Navier-Stokes Equations

Water flowing around a stone in a stream. Smoke swirling in the air. Wind sweeping over a wing. All these motions follow the same physical rules. The Navier-Stokes equations are the mathematical formulation of these rules. For every point in a liquid or gas, they tell you how fast and in which direction the material is moving there, and how that changes in the next instant. They are named after the Frenchman Claude-Louis Navier and the Irishman George Stokes, who formulated them in the first half of the 19th century.

Why the Millennium Problem is worth a million dollars

The equations have been known for almost two hundred years. Yet no one fully understands them. To this day, it remains unproven whether they yield sensible solutions in three dimensions for all time. It could be that the calculation suddenly shoots off to infinity at some point. Physically, that would make no sense, since water never becomes infinitely fast anywhere.

The Clay Mathematics Institute added the problem to its list of seven Millennium Problems in the year 2000. A solution is worth one million US dollars. Six of these seven problems remain unsolved to this day, and Navier-Stokes is one of them. Alleged proofs keep surfacing again and again, but none has withstood scrutiny.

Practically speaking, the equations are relevant nonetheless, and have been for a long time. Engineers and meteorologists work with them every day, even though the mathematical foundation is incomplete. It is known that the results hold up in practice. It just cannot be proven that they always do.

From equation to computational grid

At their core, the equations are an application of Newton’s law of force equals mass times acceleration — just applied to a material that is constantly changing shape. Several forces act on a tiny fluid parcel: the pressure from neighboring parcels, the internal friction of the material, and external forces such as gravity. Internal friction is called viscosity and describes how thick or sticky a substance is. Honey has high viscosity, air has very low viscosity.

An exact formula that can simply be calculated exists only in heavily simplified cases. That’s why a numerical approach is used: space is broken down into millions of small cells, known as a grid. For each cell, the computer calculates what happens in the next tiny time step. Then it repeats this thousands of times. This method is called Computational Fluid Dynamics, or CFD for short.

The catch is the computational cost. Turbulent flows, i.e. turbulence, contain structures ranging from the size of an aircraft down to millimeters. Anyone wanting to resolve everything needs absurdly fine grids. In practice, approximation models are therefore used for the smallest eddies. This is exactly where AI models are now stepping in too: neural networks learn from past simulations how flows typically behave, delivering approximations in seconds instead of hours.

Weather report, wind tunnel, heat pump

Every weather forecast is based on Navier-Stokes. The atmosphere is a gas in motion, and the computational grid spans half the planet. The fact that forecasts become unusable after about ten days is rooted in the nature of these equations. The tiniest differences in starting values grow into completely different outcomes. This is the famous butterfly effect.

In industry, simulations are increasingly replacing real wind tunnels. Carmakers optimize aerodynamic drag on screen before a single sheet-metal part exists. The same applies to turbine blades, ship hulls, heat sinks in data centers, and blood flow through artificial heart valves.

In tech news, you’ll mostly encounter the term in the context of supercomputers and AI weather models. Systems like GraphCast from Google DeepMind or Aurora from Microsoft predict the weather without solving the equations step by step. Instead, they have learned from decades of weather data. A common misconception is that this makes the underlying physics obsolete. In fact, the training data comes from classical simulations — without Navier-Stokes, there would be nothing to learn from.