A₁v₁ = A₂v₂ — Continuity Equation for Fluids

The continuity equation for fluid flow — why narrow pipes speed water up — with a worked example.

A_1 v_1 = A_2 v_2

Cross-sectional area times speed is constant along a pipe.

What it means

What flows in must flow out: the volume rate ( $A \times v$ ) is the same everywhere along a pipe. Squeeze the pipe and the fluid must speed up to get the same volume through — the foundation of Bernoulli's equation and Venturi meters.

Variables

Symbol	Name	Unit	Meaning
$A₁$	Area 1	m²	Pipe area at point 1
$v₁$	Speed 1	m/s	Flow speed at point 1
$A₂$	Area 2	m²	Pipe area at point 2
$v₂$	Speed 2	m/s	Flow speed at point 2

Assumptions

Incompressible fluid (constant density)
Steady flow (no buildup)
No leaks between the two cross-sections

Derivation

In steady flow, the mass entering per second ( $\rho A_1 v_1$ ) must equal the mass leaving ( $\rho A_2 v_2$ ). With constant density $\rho$ cancels, leaving $A_1 v_1 = A_2 v_2$ .

Worked example

Problem

Water flows at 1.5 m/s through a pipe of area 0.02 m², then enters a section of area 0.01 m². What is the new speed?

Given:

A₁ = 0.02 m²
v₁ = 1.5 m/s
A₂ = 0.01 m²

Write the equation: $A_1 v_1 = A_2 v_2$
Rearrange: $v_2 = A_1 v_1 / A_2$
Substitute: $v_2 = 0.02 \times 1.5 / 0.01 = 3.0\ \mathrm{m/s}$

3.0 m/s

Halving the pipe area doubles the speed — the same volume must pass each second. This is why putting your thumb over a hose makes the water jet faster.