The Navier-Stokes equations are the fundamental mathematical description of how fluids move. They govern everything from the airflow over an airplane wing to the blood pumping through your arteries, from ocean currents to the swirling cream in your coffee. First presented publicly in 1822 by the French engineer Claude-Louis Navier, and later refined by the Irish-British mathematician George Gabriel Stokes, these equations encode Newton’s second law of motion for continuous fluid substances, accounting for pressure, viscosity, and external forces. Despite being nearly two centuries old and underpinning a vast portion of modern engineering, the equations harbor one of the deepest unsolved problems in all of mathematics.
What the Equations Actually Describe
At their core, the Navier-Stokes equations express a deceptively simple idea: at every point in a fluid, the forces acting on a tiny parcel of that fluid determine how it accelerates. The forces in play are pressure pushing from neighboring fluid, the internal friction of the fluid (its viscosity), and any external forces like gravity. The equations track the velocity and pressure of the fluid at every point in space and at every moment in time. For an incompressible fluid like water at everyday speeds, this boils down to two rules: the fluid’s velocity field must change in a way that balances all those forces, and the fluid cannot be compressed, so whatever flows into a small region must flow out.
That simplicity is deceptive because the equations are nonlinear. The fluid’s own velocity appears in the term that describes how it carries momentum from place to place, which means the output of the equation feeds back into its input. This self-referential quality is what makes the equations so rich and so resistant to exact solutions. Turbulence, the chaotic swirling you see in a fast river or behind a truck on the highway, emerges naturally from the Navier-Stokes equations, but predicting its fine details remains one of the hardest problems in physics.
How the Equations Came to Be
Navier’s contribution came in two memoirs. The first appeared in the Annales de Chimie et de Physique for 1821 (printed in 1822), and both were read at the French Academy of Sciences on March 18, 1822, the date generally taken as the equation’s public debut. These memoirs introduced friction into the equations of fluid motion for the first time, an idea that the existing inviscid equations of Euler, written decades earlier, completely lacked.1SciELO / Revista Brasileira de Ensino de Física. 200 years of the Navier–Stokes equation Navier’s reasoning was rooted in a molecular picture of fluids, imagining tiny particles that attract and repel their neighbors. Stokes, working independently in the 1840s, arrived at the same equations through a more rigorous continuum-mechanics approach, treating the fluid as a smooth, deformable substance rather than a collection of molecules. Several other scientists, including Poisson and Saint-Venant, also derived versions of the equations during the intervening years, but it was Navier and Stokes whose names stuck.
The two-hundred-year gap between Navier’s original presentation and the present day has not been uneventful. Prandtl’s boundary-layer theory in 1904 showed that at high speeds, viscosity matters only in a thin layer hugging a solid surface, while the rest of the flow behaves as if frictionless. That insight made practical calculations possible long before computers existed, and it remains a cornerstone of aerodynamic design.2Topical Problems of Fluid Mechanics 2026. Prandtl’s Boundary-Layer Theory as a Basis for Improving Aerodynamic Shapes
The Million-Dollar Math Problem
In the year 2000, the Clay Mathematics Institute named seven “Millennium Prize Problems,” each carrying a one-million-dollar reward. The Navier-Stokes existence and smoothness problem is one of them, and it remains unsolved. The question sounds straightforward: given a smooth initial state for a three-dimensional fluid, do the Navier-Stokes equations always produce a smooth, well-behaved solution that exists for all future time? Or can the equations develop singularities, points where velocity or pressure blows up to infinity in finite time?3Russian Mathematical Surveys. Sixth problem of the millennium: Navier-Stokes equations, existence and smoothness
In two dimensions, the problem was resolved decades ago: smooth solutions do exist for all time. But three dimensions are a different beast. The extra spatial dimension opens up pathways for energy to concentrate in increasingly small regions, and mathematicians have not been able to rule out the possibility that the equations produce physically nonsensical infinities. Nobody has proved that singularities form, either. The problem sits in a frustrating limbo. Partial results exist: smooth solutions are guaranteed for short times, and for “weak” solutions (a relaxed notion that allows some roughness), global existence can be shown. But proving that these weak solutions are actually smooth, or that smooth solutions persist forever, has eluded every attempt so far.
