velocity

In 1934 Jean Leray proved that the Navier–Stokes equations always have solutions "in a generalized sense". Whether a solution stays smooth or can break down into a singularity he could not tell, and nobody could for ninety years. In 2000 the Clay Institute made the question one of its seven Millennium Prize problems.

In the dark chamber of the trap, thousands of charged particles dance under electric fields. At first they move freely, like a crowd in a square, but cool the cloud and the dance slows: ions begin to sense each other, their steps become constrained.

In a Paris laboratory, light was made to flow around an obstacle the way water flows around a rock in a stream. But this water is special: it feels no friction.

Water confined between two cylinders flows in smooth layers when the inner one rotates. But once the speed crosses a certain threshold, the layers crumple into toroidal rolls, like a stack of bagels. Physicists call these Taylor vortices: the flow loses stability and a regular pattern emerges.

The early universe was less like a calm pond and more like a boiling cauldron. Matter flowed, collided, and swirled against the backdrop of rapid spatial expansion. To understand how stars and galaxies emerged from this hot soup, physicists build digital simulations. This new work is the third part in a series on lattice-based cosmological modeling. The first two focused on fields; this one deals with fluids.

In calm water, the route is clear. But in turbulent flows, a robot needs a different strategy. Typical approaches demand an accurate map of currents, but in reality, you can't get one. Nature does it differently: fish feel pressure and vortices right against their bodies.

An ocean current hundreds of kilometers wide lives its own life, fed by energy from tiny eddies. Something similar happens in the atmosphere: small swirls pass their power on to huge air currents. This process, the inverse energy cascade, has been known for a long time.

In the intricate dance of active turbulence, where invisible engines spawn chaos, a soft droplet takes on the role of a detective. Its shape is not merely an outline but a sensitive cipher: it reads the hidden signals of its environment.

Quantum matter rarely treats us to the spectacle of self-motion. Where the classical world teems with starling murmurations, fish schools, and bustling bacteria, quantum thermodynamics long remained serene.

In the first moments of its life, the Universe resembled a boiling cauldron. As it cooled, phase transitions occurred — much like steam turning into water, but instead of molecules, the very state of space changed. In this cauldron, bubbles of the new phase were born, akin to bubbles in a pot coming to a boil.

Mountain river water doesn't flow smoothly: it churns, swirls into eddies, and suddenly accelerates. This chaotic motion is called turbulence. But even within it, there are hidden rhythms — rare and intense bursts that briefly alter the entire flow pattern.

A drop of thick shampoo falls into water and freezes into a strange rope. A liquid that both flows and springs back is called viscoelastic. In it, tiny particles under ultrasound behave quite differently than in water. Viscoelasticity is a material's ability to both spread like honey and return to shape like rubber.

Water going down the drain swirls into a vortex. An everyday sight in the kitchen. But this very vortex turned out to be able to reproduce two effects that physicists usually associate with black holes and the rotation of planets. The first effect is the shift of a wave as it goes around an obstacle.

In a pot of boiling water, steam bubbles rise and pop. In the quantum world of a superfluid, bubbles behave differently: they can collide and give birth to vortices — tiny swirling flows, like whirlpools in a river. Scientists modeled such a liquid using a holographic approach.

A drop of water on a table usually sits still. But make the table slightly softer on one edge and stiffer on the other, and the drop will crawl toward the stiffer side. This phenomenon is called durotaxis: the liquid itself chooses where to flow, without pumps or electricity. This trick works because the surface under the drop is not uniform.

In porous materials (soil, rocks, filters) fluids move through tiny, winding channels. When two immiscible fluids, such as water and air, are present simultaneously, their interface constantly shifts: air bubbles push water out of some pores and open paths through others.

In the cold heart of a pure crystal, electrons flow like a viscous fluid — an idea predicted by Lev Landau. But the vortices of this fluid remained invisible, like a painting inferred from the noise of a crowd. Now scientists have created a nanostring: a semiconductor tuning fork with a circular cavity.

In helium cooled to near absolute zero, the liquid stops rubbing against itself and begins to flow without any resistance.

A powder avalanche isn't just a pile of snow sliding down a mountain. It's a cloud racing downhill at car speed. Inside, snow doesn't lie still: it flies: thousands of tiny particles are caught up in air vortices. Until now, no one had seen what actually happens to each snowflake in this chaos. Now scientists have the first footage.

Lymphatic vessels are one-way streets. Through them, fluid flows only forward thanks to tiny flap valves. These valves open in just one direction, creating asymmetry, directionality, without which the flow would stop. In the lab, scientists modeled a vessel and sent compression waves along it, like a parade of gentle nudges.

A drop of ink falls into a glass of water, and ripples spread out from it. Water seems simple, but if you look inside, each molecule isn't just moving forward: it's also spinning like a tiny top. Physicists call this intrinsic rotation spin. Usually, it goes unnoticed: in calm water, the tops spin every which way, and their rotation doesn't affect the fluid's overall motion.

