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

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.

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.

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

The equations model fluids and have been shown to have solutions for all systems in two dimensions. Averaging and approximations let the equations be used in climate models, aerodynamics, and medicine, though their use in some systems produces nonsensical answers.

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.

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.

The R0 value is how epidemiologists quantify disease contagiousness

The basic reproduction number indicates the average number of people a person with a disease will infect. Measles is the most contagious disease, with an R0 of 12-18, though vaccines have significantly reduced the number of susceptible individuals.

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.

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.

Understanding R0, a measure of how contagious a pathogen is

The R0 value is a basic measure of how many people an infected person will transmit a pathogen to, on average. If R0 is less than 1, transmission dies out; if greater than 1, an outbreak is likely. Measles has one of the highest known R0 values at almost 18, meaning that after four transmission steps, more than 100,000 people could be infected.

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.

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.

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.

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.

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.

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.

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.

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.

What is the Gini coefficient?

The Gini coefficient is a measure of income inequality on a scale from 0 to 1. A value of 0 indicates perfect equality, where everyone has the same income, while a value of 1 indicates perfect inequality, where one person has all the income and...

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.

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.

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.

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.

Lava flow viscosity plays a significant role in the type of volcano formed

Thin, runny flows spread quickly before cooling, producing expansive layers that build over time into wide shield volcanoes. The ejection and fast hardening of thicker lava flows and pyroclastic material create steep slopes and the characteristic cone shape of composite volcanoes.

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.

Once cooled to temperatures just above 2 Kelvin, helium atoms exhibit a quantum property that removes internal friction, allowing the fluid to climb up walls.

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.

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.

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.

The magnetic constant sets the strength of magnetic fields

The permeability of free space, or μ₀, contributes to the speed of light in the universe—a fundamental constant. It also helps quantify how the movement of electric charges generates a magnetic field and vice versa.

Their 1901 Wind Tunnel system revolutionized the study of aeronautics

Only the second to be built in the US, the wind tunnel allowed the brothers to study how air moves around more than 200 different wing shapes. This data helped them refine a key math equation, long held as the standard, that had been based only on one wing shape.

Zero-g rides get their name from the weightless sensation of being turned upside down

Inversions are when riders are flipped upside down on rides. Zero-g rolls involve turning 360 degrees with inversions, resulting in a temporary experience of zero g-force for passengers.

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.

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.

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.

Reaching 2,400 kilometers per hour (1,500 miles per hour), Neptune's winds are the strongest in the solar system.

Ranked-choice voting redistributes ballots until a candidate wins a majority

In ranked-choice voting, voters rank candidates in order of preference. If no one wins a majority, the lowest-ranked candidate is eliminated, and their votes are reassigned until a candidate surpasses 50%, avoiding separate runoff elections.

Explore the fundamental constants of the metric system

In 2019, the International System of Units was revised, such that all seven fundamental units are defined in terms of combinations of seven fundamental constants of the universe. These include the speed of light in a vacuum, Planck's constant, Avogadro's number, and the elementary charge of protons and electrons.

General relativity describes a singularity as the final moment in time

Using Penrose diagrams—mathematical structures that compress all of space and time—physicists can observe the path of light and which events can affect one another. The space-time maps help visualize the structure of black holes and describe what happens when objects fall in.

Unlike hurricanes, tornadoes are not ranked based on measured wind speed

Technological limitations prevent direct measurements of wind speeds where tornadoes make contact with the ground. Instead, the Enhanced Fujita scale assigns a rating with a range of speeds based on how much damage a tornado caused to various types of structures. These types are more extensive than the original Fujita scale.

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.

The 'butterfly effect' is the informal name for chaos theory

In mathematics, chaos theory involves complex systems that are highly sensitive to their initial conditions. Originally discovered when analyzing weather data, slight variations in the data input into such systems create drastically different results, despite following known physical laws.

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.

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.

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.

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.

The life of Edward Lorenz, who originally used a seagull in his 'butterfly effect' metaphor

By accidentally finding that rounding one variable to its third decimal place and re-running a weather simulation produced a drastically different two-month forecast, the American mathematician established the field of chaos theory. His paper—"Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?"—introduced the well-known metaphor for the behavior of chaotic systems. (Some readers may experience a paywall.)

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.

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.

Learn how natural systems exhibit characteristics of chaos theory

In biology, some mathematical models of ecosystem populations and heart irregularities show strong sensitivity to initial conditions, which scientists can simulate to determine quantitative tipping points at which these systems become unstable. Within the social sciences, crowd behavior and the virality of social media posts have shown how human behavior fits within the framework of chaos theory.