Early in our geological history, Earth was a hellish landscape of molten oceans into which metallic impactors would sometimes collide. Geophysicists have been curious how the impactors behaved after collision: did they maintain their cohesion, or did they break up into a cloud of droplets? Here the UCLA Spinlab simulates this early planetary formation by dropping liquid gallium through a tank of viscous fluid. As the video shows, the impactor’s behavior varies strongly with size. Smaller impactors stick together as a single diapir, but, as the initial size increases, the diapir becomes unstable, eventually breaking down into a cascade of droplets – a metallic rain through an ocean of magma. (Video credit: J. Wacheul et al./UCLA Spinlab; submitted by J. Aurnou)
Month: January 2014
The Physics of a Flying-V
New research using free-flying northern bald ibises shows that during group flights the birds’ positioning and flapping maximize aerodynamic efficiency. In flight, a bird’s wings generate wingtip vortices, just as a fixed-wing aircraft does. These vortices stretch in the bird’s wake, creating upwash in some regions and downwash in others as the bird flaps. According to theory, to maximize efficiency a trailing bird should exploit upwash and avoid downwash by flying at a 45-degree angle to its leading neighbor and matching its flapping frequency. The researchers found that, on average, this was the formation and timing the flock assumed. In situations where the birds were flying one behind the next in a straight line, the birds tended to offset their flapping by half a cycle relative to the bird ahead of them–another efficient configuration according to theory. Researchers don’t yet know how the birds track and match their neighbors; perhaps, like cyclists in a peloton, they learn by experience how to position themselves for efficiency. For more information, see the researchers’ video and paper. (Photo credit: M. Unsold; research credit: S. Portugal; via Ars Technica; submitted by M. Piedallu van Wyk)
Shooting a Bullet Through a Water Balloon
This high-speed video of a bullet fired into a water balloon shows how dramatically drag forces can affect an object. In general, drag is proportional to fluid density times an object’s velocity squared. This means that changes in velocity cause even larger changes in drag force. In this case, though, it’s not the bullet’s velocity that is its undoing. When the bullet penetrates the balloon, it transitions from moving through air to moving through water, which is 1000 times more dense. In an instant, the bullet’s drag increases by three orders of magnitude. The response is immediate: the bullet slows down so quickly that it lacks the energy to pierce the far side of the balloon. This is not the only neat fluid dynamics in the video, though. When the bullet enters the balloon, it drags air in its wake, creating an air-filled cavity in the balloon. The cavity seals near the entry point and quickly breaks up into smaller bubbles. Meanwhile, a unstable jet of water streams out of the balloon through the bullet hole, driven by hydrodynamic pressure and the constriction of the balloon. (Video credit: Keyence)
“Porgrave”
Artist Sandro Bocci uses macro imagery of fluids in his new piece “Porgrave” to create scenes reminiscent of celestial landscapes and the first moments of life. Surface tension, the Marangoni effect, and diffusion create pulsating motion in some frames whereas immiscible liquids form untouchable islands in others. “Porgrave” reminds me of work by Pery Burge and Julia Cuddy as well as sequences from films like 2001 and The Fountain, both of which created some of their effects with macro photography of fluids. Still images from “Porgrave” are available on Bocci’s site. (Video credit and submission: S. Bocci)
ETA: This article originally misprinted the artist’s name as “Sandro Bocchi” and has been updated with the correct spelling.
Solution to a Millennium Prize Problem?
Reports emerged this weekend that Kazakh mathematician Mukhtarbay Otelbaev has published a proposed solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize problems offered by the Clay Mathematics Institute. Today I want to explain some of the background of this problem, what is known about Otelbaev’s proposed solution, and what a solution would mean for fluid dynamics.
The Navier Stokes Equation
The Navier-Stokes equation is one of the governing equations of fluid dynamics and is an expression of conservation of momentum in a fluid. With the exception of a few very specific and simplified cases, there is no known general solution to equation. Instead, the equation, or a simplified model, is solved numerically using supercomputers as part of direct numerical simulation (DNS) or other forms of computational fluid dynamics (CFD). These methods allow scientists and engineers to solve the equations of fluid motion for practical problems from flow through a pipe to flow around a re-entering spacecraft.
Existence and Smoothness
Although the Navier-Stokes equation has been known for more than 150 years and can be solved numerically for many situations, some basic mathematical aspects of the equation have not yet been proven. For example, no one has proven that a general solution always exists in three-dimensions and that the energy of such a solution is bounded at all points. Colloquially, this is known as the Navier-Stokes existence and smoothness problem. The Clay Mathematics Institute has a very specific problem statement (PDF) asking for a proof (or counter-proof) of the existence and smoothness of the Navier-Stokes equation for an incompressible fluid in three-dimensions. Otelbaev contends that he has provided such a proof.
Otelbaev’s Proposed Solution
Mukhtarbay Otelbaev is an experienced mathematician with numerous papers addressing related mathematical problems. His latest paper, entitled “Existence of a strong solution to the Navier-Stokes equation,” is freely available online (PDF, in Russian, with an English abstract at the end). There is an ongoing project to translate the paper into English, and mathematicians are already evaluating the validity of this proposed solution. From what I can gather of the paper, it specifically address the Millennium Prize problem and presents Otelbaev’s proposed solution for the existence and smoothness of an incompressible fluid in three dimensions with periodic boundary conditions.
