The Navier-Stokes equations are fundamentally accurate but unsolved in 3D

The Navier-Stokes equations include the fluid version of Newton's second law

The first equation is the fluid version of the conservation of mass, which prevents incompressible fluids from gaining or losing matter. The second equation replaces acceleration and inertia, which is resistance to acceleration, with fluid pressure and viscosity.

Turbulence has been called the most important unsolved problem in classical physics

Despite being the most common type of fluid flow in our world, scientists have struggled to model turbulent flow accurately. Though behaving unpredictably, understanding inertia and viscosity can help determine whether turbulence will occur.

Climate models use complex equations and supercomputers to predict future climate

The equations apply the laws of thermodynamics, fluid motion, and radiative transfer to a 3D grid along Earth's surface. After solving them for each cell in the grid over a time interval, the results carry over to neighboring cells, and the process repeats into the future.

The Reynolds number is the ratio of viscous and inertial forces in fluids

It can approximate fluid behavior in simple systems by non-dimensionalizing—removing the units within—the Navier-Stokes equations. A value close to zero indicates that the fluid will exhibit laminar flow, while a large number suggests that turbulence will occur.

When experiencing turbulence, fluid elements undergo stretching and folding

Simulations show patches of fluid particles, such as splashes of milk, bend under turbulent flow, creating curvature that grows linearly then exponentially. This accelerated deformation enhances folding and mixing faster than other methods, such as stirring or beating.

Researchers identify a hidden instability pattern in simple ocean wave models

Stokes waves—a solution to the mathematics of water waves—were once assumed to be stable. Italian mathematicians have proven disturbances cause these waves to collapse and such collapses follow an infinite repeating pattern. To complete the proof, the team offered a $100 donation to solicit online help.

Vincent Van Gogh's 'The Starry Night' accurately depicts turbulent flow

Analysis of the swirling patterns in the painting found an alignment with the mathematics used to describe turbulence, despite Van Gogh's unfamiliarity with the science. The depiction may indicate he possessed an uncanny ability to capture nature's details.

Explore a simulation of ideal gas behavior, which can be explained by kinetic molecular theory

The theory describes particles as moving in random directions, with a distribution of velocities determined by the gas's temperature. The particles collide elastically like billiard balls, are negligibly small, and do not exert forces on one another. The gas laws can be derived from these principles using statistics or observed via this interactive.

View the Lorenz attractor, a graphical solution to chaotic equations that resembles a butterfly

The discovery of chaos theory came from Edward Lorenz's computer model of the atmosphere, which used three equations to quantify atmospheric circulation. However, the chaotic evolution of this system creates wandering figure-8-like shapes—some resembling butterfly wings—as it repeatedly attempts to settle on one of two values along unique trajectories.

View 3D renderings of atomic orbitals created with quantum mechanics

Scientists use the Schrödinger equation to determine how a system governed by a wavefunction—an equation that predicts the probability of certain outcomes—evolves with time. Computers can animate atomic structure and behavior by depicting the wavefunction as a fluid that guides a particle, like water moving a speck of dust.

Using wind tunnels, the brothers corrected a centuries-old error used to measure lift

When their early gliders failed, the Wright brothers scrutinized inherited data analysis. Smeaton’s coefficient had been used to calculate the density of air (0.005) for centuries. The Wright’s data validation through the use of small, laboratory wind tunnels showed the correct number was 0.0033.

Uncertainties in measurements make finding gravity's Newtonian limit unclear

Although Newton's "universal" law is unreliable in strong gravitational fields when compared with the field equations of general relativity, it is still taught for its accuracy on everyday scales. However, gravity's effect on small scales is too weak to be measured reliably, its accuracy at atomic scales remains unknown.

View a breakdown of the mathematics behind the Standard Model

The Lagrangian version of the theory—related to the system's energy—presents a vast mathematical equation, with different components representing various force interactions and types of particles, including ghost particles to cancel out mathematical redundancies.

Explore a visualization of curved spacetime under general relativity

While depictions involving bends in elastic fabrics are common to illustrate gravity, they are circular representations that rely on gravity in the real world to depict gravity in the model. The river model instead shows how bends in spacetime convert speed through time into speed through space, producing acceleration.

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

Using generative AI to create video introduces physics-compatibility challenges

Models involved in generating video from text must maintain spatial and temporal coherence while adhering to the laws of reality to produce smooth, logical motion. Ensuring this consistency creates long-term dependencies with high computational times.

Simulation of the coronavirus inside an aerosol droplet

A new simulation shows how the virus survives inside tiny airborne particles of water — and gives clues about how the Delta variant became dominant.

How 'vibes' factor into scientists' decisions about using certain units

Although there are absolute quantities from which units can be defined, such as the speed of light and the corresponding light year for distance, the usefulness of units comes from their ability to easily communicate values, rather than how fundamental their origins are. For example, while Kelvin measures absolute temperature, Celsius is scaled to the temperatures of boiling and freezing water, making them more practical for everyday usage.

Swimming in schools helps some fish conserve energy in turbulent flows

Scientists studying groups of giant danios swimming in turbulent waters found that the fish used up to 79% less energy than they did when swimming individually in the same conditions. The same fish were found to use the same amount of energy in turbulent and calm water when in schools.

