturb it, it will roll down the hill because its original position was unstable. If, on the other hand, the ball is in a depression, then you can prod the ball and it will eventually settle back down into its original place because that position was stable. Another way of looking at it is that, in the unstable case, the disturbance–how far the ball is from its original position–grows uncontrollably. In the stable case, on the other hand, the disturbance can be initially large but eventually decays away to nothing.
There are many ways to disturb a laminar flow–surface roughness, vibrations, curvature, noise, etc., etc. These disturbances enter the flow and they can either grow (and become unstable) or decay (because the flow is stable to the disturbance). Just as one can look at the stability of a pendulum, one can mathematically examine the stability of a fluid flow. When one does this for water flowing down an incline, one finds that the flow is quite unstable, even in the ideal case of a pure, inviscid fluid flowing down a smooth wall.
The reason that one sees distinctive waves with a particular wavelength (assuming that they aren’t caused by local obstructions) is directly related to this idea of instability. Essentially, the waves are the disturbance, having grown large enough to see. One could imagine that any wavelength disturbance is possible in a flow, but mathematically, what one finds, is that different wavelengths have different growth rates associated with them. The wavelength we observe is the most unstable wavelength in the flow. This is the wavelength that grows so much quicker than the others that it just overwhelms them and trips the flow to turbulence. This is very common. For example, you can see distinctive waves showing up before the flow goes turbulent in both this mixing layer simulation and this boundary layer flow.
(Image credits: anataman, mo_cosmo; also special thanks to Garth G. who originally asked a similar question via email)
Behind the Science
FYFD features lots of science, but this new video gives you a chance to see the scientists, too! It’s a behind-the-scenes look at the American Physical Society Division of Fluid Dynamics meeting that took place in San Francisco recently. You may recognize some of the stories, but I guarantee there’s new stuff, even if you were there! Special thanks to everyone who helped me make the video; I had a blast doing this. (Video credit: N. Sharp)
Crown Sealing
Objects falling into a liquid pool create a beautiful splash, and, in this beautiful, award-winning video, the Splash Lab explores a peculiar instability that occurs just as the splash closes. The buckling instability they describe involves distinctive ridges that form along the splash’s ejecta sheet as it domes over and closes. The number of ridges depends both on the object size and the liquid’s properties. (Video credit: J. Marston et al.)
Piazza del Popolo
The lions of the fountain in Rome’s Piazza del Popolo eject a turbulent sheet of water. Random fluctuations in the water sheet cause holes to form. Driven by surface tension, these holes grow and merge, leaving behind ligaments of water which quickly break up into a spray of unevenly-sized drops. (Image credit: E. Villermaux)
The Rayleigh-Taylor Instability
What’s this? An FYFD video?! Yes, at long last, I’ve begun filming some videos of my own. This first one takes a look at the Rayleigh-Taylor instability and all that action that goes on in your coffee cup. I hope to bring you more FYFD-produced videos in the future, including some videos from the American Physical Society Division of Fluid Dynamics conference in San Francisco next week. What kind of topics would you guys be interested in for the future? (Video credit: N. Sharp)
Momentary Crown
When a drop falls on a liquid film, its impact drives a thin liquid sheet called the ejecta upward and outward from the point of impact. Within milliseconds, tiny perturbations develop in the ejecta and begin growing exponentially. These become the distinctive spikes of the crown. The momentum from the impact drives the ejecta and spikes further outward until it overcomes surface tension’s ability to hold the liquid crown together. Tiny droplets escape the crown before the ejecta comes crashing down. The whole process takes only a few hundred milliseconds from start to finish. (Photo credit: S. Jung et al.)
Kelvin-Helmholtz Clouds
When differing layers of fluid move past one another, friction between them causes shear. This shear quickly transforms a simple flat interface between fluid layers into a wavy unstable boundary that resembles a series of breaking ocean waves. This effect is known as the Kelvin-Helmholtz (KH) instability. In the atmosphere, this instability causes air layers with differing temperatures and moisture content to form wave-like clouds where the two layers meet. Other examples of the effect are widespread. On earth, many ocean waves are generated by wind shearing the water; elsewhere in our solar system, the cloud bands of Jupiter are lined with spinning eddies from the KH instability. (Photo credit: H. Bondo)
