Euler Equation (Inviscid Flow)

Equivalent forms

a = -\frac{1}{\rho}\frac{dp}{dx}

\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v} = -\frac{1}{\rho}\nabla p + \mathbf{g}

Newton's second law for an ideal fluid — the parent of Bernoulli's equation.

Symbol	Name	SI unit	Dimension	Range
$\rho$	Density Fluid mass density	kg/m³	M·L⁻³	0.1 – 2000
$\frac{dp}{dx}$	Pressure Gradient Pressure change per unit length along the streamline	Pa/m	M·L⁻²·T⁻²	-5000 – 5000
$a$	Parcel Accelerationoutput Acceleration of the fluid parcel along the streamline	m/s²	L·T⁻²	-5000 – 5000

Unit systems

Symbol	SI	CGS	Imperial
$\rho$	kg/m³	g/cm³	slug/ft³
$\frac{dp}{dx}$	Pa/m	dyn/cm³	psi/ft
$a$	m/s²	cm/s²	ft/s²

Where it holds

Valid for inviscid (frictionless) flow away from boundary layers and shocks. Excellent for the bulk of high-Reynolds-number flows; fails inside thin viscous layers next to walls where the no-slip condition matters.

Dimensional analysis

$[(1/\rho )(dp/dx)] = (M^{-1}\cdot L^{3})(M\cdot L^{-2}\cdot T^{-2}) = L\cdot T^{-2} \checkmark$ (acceleration)

Discovery

Leonhard Euler · 1757

Euler published the general equations of inviscid fluid motion in his memoir on fluid mechanics, founding the field of theoretical hydrodynamics nearly a century before viscosity was added by Navier and Stokes.

Try this

Strip away friction entirely — what does a 'perfect' fluid do?

Air (density 1.2 kg/m³) accelerates along a streamline from rest, driven by a pressure gradient of 600 Pa/m. What is the fluid parcel's acceleration?

Research status: stable

Real-world applications

Airfoil and propeller lift via potential-flow theory
Nozzle and diffuser design
Water waves and free-surface flows
Astrophysical and gas-dynamic modeling

Common misconceptions

Euler flow predicts zero drag on any body (d'Alembert's paradox) — real drag comes from the neglected viscosity and the wake
Euler's equation is not restricted to steady flow; the unsteady term $\partial v/\partial t$ is included
It applies to compressible flow too, when paired with an energy equation and equation of state

Experimental verification

Potential-flow predictions for streamlined bodies (lift on airfoils via Kutta condition) match wind-tunnel data closely; flow over weirs and nozzles obeys Euler/Bernoulli to high accuracy where viscosity is negligible.

Derivation

Take the Navier-Stokes equation and

set \mu = 0

.

The viscous term

\mu \nabla ^{2}v

vanishes, leaving

\rho Dv/Dt = -\nabla p + \rho g

.

Equivalently, apply Newton's second law to a fluid element with only pressure forces (the net force is the pressure gradient times volume) and the body force of gravity.

Limiting cases

dp/dx \to 0

⟶

a \to 0

(with no gravity)No pressure gradient means no net force along the streamline; the parcel coasts.

Steady, incompressible, along a streamline⟶ Bernoulli's equationIntegrating Euler's equation along a streamline gives

p + \tfrac{1}{2}\rho v^{2} + \rho gh

= constant.

\mu

added back⟶ Navier-Stokes equationRestoring viscous stresses recovers the full viscous momentum balance.

What if…

What if the pressure gradient is adverse (pressure rising downstream)?

The parcel decelerates; in a real viscous flow a strong enough adverse gradient causes boundary-layer separation and stall.

What if you integrate along a streamline?

You obtain Bernoulli's equation — Euler's equation is its differential parent.

1

Acceleration of an air parcel

Given ·

\rho:: 1.2
\frac{dp}{dx}:: 600

Find · a

Steps

Along a streamline, $a = -(1/\rho ) dp/dx$
$a = -(1/1.2) \times 600$
$a \approx -500\,\mathrm{m/s}^{2}$ (the parcel decelerates moving into rising pressure)

Result ·

a = -(1/\rho )(dp/dx) = -(1/1.2)(600) = -500\,\mathrm{m/s}^{2}

Navier-Stokes Equation→Bernoulli's Equation→Continuity Equation→