Understanding why suction cups rely on Earth's atmosphere to work

When pressed against a surface, air is squeezed out of a sealed volume, causing the internal pressure to drop below the surrounding atmospheric pressure, and this imbalance creates the "sticking." For this reason, astronauts do not use suction cups or related technologies to stay on spacecraft during spacewalks.

This flat map is still wrong, but it's the least wrong of all flat maps

While flat maps can never be as perfectly scaled as a globe, they're far more practical than carrying around a sphere. Researchers set out to minimize six major errors of flat maps, including boundary cuts and skewed proportions, by simply squashing...

For the same cost, turbines with three blades are the most resilient

Odd-numbered blade values more evenly distribute wind loads across the face of rotors, reducing bending and twisting forces on the turbine that cause wear. Modeling shows that three blades optimize efficiency and cost for everyday wind speeds.

Why the ratios of helium isotopes are analyzed to investigate ocean circulation

Volcanic activity on the seafloor produces high ratios of helium-3 to helium-4, which can be tracked in ocean samples more easily than other elements because helium is nonreactive. Water from deep ocean currents can also be distinguished from air-exposed surface water sources by comparing these ratios, since atmospheric helium ratios are much lower.

Hurricanes, which rely on air consistently spinning in the same direction, cannot cross the equator because air curves in opposite directions north and south of it.

Explore the heart in this interactive 3D model

The heart is the fleshy muscle at the heart (pun intended) of the circulatory system. The heart is segmented into four chambers: two for intake (the right side) and two for output (the left side). Controlled via the autonomic nervous system, also known as the involuntary nervous system, the heart typically beats 60 to 100 times per minute, sending nutrient- and oxygen-rich blood throughout the body.

Scientific progress works toward absolute truth, but can never reach it

Through the acquisition of scientific knowledge, scientists construct explanations of the world, which are rigorously tested and verified until they narrow down to a leading theory of the natural world. However, because of how science works, these ideas will never become factual, as they must always remain open to be iteratively revision based on new data.

Hurricanes spin due to differences in air and ground speed as they move across Earth

The movement of an object across latitudes causes it to take a curved trajectory because its speed no longer matches that of the ground beneath it. This phenomenon, called the Coriolis effect, explains why hurricanes spin counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere.

According to the physics of black holes, our 4D reality may be a hologram—a projection of information coded on a 2D surface on the edge of the universe.

While the sun's gravity keeps the planets in orbit, their gravity also pulls on the sun, causing it and everything else in the solar system to orbit around a mutual balance point called the barycenter—the true center of the solar system.

Scientists can determine Earth's past climate conditions by analyzing air trapped in microscopic bubbles within ice cores—columns of compressed snow layers that build up over hundreds of thousands of years.

The boiling temperature of water decreases with increasing elevation because the molecules need less energy to overcome the lower atmospheric pressure associated with thinner air.

Most of the observable universe's volume consists of voids and supervoids—enormous regions of space with little to no galaxies, stars, or light.

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

The deadliest aspect of volcanic eruptions is often not lava, but pyroclastic flows—avalanches of ash, gas, and rock that can move more than half the speed of sound.

English mathematician Ada Lovelace wrote the first computer algorithm a century before computers existed and hypothesized that computing devices would one day be able to compose music—foreseeing the generative AI systems of today.

Pure water does not conduct electricity because this flow of energy requires the movement of charged particles, and water is a neutral molecule.

After 25 people used virtual reality headsets to practice flying with virtual wings—flapping their arms—their visual cortex responded to the wings almost as if they were real arms, treating them as body parts.

Objects can be 3D printed with light by projecting 2D computerized tomography (CT) scans onto rotating cylinders of resin, which solidify upon exposure.

Clever Hans was a 1900s horse that was believed to be able to solve math equations, tell time, and read and comprehend German, until a psychology student discovered how he got his answers.

The American Society of Civil Engineers' infrastructure report card gave the US a C in 2025, its highest grade since the project launched in 1998.

How the most famous Ebola book, The Hot Zone, which sold 3.5 million copies and was translated into 30 languages, got aspects of the disease so wrong.

To reduce wrinkles in clothing, research recommends gentle wash cycles with more water and tumble drying with smaller loads and longer times.

Waves off the coast of Nazaré, Portugal, can reach over 100 feet and are caused by the area's deep underwater canyon geography

The 19th-century painters of the Hudson River School were among the first American romantics, with portraits that displayed the natural beauty of the young country.

SpongeBob exemplifies Aristotle's philosophy for living well because he understands that happiness doesn't come from external factors like money, recognition, or power.

Play the role of a student in a game where you try to navigate college and remain debt-free, selecting different income levels, high school experience, race, ethnicity, and more.

Gerrymandering works through two techniques: cracking opposing voters across districts to dilute their power, and packing them into a few to waste their votes in lopsided majorities.

Bone marrow (soft tissue found inside certain bones) generates roughly 500 billion new blood cells each day

The word 'jazz' evolved from 'jasm,' 19th-century American slang for 'energy'

See a 3D reconstruction of Tenochtitlan, the Aztec capital that was home to an estimated population of 200,000 to 400,000.