What It Means
As with any announcement of a major technical breakthrough, skepticism is warranted while experts evaluate the proposal. If the mathematical community upholds the validity of Otelbaev’s proof, he may be offered the Millennium Prize and other honors. More importantly, his solution could lead to a better understanding of the nature of the equation and the flows it describes. It is not, in itself, a general solution to the Navier-Stokes equation, but it may be a stepping stone in the path toward one. In the meantime, scientists and engineers will continue to rely on a combination of theory, experiment, and computation to progress our understanding of fluid dynamics.
For More
The story of Otelbaev’s proof and the community’s evaluation of its validity is on-going. You can follow @fyfluiddynamics and the #NavierStokes hashtag on Twitter for updates and commentary. I’d like to specially thank Catriona Stokes, Praveen C, David Sarma, and Glenn Carlson for their helpful links and observations as this story develops.
Fluids Round-up – 11 January 2014
It’s a big fluids round-up today, so let’s get right to it.
- Over at txchnologist, there’s a great article on controlling combustion instabilities in rocket engines with sound.
- Quanta Magazine asks if knot theory can help unravel turbulence. (submitted by iamaponyrocket)
- SciAm takes a look at how FIFA finally got their aerodynamics right so that their video game football (soccer) balls fly correctly.
- The Smithsonian considers an important question: can you fry foods in space?
- The Navy unveiled a fantastic new facility for simulating ocean waves (via J. Ouellette)
- At SciAm, there’s a nice explanation of the polar vortex and its effects on recent freezing weather. For additional background, check out this excerpt from a presentation by meteorology professor Jennifer Francis. (via Nicholas Travers)
- Cold weather also brings a host of new viral videos; NatGeo explains some of the science behind instant snow, ice fog, and frozen bubbles. See also: our own explanation of the instant snow phenomenon.
- io9 looks at the physics of knuckleballs.
- Over at Wired, Rhett Allain questions whether dwarves should stand in floating barrels. Also on the subject of The Hobbit, here’s an analysis of fire-breathing in dragons.
- At SciAm, Kyle Hill explains how inertia lets one pour a drink toward the sky.
- SciAm reports on a manufacturing process for superhydrophobic paper.
- I don’t know what banking has to do with a pool of non-Newtonian fluids, but this Malaysian ad sure makes it look fun. (via physicsphysics and jmlinhart)
- Wired has a great write-up on the mantis shrimp, which kills its prey with cavitation.
- io9 tackles explaining one of the most vexing brain teasers in fluid dynamics, the Feynman sprinkler.
- Finally, today’s lead image comes from our friends at Think Elephants, who study elephant intelligence over in Thailand and occasionally capture the animals’ mastery of fluid dynamics. Be sure to check them out and follow them on Twitter and Facebook.
(Photo credit: Think Elephants International/R. Shoer)
Acoustic Levitation in Three Dimensions
Acoustic sound is a form of pressure wave propagating through air or another fluid. Place a speaker opposite a plate, and its sound will reflect off the surface. The original pressure wave and its reflection form a standing wave. With intense enough sound waves, the acoustic radiation pressure can be large enough to counter the force of gravity on an object, causing it to levitate. We’ve shown you several examples of acoustic levitation before, including squished and vibrating droplets and applications for container-free mixing. Today’s video, however, shows the first acoustic levitation system capable of manipulating objects in three dimensions, an important step in developing the technology for application. (Video credit: Y. Ochiai et al.; via NatGeo)
Hydrophobia
On a recent trip to G.E., the Slow Mo Guys used their high-speed camera to capture some great footage of dyed water on a superhydrophobic surface. Upon impact, the water streams spread outward, flat except for a crownlike rim around the edges. Then, because air trapped between the liquid and the superhydrophobic solid prevents the liquid from wetting the surface, surface tension pulls the water back together. If this were a droplet rather than a stream, it would rebound off the surface at this point. Instead, the jet breaks up into droplets that scatter and skitter across the surface. There’s footage of smaller droplets bouncing and rebounding, too. Superhydrophobic surfaces aren’t the only way to generate this behavior, though; the same rebounding is found for very hot substrates due to the Leidenfrost effect and very cold substrates due to sublimation. As a bonus, the video includes ferrofluids at high-speed, too. (Video credit: The Slow Mo Guys/G.E.)
Aurora From Space
An aurora, as seen from the International Space Station, glows in green and red waves over the polar regions of Earth. These lights are the result of interactions between the solar wind–a stream of hot, rarefied plasma from the sun–and our planet’s magnetic field. A bow shock forms where they meet, about 12,000-15,000 km from Earth. The planet’s magnetic field deflects much of the solar wind, but some plasma gets drawn in along field lines near the poles. When these energetic particles interact with nitrogen and oxygen atoms in the upper atmosphere, it can excite the atoms and generate photon emissions, creating the distinctive glow. Similar auroras have been observed on several other planets and moons in our solar system. (Photo credit: NASA)
Impacting a Viscous Pool
Whenever a hollow cavity forms at the surface of a liquid, the cavity’s collapse generates a jet–a rising, high-speed column of liquid. The composite images above show snapshots of the process, from the moment of the cavity’s greatest depth to the peak of the jet. The top row of images shows water, and the bottom row contains a fluid 800 times more viscous than water. The added viscosity both smooths the geometry of the process and slows the jet down, yet strong similarities clearly remain. Focusing on similarities in fluid flows across a range of variables, like viscosity, is key to building mathematical models of fluid behavior. Once developed, these models can help predict behaviors for a wide range of flows without requiring extensive calculation or experimentation. (Image credit: E. Ghabache et al.)
