Principle

Bernoulli's equation

P + \frac{1}{2} ρ v^{2} + ρ g h = constant

Statement

Along a streamline in steady, non-viscous flow, pressure, dynamic pressure, and height together are conserved.

Symbols and units

$P$	pressure	Pa
$ρ$	fluid density	kg/m³
$v$	flow speed	m/s
$h$	height	m
$g$	gravitational acceleration	9.8 m/s²

When it applies

Steady, non-viscous, incompressible flow along a single streamline.

Don't use when

Assumes steady, non-viscous, incompressible flow along a single streamline — does not hold across turbulent or viscous flow.

Derivation

Apply the work-energy theorem to a small parcel of fluid moving along a streamline: the net work done by pressure differences equals the parcel's change in kinetic and potential energy.

W_{net} = Δ K + Δ U

Dividing through by the parcel's volume converts each energy term into a pressure-like quantity, giving the conserved combination:

P + \frac{1}{2} ρ v^{2} + ρ g h = constant

Worked examples

Torricelli's law: speed of efflux from a tank

An open tank with a small hole a depth h below the surface; at both the top surface and the hole, pressure is atmospheric, and the surface speed is negligible:

P_{0} + ρ g h = P_{0} + \frac{1}{2} ρ v^{2}

Solving for the exit speed:

v = 2 g h

Identical in form to the speed of a freely falling object dropped through the same height h.

Watch out for

Assumes steady, non-viscous, incompressible flow along a single streamline — it does not hold across turbulent or viscous flow.

Connects to

Work-energy theorem Hydrostatic pressure