This is not a purely academic curiosity. If singularities do form, it would mean the equations fail to describe the real world at those moments, since physical fluids do not produce infinite velocities. That would signal a fundamental gap in our mathematical model of something as commonplace as water.
Solving the Equations in Practice
Engineers and scientists cannot wait for the Millennium Prize to be settled. They need answers now, for designing jet engines, predicting hurricanes, and simulating blood flow. The practical workaround is computational fluid dynamics, or CFD: chop space into a fine grid of cells, approximate the equations at each cell, and march the solution forward in small time steps. This approach transforms the continuous equations into a massive system of algebraic equations that a computer can crunch through.
The accuracy of the result depends heavily on how you approximate the equations on that grid. Comparisons of different schemes for airfoil simulations have found that using higher-order approximations for all terms, not just the main flow variables, substantially outperforms lower-order methods, producing less than two percent error in lift and drag on grids with fewer than 18,000 nodes even for transonic flows.4Journal of Computational Physics. Comparison of Several Spatial Discretizations for the Navier–Stokes Equations Other approaches combine boundary element methods with finite difference approximations to handle two-dimensional flow problems, converting the spatial equations into a system that can be stepped forward in time using standard numerical techniques.5Engineering Analysis with Boundary Elements. Numerical solution of 2D Navier–Stokes equation discretized via boundary elements method and finite difference approximation
Even with modern supercomputers, fully resolving every swirl and eddy in a turbulent flow remains impossibly expensive for most real-world problems. A simulation of the airflow around a full-sized car at highway speed, resolving every detail of the turbulence, would require a grid so fine and time steps so small that the computation could take years. That constraint drives the field toward turbulence modeling, where the small-scale chaotic motion is approximated rather than directly simulated.
Dealing with Turbulence
Turbulence modeling is where the rubber meets the road for most engineering applications of the Navier-Stokes equations, and it comes in tiers of fidelity and cost.
The most common industrial approach is RANS, which stands for Reynolds-Averaged Navier-Stokes. The idea is to split every flow variable into a time-averaged part and a fluctuating part, then solve equations only for the averages. The effect of the fluctuations gets lumped into extra terms that need a separate model, and a variety of these models exist with different strengths and weaknesses. RANS is fast enough to run on a workstation and gives useful results for many design problems, but it can miss important details, especially in flows with separation or strong unsteadiness.
A step up is Large Eddy Simulation, or LES, which directly resolves the larger turbulent structures and only models the smallest ones. LES is more expensive than RANS but captures much more of the flow physics. Comparisons in biomedical settings, for example simulating blood flow through a narrowed artery (a stenosis), have found significant differences in the turbulent structures predicted by RANS and LES models, though certain RANS variants can approximate LES results reasonably well across a cardiac cycle.6International Journal of Heat and Fluid Flow. Comparison of RANS and LES turbulent flow models in a real stenosis
At the top of the fidelity ladder sits Direct Numerical Simulation, or DNS, which resolves every scale of turbulence and uses no models at all. DNS is essentially a brute-force solution of the Navier-Stokes equations. Studies comparing all three tiers for flow through packed beds of spheres have shown that RANS and LES can predict velocity and vorticity with reasonable accuracy when validated against DNS, though certain RANS model families underpredict turbulence by several orders of magnitude in wall-dominated flows at lower flow speeds.7AIChE Journal. Particle‐resolved turbulent flow in a packed bed: RANS, LES, and DNS simulations DNS is reserved for research and relatively simple geometries because the computational cost scales steeply with the level of turbulence in the flow.
Machine Learning Meets Fluid Mechanics
A newer approach to solving the Navier-Stokes equations bypasses traditional grid-based computation altogether. Physics-informed neural networks, or PINNs, embed the governing equations directly into the training process of a neural network. Instead of learning from a massive dataset of pre-computed solutions, the network learns to produce outputs that satisfy the Navier-Stokes equations at sampled points in the domain. The equations themselves act as the training signal.
