The Continuity Equation — Why Fluids Speed Up in Narrow Pipes

High School⏱ 8 min readFluid Mechanics

The continuity equation A₁v₁ = A₂v₂ explained: why fluids speed up in narrow pipes, conservation of mass, and worked examples.

Quick answer

The continuity equation says that for steady flow of an incompressible fluid, the volume passing any cross-section per second is constant: A₁v₁ = A₂v₂. Narrow the pipe and the fluid must speed up.

$A_1 v_1 = A_2 v_2$

Unit: area: m²; speed: m/s; flow rate Q: m³/s

Example: A pipe narrows from 0.10 m² to 0.025 m². Water entering at 1 m/s exits at 4 m/s — area shrank 4×, so speed grew 4×.

What you will learn

State the continuity equation and explain what it conserves
Use A₁v₁ = A₂v₂ to find unknown speeds or areas
Explain the garden-hose effect from mass conservation
Describe where the simple form breaks down

The garden-hose effect

Put your thumb over a garden hose and the water jets farther. You did not add water — you shrank the opening. The same volume per second now escapes through a smaller hole, so it must leave faster.

That is the continuity equation, and at heart it is just conservation of mass wearing a fluid disguise. In steady flow, mass cannot pile up inside the pipe: whatever mass enters one end per second must leave the other end per second. For a fluid whose density does not change (a good approximation for water and slow-moving air), equal mass per second means equal volume per second.

The equation

Volume flow rate is area times speed: $Q = Av$ . Setting the flow rate equal at two cross-sections gives:

$A_1 v_1 = A_2 v_2$

Halve the area, double the speed. Shrink the area to one quarter and the speed quadruples. The relationship is exact and inverse — which makes it a favorite exam question because the arithmetic is simple but the reasoning must be right.

Note the hidden assumption: incompressible fluid. If the fluid can bunch up (like air near the speed of sound), density joins the equation: $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$ .

A_1 v_1 = A_2 v_2

A = cross-sectional area, v = average flow speed across it

Worked mini-example: branch pipe speed

A water main with cross-section $0.10\ \mathrm{m^2}$ carries water at $1.0\ \mathrm{m/s}$ into a branch pipe of $0.025\ \mathrm{m^2}$ . What is the speed in the branch?

Continuity: $A_1 v_1 = A_2 v_2$
Solve for $v_2$ : $v_2 = (A_1/A_2) v_1$
Substitute: $v_2 = (0.10 / 0.025) \times 1.0 = 4 \times 1.0$

Answer: $4.0\ \mathrm{m/s}$ . The area ratio of 4 forces a 4× speedup.

Sanity check: flow rate $Q = 0.10 \times 1.0 = 0.10\ \mathrm{m^3/s}$ before and $0.025 \times 4.0 = 0.10\ \mathrm{m^3/s}$ after. Equal — mass is conserved.

Key idea

Continuity is a constraint, not a cause. It tells you speeds must be related this way; the pressure differences (Bernoulli) are what actually push the fluid to those speeds.

Teaming up with Bernoulli

Continuity plus Bernoulli is the standard one-two punch of pipe-flow analysis: continuity finds the speeds from the geometry, Bernoulli turns the speed change into a pressure change. Together they explain Venturi meters, carburetors, and why partially closing a valve changes pressures throughout a plumbing system.

Watch the limits: the simple form fails for compressible high-speed gas flow, for unsteady surging flow (like water hammer when a valve slams shut), and where the flow splits into multiple branches — there, you apply continuity at each junction: flow in equals the sum of flows out.

Variables and units

Symbol	Name	Unit	Meaning
$A$	Area	m²	Cross-sectional area of the pipe
$v$	Flow speed	m/s	Average flow speed through that cross-section
$Q$	Volume flow rate	m³/s	Volume passing per second

Common mistakes

✗ Water speeds up in a nozzle because it gets 'squeezed' like toothpaste.

Squeezing is not the mechanism — conservation of mass is. The same mass must exit per second as enters; a smaller area forces higher speed to move that mass through. Toothpaste is actually a bad analogy because you are adding pressure by squeezing the tube.

✗ The continuity equation alone tells you the pressure change.

Continuity only relates areas and speeds. To find pressures you also need Bernoulli's equation (energy) — continuity gives the speeds, Bernoulli converts the speed change into a pressure change.

✗ A narrower pipe always means faster flow everywhere downstream.

The speed is higher only within the narrow section. Downstream, where the pipe widens again, the fluid slows back down (assuming the same total flow rate).

Real-world applications

Garden hose nozzles speed water up by shrinking the exit area — same flow rate, smaller opening, faster jet.
Blood narrows through a clogged artery, so blood speeds up there — doctors hear this as a whooshing murmur.
River rapids form where the channel narrows; the same water must move faster through the gap.
Respiratory airways: air speeds up in constricted bronchioles, which is part of why asthma feels the way it does.

Frequently asked questions

What is the continuity equation?

For incompressible flow: A₁v₁ = A₂v₂ — the cross-sectional area times the flow speed is constant along a pipe. What flows in must flow out.

Why does water speed up in a narrow pipe?

Because of conservation of mass. The same mass per second must pass every cross-section; with less area available, the fluid has to move faster to get the same mass through.

Does continuity work for gases too?

For gases at high speeds, density changes matter and the full form is ρ₁A₁v₁ = ρ₂A₂v₂. But for liquids, and for air moving much slower than sound, density is nearly constant and A₁v₁ = A₂v₂ works well.

Does more water come out of a narrow nozzle?

The volume flow rate Q = Av actually stays the same (for incompressible flow) — the speed increases while the volume per second does not. More water per second would require a bigger supply upstream.