Principle

Continuity equation

A_{1} v_{1} = A_{2} v_{2}

Statement

The product of cross-sectional area and flow speed is the same at every point along a pipe.

Symbols and units

$A_{1}, A_{2}$	cross-sectional areas	m²
$v_{1}, v_{2}$	flow speeds	m/s

When it applies

Steady, incompressible flow through a pipe with no leaks.

Don't use when

Assumes steady, incompressible flow with no leaks — not valid for compressible flow (e.g. gas at high speed) or unsteady flow.

Derivation

Mass flowing past any cross-section per unit time must be the same everywhere along the pipe, for an incompressible fluid with no leaks:

ρ A_{1} v_{1} = ρ A_{2} v_{2}

Density cancels, since it is the same fluid throughout:

A_{1} v_{1} = A_{2} v_{2}

Worked examples

Speed change in a narrowing pipe

Water flows at 2.0 m/s through a pipe of area 0.010 m², narrowing to 0.004 m²:

v_{2} = \frac{A _{1} v _{1}}{A _{2}} = \frac{( 0.010 ) ( 2.0 )}{0.004} = 5.0 m/s

Watch out for

A narrower section of pipe means faster flow, not slower — a frequent source of confusion.