---
tag: 18KXZHVF4M
kind: concept
name: Navier–Stokes equations
msc: [35Q30, 76D05]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Navier–Stokes_equations
same_as:
  wikidata: Q201321
  wikipedia: Navier–Stokes equations
---

## Summary

The Navier–Stokes equations describe the motion of a viscous fluid, such as water or air.
They are Newton's second law for each small parcel of fluid, with forces from pressure, internal friction and external sources.
In the incompressible case they form a nonlinear system for the velocity field and the pressure.
Whether smooth solutions in three dimensions stay smooth for all time is one of the Millennium Prize Problems.

## Explanations

### L1 human

Think of a fluid as made of many tiny parcels, each with a velocity.
A parcel speeds up or slows down because of three effects.
Differences in pressure push it, friction with its neighbours drags it, and outside forces such as gravity act on it.
The Navier–Stokes equations write this balance as a formula, together with the rule that an incompressible fluid neither piles up nor thins out.
A simple solution is the shear flow $u = (y, 0, 0)$ with constant pressure, where layers of fluid slide past each other at speeds that grow with height.
The equations are used to design aircraft, predict weather and model blood flow.