A droplet of liquid metal glides along a narrow channel like mercury from a broken thermometer. It needs no motor — the fuel is a chemical reaction with the surrounding liquid. But what happens when the droplet hits a wall? Previously, such machines were studied in open spaces, and in tight channels they behave differently.

The Sun's corona looks like a shaggy dandelion: bright petal-like loops hold hot gas near the surface, while long, tenuous jets escape into space between them. These jets, the heliospheric current sheet, stretch across the entire solar system.

A cell is a city with gates. Each gate is a protein channel in the membrane: through it, ions, sugars, and signaling molecules pass in and out. Some gates are almost always open, while others open and close randomly, like subway doors that let a passenger through one moment and slam shut in their face the next.

Cilia are tiny hairs on cell surfaces. They beat like oars, moving fluid. In the lungs, they push dust and microbes out. But their lengths vary: in humans, they are thousandths of a millimeter; in comb jellies, several millimeters. How do such different cilia work together? Scientists built a machine that jerks fluid sharply, as if a cilium had struck. They filmed it with a high-speed camera.

In turbulent air, droplets don't fly in straight lines: they get tossed around like dust motes in a beam of light. Physicists have built a setup that tracks each droplet, which is about a tenth of the thickness of a human hair. A camera captures their motion, and a computer reconstructs their trajectories.

A thin stream of water on a shower glass gathers into winding rivulets that quiver and shift before your eyes. Such a liquid film flowing down a vertical surface is a classic example of chaos in nature: it all flows downward, sure, but the surface lives its own unpredictable life.

Liquid helium at ultra-low temperatures enters a state of superfluidity: it flows without friction, like a perfect lubricant.

A sudden gust hits a wing, and a huge vortex swirls above it. Such a vortex can abruptly change lift, making the plane's nose dip. Engineers have long sought ways to predict these vortices, but complexity gets in the way: fast airflow is full of small chaotic eddies.

The lymphatic system is like a city's network of canals, through which water slowly moves, carrying away waste from tissues. But it has no single pump-heart. Instead, along the vessels are placed small station-locks — lymphangions. Each such lock squeezes itself like a fist and pushes fluid onward. How they coordinate their work has long remained a mystery.

A pot of water heated from below is a simple example of turbulence. Hot fluid rises, cold fluid sinks, and the flow spontaneously breaks into rotating cells that look like horizontal rolls. In a new study, physicists took a wide rectangular container (ten times longer than its height) and tracked how many such rolls appear.

A drop falls into a glass, and a tiny jet, a thin fountain, shoots up from the surface for an instant. Something similar happens with a bubble stuck to the wall of a tube when you yank the tube downward. Water around the bubble rushes in, and a jet fires from its top. The speed of this jet depends on the bubble's shape — how much it bulges, like an umbrella dome.

Hot plasma in a future fusion reactor is like a packed dance floor: billions of particles whirl in a complex rhythm, colliding and changing speed. Figuring out how tangled this dance is — no easy task. Scientists have found a way to measure its complexity without watching every move.

A sheet of paper released from a window flutters, tumbles, and sometimes pauses flat for a second. Until now, to honestly predict such behavior, you had to run millions of equations on supercomputers for every drop of air. That's like simulating every atom of a cash register to count change at a store. The new approach is smarter.

A young star is surrounded by a flat, rotating disk of gas and dust — the construction site for future planets. The disk lives its own life: gas flows toward the star, dust clumps together, and from outside, harsh ultraviolet light from neighboring hot stars presses down on it all, like wind blowing leaves off a path.

The wind blows steadily, but suddenly a gust hits, and leaves swirl up in a vortex. Inside such a gust, the air speed changes not smoothly but in jumps — sometimes sharply forward, sometimes sideways. These rare and strong bursts are called intermittency, and they are the main mystery of turbulence.

Water in a stream bends around rocks, but the math of that motion hides in a black box: that's how ordinary neural networks work. They give an answer but don't say why. Engineers can't trust a prediction if they can't see its reasoning. Fluid dynamics now has a transparent model.

A microscopic gel bead or a tiny oil droplet squeezes through a narrow channel whose walls rhythmically contract and relax. The particle flattens and then recovers its shape, and by how quickly it does so, you can infer its internal elasticity and viscosity.

Water swirls into a ring, and that ring flies through the bulk without breaking apart. Such vortex rings, from smoke rings to underwater currents, follow laws that are hard to capture on paper. Usually, the flow is divided into millions of cubic cells, and each one is tracked.

Water in a puddle after rain swirls differently than in a deep lake. A thin layer of fluid feels the bottom, and that changes the nature of its vortices. Scientists created such a layer in the lab: they poured in an electrolyte, ran a current and a magnetic field, and the fluid began to move.