CHAPTER02ARTICLE04

Fluid mechanics / Form governing equations

FLUID MECHANICS · 02–04 / ADVANCED

Euler equation

Derive the inviscid momentum equation relating pressure gradient and body force to fluid-particle acceleration, and define where it is valid.

13 min read2026-08-13Definitions, equations & units checkedJA version

Abstract

The Euler equation is the local momentum balance for an inviscid fluid. Pressure gradients and body forces such as gravity accelerate fluid particles; under steady conditions, integration along a streamline leads to Bernoulli's equation.

Understand it in 30 seconds

The Euler equation applies Newton's second law to a fluid particle while neglecting viscous friction. Pressure force and body forces such as gravity produce the particle acceleration.

The acceleration is material: it includes both change at a fixed point and change caused by motion into a region with a different velocity.

Forces on a differential fluid element

Consider a differential element of volume $\mathrm dV$ and density $\rho$ . Pressure acts over its entire surface, but the small difference between opposite faces gives the resultant $-\nabla p$ $\mathrm dV$ .

Let $\boldsymbol{b}$ be body force per unit mass. Newton's second law uses mass $\rho\,\mathrm dV$ and material acceleration $\mathrm D\boldsymbol{u}/\mathrm D t$ .

Equation (1)Force balance on the element

\rho\,\mathrm dV\frac{\mathrm D\boldsymbol{u}}{\mathrm D t}=-\nabla p\,\mathrm dV+\rho\boldsymbol{b}\,\mathrm dV

The Euler equation

Dividing Equation (1) by $\mathrm dV$ gives the local momentum equation for an inviscid fluid [1,2]. This form does not itself require constant density.

Expanding the material derivative separates local and advective acceleration. Advective acceleration may remain in a steady flow.

Equation (2)Euler equation

\rho\frac{\mathrm D\boldsymbol{u}}{\mathrm D t}=-\nabla p+\rho\boldsymbol{b}

Equation (3)Expanded material acceleration

\rho\left\{\frac{\partial\boldsymbol{u}}{\partial t}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}\right\}=-\nabla p+\rho\boldsymbol{b}

Symbols & units

Every term has the dimensions of force per unit volume, $\mathrm{N/m^3}$ . Dimensional checking helps catch a missing density or pressure gradient.

Principal symbols
Symbol	Meaning	SI unit
$\rho$	Density	$\mathrm{kg/m^3}$
$\boldsymbol{u}$	Velocity vector	$\mathrm{m/s}$
$\boldsymbol{p}$	Pressure	$\mathrm{Pa}$
$\boldsymbol{b}$	Body force per unit mass	$\mathrm{m/s^2}$
$\frac{\mathrm D\boldsymbol{u}}{\mathrm D t}$	Fluid-particle acceleration	$\mathrm{m/s^2}$
$\nabla p$	Pressure gradient	$\mathrm{Pa/m}$

Derivation from momentum conservation

The Cauchy momentum equation represents surface force with the stress tensor $\boldsymbol{\sigma}$ . Splitting $\boldsymbol{\sigma}=-p\boldsymbol{I}+\boldsymbol{\tau}$ produces pressure and viscous contributions.

Where $\nabla\cdot\boldsymbol{\tau}$ is negligible compared with inertia and pressure force, the Euler equation follows. It is momentum conservation under the inviscid approximation, not a separate conservation law.

Equation (4)Cauchy momentum equation

\rho\frac{\mathrm D\boldsymbol{u}}{\mathrm D t}=-\nabla p+\nabla\cdot\boldsymbol{\tau}+\rho\boldsymbol{b}

Equation (5)Inviscid approximation

\nabla\cdot\boldsymbol{\tau}\approx\boldsymbol{0}

Validity and limitations

Inviscid means that viscous force is negligible in the region of interest, not necessarily that the physical viscosity is exactly zero.

Good candidates: High-Reynolds-number regions away from walls, jet cores and outer flows.
Poor candidates: Boundary layers, lubrication, small-duct flow and problems controlled by wall friction or dissipation.
Compressibility: Compressible use requires continuity, energy and an equation of state.
Rotation: The Euler equation also applies to rotational inviscid flow; irrotationality is not required [3].

Worked example: acceleration by a pressure gradient

For horizontal inviscid flow, take $\rho$ = 1000 $\mathrm{kg/m^3}$ and $\mathrm D\boldsymbol{u}/\mathrm D t$ = 2.0 $\mathrm{m/s^2}$ . Neglect body force along the flow.

The required pressure gradient is −2000 $\mathrm{Pa/m}$ : pressure drops by about $2.0\ \mathrm{kPa}$ over each metre in the acceleration direction.

Equation (6)Streamwise pressure gradient

\frac{\mathrm dp}{\mathrm dx}=-\rho\frac{\mathrm Du}{\mathrm Dt}=-(1000)(2.0)=-2000\ \mathrm{Pa/m}

Connection to Bernoulli's equation

For steady flow with gravity as the only body force, integrate Equation (2) along a streamline. With constant density, this gives Bernoulli's equation.

Bernoulli's equation is therefore an integrated consequence of the Euler equation under added assumptions. In irrotational flow, the same constant can apply across streamlines.

Equation (7)Bernoulli equation along a streamline

p+\frac{1}{2}\rho u^2+\rho gz=\text{constant}

Common mistakes

Check separately whether viscosity is negligible, whether the flow is steady, whether density is constant and which body forces are included.

Steady means zero acceleration: Advective acceleration $(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ can remain.
Inviscid predicts wall friction: Wall shear is viscous and cannot be obtained directly.
Euler equals Bernoulli: Euler is a local differential equation; Bernoulli is an integrated relation with additional assumptions.
Pressure must fall downstream: Deceleration or gravity can produce a downstream pressure rise.

Applications & next topic

The Euler equation underpins outer flow over wings, compressible waves, nozzles, ideal-fluid vortices and inviscid CFD. NASA notes that it can model some outer flows but not boundary-layer growth [3].

The Navier–Stokes equation restores viscous stress and can represent momentum diffusion and wall friction.

References

Sources accessed 13 August 2026. Independent expert review has not yet been completed.