---
tag: RW0JK9VDZE
kind: concept
name: Euler equations
msc: [35Q31, 76B03]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)
same_as:
  wikidata: Q375175
  wikipedia: Euler equations (fluid dynamics)
---

## Summary

The Euler equations describe the motion of an ideal fluid, that is, a fluid without viscosity.
They express conservation of mass and Newton's second law for each parcel of fluid, with pressure as the only internal force.
The incompressible Euler equations are the case $\nu = 0$ of the [[Navier–Stokes equations|18KXZHVF4M]].
They were among the first partial differential equations written down, and many questions about them are still open.

## Explanations

### L1 human

Imagine a fluid with no internal friction at all, so that layers can slide past each other freely.
Each small parcel of such a fluid is pushed only by differences in pressure around it.
The Euler equations state this, together with the rule that fluid is neither created nor destroyed.
A simple example is water spinning as a rigid body in a bucket: the velocity is $(-y, x, 0)$ and the pressure grows like $(x^2 + y^2)/2$ away from the axis.
That higher pressure at the rim is what bends the water surface into a curve.