The Navier-Stokes equations are a mathematical expression of mass conservation and Newton's second law of motion—force equals mass times acceleration—applied to continuous fluids.

Based on accepted laws of physics, the Navier-Stokes equations are fundamentally accurate, but it is unclear whether they always provide smooth, sensible solutions in 3D.

Although fluid motion has been modeled by the Navier-Stokes equations for decades, mathematicians have long sought to prove whether the equations always provide realistic solutions.

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.

Newton's laws describe how objects resist acceleration and respond to forces

Newton’s first law defines inertia as the tendency of an object to maintain its velocity. His second law connects external force, inertia, and acceleration, while his third law states that forces come in pairs—every action has an equal and opposite reaction.

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.

Explore a simplified breakdown of the Einstein field equations

Often depicted as one formula, Einstein's equation represents a suite of 16 nonlinear partial differential equations, which can be simplified to six unique ones, that relate the geometry of spacetime to the presence of mass and energy.

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.

Mathematics proves turbulence's chaos creates super diffusion

Turbulent flow is responsible for the rapid scattering of dandelion seeds, volcanic ash, and untied balloons. Its swirling behavior accelerates particle scattering faster than other mixing methods, as particles acquire energy from the interactions of eddies.

Understanding turbulence, which is characterized by growing chaos at multiple scales

As energy is lost in fluids undergoing turbulent, non-linear flow, particles will collectively form eddies, or whirlpools, of various sizes that mix together. The energy cascade of larger eddies becoming smaller ones before dissipating completely varies based on the initial conditions of the fluid, a characteristic feature of chaotic systems.

Conservation laws identify physical properties that remain unchanged in systems

The total amounts of properties like energy, charge, and angular momentum within a system have been experimentally observed to remain constant, even as changes occur. Particle interactions also conserve specific quantities, such as baryon and lepton numbers. Inconsistencies in these values within isolated systems have historically hinted at the existence of new physics.

Modified Newtonian dynamics revises gravity to eliminate the need for dark matter

Rather than imagining that an undiscovered particle is responsible for discrepancies in the orbits of stars and galaxies, MOND suggests that the mathematics underlying Newton's laws of gravity and force change beyond a certain scale. However, the suggested revisions struggle to explain relativistic effects involving gravity, including gravitational lensing and gravitational waves.

View the transition from laminar to turbulent flow

Using a dye released in the water, streamlines, which are parallel to one another and consistent at low flow velocities, become visible. As the water moves more quickly, viscosity cannot suppress the velocity change, and streamlines become chaotic, creating turbulence.

Laminar flow moves in smooth, parallel layers, while turbulent flow is chaotic

In laminar flow, particles that make up a fluid move in consistent paths that do not cross each other. Contrastingly, differences across the speed of these particles cause them to mix, creating turbulence when viscosity cannot suppress the differences.

Classical mechanics describes the motion of and forces on everyday objects

Newton's laws form the core of classical mechanics, suitable for everyday objects and speeds. The term is relatively new, distinguishing the branch of physics from quantum and relativistic mechanics.

Horsehead-shaped features seen on the sun and the clouds of Earth and Jupiter are examples of Kelvin-Helmholtz instabilities, where a faster-moving fluid rolls atop a slower-moving fluid.

The centrifugal force is associated with the feeling of being pushed outward when moving in a circle, but it is a fictitious force arising from objects' tendency to move in straight lines.

Virtual particles are bookkeeping tools used to describe force interactions

Physicists mathematically describe the electromagnetic, weak nuclear, and strong nuclear forces as the result of these particles being exchanged between particles of matter. Despite not being real, observable particles, this method has provided experimental predictions with an accuracy of over 12 decimal places.

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.

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.

Viscosity is a fluid's resistance to flow, which varies with temperature

Liquids become less viscous as temperature increases, while gases become more viscous and therefore flow more slowly. The increased gas collisions at higher temperatures contribute to greater friction between streamlines, increasing viscosity.

Friction is the force that resists the sliding of surfaces along one another

Friction always acts to resist movement and is determined by how rough the surfaces are and how hard they are pressed together. In the case of fluids, friction arises from the relative motion between adjacent fluid layers, which act like sliding surfaces.

Winter Olympic sports highlight various laws and principles of physics

Figure skaters leverage the conservation of angular momentum during spins by tightening their bodies, and the skintight suits used in ski jumping help reduce drag. Centripetal force allows speedskaters to make tight turns, while friction on skis helps ski mountaineers climb slopes. No action can violate the conservation of energy.