Recent work has shown that PINNs can achieve accuracy comparable to traditional CFD methods, with velocity errors below one percent in many test cases, while significantly reducing computational cost.8Engineering Applications of Artificial Intelligence. Using Physics-Informed neural networks for solving Navier-Stokes equations in fluid dynamic complex scenarios Specialized architectures like NSFnets have been developed for incompressible flows in both velocity-pressure and vorticity-velocity formulations, extending the approach to turbulent regimes.9Journal of Computational Physics. NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations Other researchers have pursued hybrid strategies, combining machine learning with traditional discretization. One approach, DiscretizationNet, uses a convolutional neural network that implements finite-volume discretization inside its computational graph, achieving faster convergence and better stability for nonlinear, coupled problems.10Computer Methods in Applied Mechanics and Engineering. DiscretizationNet: A machine-learning based solver for Navier–Stokes equations using finite volume discretization
The promise here is real but the limitations are equally real. PINNs tend to perform best at moderate flow speeds and in relatively simple geometries. At higher speeds and more complex scenarios, accuracy can degrade, with errors ranging across several orders of magnitude depending on the problem.11arXiv. Solving Navier-Stokes Equations Using Data-free Physics-Informed Neural Networks With Hard Boundary Conditions The field is advancing quickly, but traditional CFD is not going anywhere soon for safety-critical engineering work.
Beyond Wind Tunnels and Airplane Wings
When people hear “Navier-Stokes,” they often think of aerodynamics, but the equations describe any fluid, and their applications reach into domains you might not expect.
In biomedical engineering, the Navier-Stokes equations are coupled with structural mechanics equations to simulate how blood interacts with the flexible walls of arteries and heart valves. These fluid-structure interaction simulations solve the flow equations alongside elasticity equations for the vessel walls, capturing the two-way feedback between blood pressure and arterial deformation.12PubMed Central. Modeling Dynamics of the Cardiovascular System Using Fluid-Structure Interaction Methods Surgeons and device designers use these models to predict how a stent will change blood flow patterns or where dangerous stress concentrations might develop in an aneurysm wall.
In oil recovery, surface tension gradients drive a phenomenon called the Marangoni effect, where fluid flows along an interface from regions of low surface tension to regions of high surface tension. This motion is governed by the Navier-Stokes equations with an added surface-tension-gradient force term, and understanding it is critical for several enhanced oil recovery techniques.13PubMed Central. Impact of the Marangoni phenomenon on the different Enhanced Oil Recovery methods
In geophysics, the equations are modified with a Coriolis force term to account for the Earth’s rotation, which deflects large-scale atmospheric and oceanic flows. Weather prediction models and climate simulations are, at bottom, numerical solutions of the Navier-Stokes equations with added thermodynamics, moisture physics, and radiation, running on grids that can span the entire globe.14Comptes Rendus de l’Académie des Sciences – Series I – Mathematics. Navier-Stokes equations with Coriolis force and vanishing vertical viscosity
Shock Waves and Compressible Flows
The version of the Navier-Stokes equations most commonly encountered in textbooks assumes the fluid is incompressible, a good approximation for water and for air moving at speeds well below the speed of sound. But at higher speeds, density changes become important. A jet engine exhaust, a re-entering spacecraft, or a supersonic bullet all involve compressible flow, and the equations become considerably more complex. Density is no longer a known constant; it becomes another unknown that the equations must solve for, intertwined with pressure and temperature through thermodynamic relationships.
Compressible flows can produce shock waves, abrupt jumps in pressure, density, and temperature that form when an object moves faster than sound or when a fast flow encounters an obstacle. Mathematically, shocks are discontinuities, and handling them numerically is tricky because the equations assume smooth variations. Recent mathematical work has shown that planar shock waves in three-dimensional compressible Navier-Stokes equations are nonlinearly stable under certain conditions, meaning that small perturbations to the shock do not cause it to fall apart, provided the shock strength and initial disturbances are small enough.15Nonlinearity. Nonlinear stability of planar shock wave to 3D compressible Navier–Stokes equations in half space with Navier boundary conditions These stability results are important because they give engineers mathematical confidence that the shock-wave structures they compute are not numerical artifacts.
When the Equations Stop Working
The Navier-Stokes equations rest on the assumption that a fluid is a continuous substance, that you can always zoom in further and still find a smooth distribution of velocity, pressure, and density. That assumption breaks down in two directions: when things get extremely small, and when things get extremely cold.
