A clear introduction to the partial differential equations of fluid motion, from simple intuition to mathematical form
What the Navier-Stokes equations are
The Navier-Stokes equations are a system of partial differential equations that describe the motion of viscous fluids such as water and air.
They are used to describe water in a pipe, air around an airplane wing, blood in an artery, and countless other flows.
At a high level, they say: a fluid changes its motion because of pressure, viscosity, and any forces acting on it. Pressure pushes fluid around, viscosity smooths out sharp differences in motion, and external forces such as gravity can drive the flow.
These equations are not just a physics slogan. They are the working language of much of fluid dynamics, engineering, and computational simulation.
The Navier-Stokes equations in simple terms
In their simplest common form, the equations look like this:
$$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u + f$$
$$\nabla \cdot u = 0$$
Here:- $u$ is the fluid's velocity- $p$ is the pressure- $\nu$ is the kinematic viscosity- $f$ is any external force, such as gravityThe left side describes how the velocity changes in time and how the fluid transports its own motion. The right side contains the forces that push and smooth the flow.
Where the Navier-Stokes equations come from
The equations come from a simple idea: apply Newton's second law to a tiny parcel of fluid. The mass of that parcel times its acceleration must equal the total force acting on it.
For a viscous fluid, those forces come mainly from pressure and internal friction. When you write that balance carefully at every point in the fluid, you get the Navier-Stokes equations.
So the equations are not arbitrary. They are a continuum-mechanics version of force equals mass times acceleration. For the full step-by-step derivation, see How the Navier-Stokes Equations Are Derived.
Why the Navier-Stokes equations are hard
The difficult part is the nonlinear term $(u \cdot \nabla)u$. The fluid does not just respond to outside forces; it also pushes itself around. That feedback is what makes turbulence and chaotic-looking motion possible.
In two spatial dimensions, the equations are much better behaved. In three dimensions, we still do not know whether every smooth starting flow stays smooth forever.
That is why these equations are famous far beyond engineering: they lead directly to the Navier-Stokes Millennium Problem.
What they are used for
The Navier-Stokes equations are used every day in science and engineering. Typical applications include:- airflow around wings and vehicles- weather and climate models- ocean circulation- industrial fluid transport- blood flow and other biological transport problemsIn practice, people solve approximations of these equations numerically, often with additional modeling assumptions. That practical success is one reason the remaining mathematical questions are so striking.
What to read next
If your main question is whether the problem is solved, start with Is the Navier-Stokes Problem Solved?.
If you want the broad mathematical stakes, continue to The Millennium Problem.
If you want to understand how Navier-Stokes differs from the inviscid Euler equations and why viscosity matters, read Euler vs. Navier-Stokes.
If you want the physical intuition behind turbulence and small scales, read Reynolds Number, Turbulence, and Why Small Scales Matter.
If you want the main obstacles, go to Why It's Hard.
Originally published on navier-stokes.org. The site covers the Navier-Stokes existence and smoothness problem with toggleable simple/rigorous explanations.
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