Over the past few years, researchers have been exploring the dynamics of droplets bouncing on a vibrating fluid. These systems display many behaviors associated with quantum mechanics, including wave-particle duality, single-slit and double-slit diffraction, and tunneling. A new paper examines the system mathematically, showing that the droplets obey many of the same mathematics as quantum systems. In fact, the droplet-wave system behaves as a macroscopic analog of 2D quantum behaviors. The implications are intriguing, especially for teaching. Now students of quantum mechanics can experiment with a simple apparatus to understand some of the non-intuitive aspects of quantum behavior. For more, see the paper on arxiv. (Image credit: D. Harris and J. Bush; research credit: R. Brady and R. Anderson)
Category: Research
Simulating Early Planetary Impacts
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)
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)
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.
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)
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.)
What Makes Squids Fast
Cephalopods like the octopus or squid are some of the fastest marine creatures, able to accelerate to many body lengths per second by jetting water behind them. Part of what makes its high speed achievable, though, is the way the animal changes its shape. In general, drag forces are proportional to the square of velocity, meaning that doubling the velocity increases the drag by a factor of four. The energy necessary to overcome such large drag increases generally prevents marine animals from going very fast (compared to those of us used to moving through air!) But drag is also proportional to frontal area. Like the bio-inspired rocket in the video above, jetting cephalopods begin their acceleration from a bulbous shape and then shrink their exposed area as they accelerate. Not only does this shape change help mitigate increases in drag due to velocity, it prevents flow from separating around the animal, shielding it from more drag. The result is incredible acceleration using only a simple jet for thrust. For example, the octopus-like rocket in the video above reaches velocities of more than ten body lengths per second in less than a second. (Video credit: G. Weymouth et al.)
Impacts on Sand
Granular materials like sand are sometimes very fluid-like in their behaviors. The high-speed video above shows a ball bearing being dropped into packed sand. Many features of the splash are fluid-like; the initial impact creates a spreading crownlike splash, followed by a strong upward jet that eventually collapses back into the medium. At the same time, many of the impact characteristics are decidedly non-fluidic. Sand has no surface tension, so both the crown and the jet readily break up into small particles. The granular jet is very narrow and energetic, reaching heights greater than the impacter’s drop height. Interestingly, the column begins collapsing on its lower end before the jet even reaches its highest peak. This may be due to the lower energy of the sand particles that were ejected later in the crater formation process. (Video credit: J. Verschuur, B. van Capelleveen, R. Lammerink and T. Nguyen)
The Science of Champagne
Champagne owes much of its allure to its tiny bubbles. Unlike other wines, champagne undergoes a secondary fermentation in the bottle, during which the yeasts in the wine consume sugars and produce carbon dioxide, which dissolves into the wine. When opened, the carbon dioxide can begin to escape. Bubbles form in the glass around imperfections, either due to intentional etching of the glass or impurities left behind by cleaning. Once formed, trails of bubbles rise to the surface, swelling as more dissolved carbon dioxide is absorbed into each bubble. The bubbles then cluster near the surface of the champagne, occasionally popping and creating a flower-like distortion of the surrounding bubbles. The gases within the bubbles contains higher concentrations of aromatic chemicals than the surrounding wine, and the bursting of each bubble propels tiny droplets of these aromatics upwards, carrying the scent of the champagne to the drinker. For more beautiful champagne photos, I recommend this LuxeryCulture article; for more on the science of champagne, see Chemistry World’s coverage. Happy 2014! (Image credits: G. Liger-Belair et al.)
Huddling Penguins and Traffic Jams
Male emperor penguins have the unenviable task of incubating their eggs in temperatures as cold as -50 deg Celsius and winds of up to 200 km/h. To stay warm, the penguins form huddles of up to thousands of individuals. Observations in the wild show that these huddles move in a stop-and-go fashion, with changes propagating through the penguins like waves. Researchers adapted a model used for heavy traffic flow to describe the penguins’ motion. They found that motions like those found in observed penguin huddles could be initiated by slight movements of any penguin in the model huddle, regardless of its position; in other words, the huddle has no leader. They also found that the wave that travels through the penguins can align the huddle to uniform density or help two huddles merge. To learn more, check out the researchers’ video or their paper. (Video credit: D. Zitterbart et al./New Scientist; via J. Ouellette)