At the small end, consider the air surrounding a spacecraft during re-entry at very high altitudes, or the gas inside a microelectromechanical device. When the mean free path of the gas molecules (the average distance a molecule travels between collisions) becomes comparable to the size of the flow features, the continuum assumption fails. This regime is characterized by a high Knudsen number, the ratio of the mean free path to the relevant length scale. Modified versions of the Navier-Stokes equations have been developed that attempt to extend their validity into this transitional regime, but beyond a certain point, you need to abandon continuum equations entirely and switch to methods that track individual molecular behavior.16International Journal of Heat and Mass Transfer. Modeling of Navier–Stokes equations for high Knudsen number gas flows
At the cold end, superfluids like helium-4 cooled below about 2.17 kelvin behave in ways that the Navier-Stokes equations were never designed to handle. A superfluid has zero viscosity, which already removes a central term from the equations. Worse, at finite but very low temperatures, superfluid helium behaves as two interpenetrating fluids: a superfluid component with no viscosity and a normal component that does have viscosity. Quantum turbulence in these systems, the stochastic motion of quantized vortex lines, has been studied intensely for over fifty years, yet no satisfactory phenomenological framework captures the full variety of experimental observations at the level of detail achieved by the Navier-Stokes equations for ordinary fluids.17PubMed Central. Phenomenology of quantum turbulence in superfluid helium The Navier-Stokes equations describe classical fluids beautifully, but they were built for a world where viscosity exists and matter is continuous, and when those conditions vanish, so does their authority.
Non-Newtonian Fluids and the Limits of the Standard Model
Even within the continuum world, the standard Navier-Stokes equations assume a specific relationship between how fast a fluid is being deformed and how much internal stress that deformation produces. This relationship is linear: double the rate of deformation, and you double the stress. Fluids that obey this rule are called Newtonian, and they include water, air, and most simple liquids.
Many real fluids are not Newtonian. Ketchup, blood, polymer melts, wet concrete, and the mantle of the Earth all exhibit nonlinear stress-deformation relationships. Some get thinner when you shear them faster (shear-thinning), some get thicker (shear-thickening), and some will not flow at all until a threshold stress is exceeded (yield-stress fluids like toothpaste or Bingham plastics). Generalizing the Navier-Stokes equations to these materials requires replacing the simple viscosity constant with a more complex constitutive law, and this opens up a host of new mathematical challenges. Recent work has established the existence of weak solutions for steady, compressible non-Newtonian Navier-Stokes systems, including models relevant to Herschel-Bulkley fluids, a class that includes materials with both a yield stress and power-law viscosity behavior.18arXiv. Weak solutions to the Navier-Stokes equations for steady compressible non-Newtonian fluids
Relativistic Navier-Stokes
Pushing the equations in the opposite direction from quantum superfluids, physicists working on high-energy phenomena need versions of the Navier-Stokes equations that are compatible with Einstein’s special relativity. This matters for the quark-gluon plasma produced in heavy-ion collisions at facilities like CERN, an exotic state of matter that existed microseconds after the Big Bang and behaves as a nearly perfect fluid. It also matters for modeling accretion disks around black holes and neutron star mergers.
For decades, the naive relativistic generalization of the Navier-Stokes equations was known to be pathological: it permitted signals to travel faster than light and was unstable. Recent theoretical advances have shown that these problems can be resolved by choosing suitable definitions of the hydrodynamic variables (temperature, velocity, chemical potential) in non-equilibrium states. Under these redefinitions, the relativistic Navier-Stokes equations become both stable and causal.19Journal of High Energy Physics. Stable and causal relativistic Navier-Stokes equations Follow-up work performing real-time numerical evolutions of these equations for a conformal fluid has provided quantitative evidence that the choice of redefinition does not affect the physics at first order, as long as the system stays within the regime where the equations are valid. Even for systems only marginally within that regime, mimicking conditions in the quark-gluon plasma, the first-order physics proved robust.20Journal of High Energy Physics. Field redefinitions and evolutions in relativistic Navier-Stokes
The fact that a set of equations written to describe water flowing through pipes in nineteenth-century France can, with appropriate generalization, capture the behavior of quark-gluon plasma at trillions of degrees is one of the more striking examples of how deeply the Navier-Stokes framework is woven into our description of the physical world. The equations have outlived the molecular picture Navier used to derive them, survived the quantum revolution, and adapted to relativistic speeds. Two centuries on, they remain both our best tool for understanding fluid motion and one of mathematics’ most humbling open questions.