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.

Within Einstein's picture of the universe, gravity is technically not a force because forces cause accelerations that deviate objects from their natural path, and moving solely due to gravity is natural.

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.

Accurate climate models incorporate calculus to account for complex feedback loops

The simplest models assume uniformity across the Earth, with incoming energy from the sun being offset by the planet's albedo, or reflectiveness, which prevents the radiation from being fully absorbed. Incorporating elements such as atmospheric layers, latitude bands, geographic variability, and emissivity yields coupled differential equations quantifying Earth's interconnected systems.

According to the equivalence principle, objects in a closed space cannot tell the difference between whether they are accelerating or experiencing a uniform gravitational field.

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.

Quantum mechanics and general relativity—two of the most successful explanations of the physics seen across the universe—are logically incompatible with one another because they use contradictory models for space and time.

The underlying fabric of the universe—spacetime—changes shape in the presence of mass and energy, which affects measurements of space and time.

While special relativity describes motion along a flat underlying fabric of the universe called spacetime, general relativity extends this theory to explain motion through curved spacetime.

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.

Einstein created a mathematical term for anti-gravity to keep the universe static

General relativity predicted that the gravity of all the matter in the universe would cause it to collapse, contradicting the belief at the time that the universe was constant and unchanging. Albert Einstein added the cosmological constant to his equations, which acted like anti-gravity, to "fix" his math.

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.

Explore interactive simulations of strange attractors, the graphs that help visualize chaos

Although simple, linear equations reach an explicit value for a given input, such as *f(x) \= x* + 1 equaling 4 when *x* equals 3, some chaotic, nonlinear systems tend to evolve towards multiple values simultaneously. These simulations allow users to observe the unusual mathematical artifacts created by systems as their outputs evolve toward these distinct values.

Turbulence is needed to make airplanes more stable and golf balls fly farther

Vortex generators—small fins placed along airplane wings—create small eddies that prevent airflow separation, maintaining lift and preventing stalls. Golf balls have dimples that create a turbulent boundary layer on their surface, minimizing drag forces.

Turbulence is crucial to the formation of stars in molecular clouds

Within simulations of these clouds, gravity, jets, supernovae, and galactic motion-induced stirring have been shown to develop turbulence that creates shocks. These turbulent shocks move quickly in low-density regions and slow in high-density areas, forming lumps of matter with sufficient gravitational force to collapse and initiate fusion, birthing a star.

Learn why modern scientific patterns have been described as theories rather than laws

After confirming that the accuracy of some laws of physics is limited to certain domains, science became more hesitant to categorize identified mathematical relationships, such as quantum mechanics, as unchanging laws. Instead, these recent, complex frameworks are designated as theories to acknowledge that they are conditional and may be part of a larger, more complete future framework.

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

An explanation of isotopes, which can help scientists trace food, water, and farming choices

Atoms of the same element have the same number of protons, but experiments showed that atoms of the same element can move through electromagnetic fields as if they had different masses. These differences in mass come from variations in the number of neutrons, which contribute to the stability of atoms by providing a buffer to the repulsion between protons.

Gravity, electromagnetism, and two nuclear forces make up nature’s fundamental forces

While gravity is the most familiar force, it is also the weakest and the only one without a particle to mediate the force. The weak nuclear force is responsible for radioactive decay and transmutation of elements, while the strong nuclear force holds quarks together.

The solar system's planets orbit in the same direction in a nearly flat disc because the particles that made up the gas cloud from which the system formed had an initial, overall plane of rotation.

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.

Discoveries in electromagnetism, which suggested that light traveled at a single speed, became the catalyst for overturning Isaac Newton's view that all speeds are relative and could appear to change based on one's perspective.

Special relativity shows how motion at high speeds reshapes time, space, and energy

Einstein’s 1905 special theory of relativity revealed that the speed of light is absolute in the universe, forcing time and space to vary with one's speed. This idea led to the famous equation E=mc2, which shows that mass and energy are interchangeable.

Beyond a certain pressure and temperature, matter can become a supercritical fluid—an exotic state of matter that simultaneously exhibits properties of gases and liquids.

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.

Observations of colliding neutron stars by the Laser Interferometer Gravitational-Wave Observatory in 2017 confirmed that gravity travels at the speed of light.

What is gravity made of?

Our world at very small scales is governed by quantum mechanics, in which forces and energy come in discrete packets. At large scales, our world is governed by gravity, which moves the planets and stars and can create waves in spacetime. To merge...