Fluid continuity and flow rate

For an incompressible fluid flowing through a pipe, the same volume must pass every cross-section each second, so the fluid speeds up wherever the pipe narrows and slows down wherever it widens.

Practise this concept

Water does not pile up inside a pipe and it does not vanish. If ten litres enter one end each second, ten litres must leave the other end each second, and ten litres must cross every cross-section in between. That is the continuity equation, and written out it says A·v is the same everywhere along the pipe: cross-sectional area times flow speed.

The immediate consequence runs against intuition. Squeeze the pipe and the fluid speeds up, because the same volume per second now has to get through a smaller opening. Halve the area and the speed doubles. Everyone has used this — putting a thumb over the end of a garden hose does not reduce the water delivered, it accelerates the same flow into a jet. The narrowing constrains the area, not the volume passing per second.

Keeping flow rate and speed distinct is the other half of the idea. Flow rate is a volume per second, measured in cubic metres per second; speed is a distance per second. A wide slow river and a narrow fast stream can carry identical flow rates. Continuity says nothing about how fast the fluid must move — only that whatever the speed and area do, their product holds constant, provided the fluid is incompressible and nothing leaks out along the way.

Key ideas

Requires: Pressure in fluids

Unlocks: Bernoulli's equation

Formulas

A_1 v_1 = A_2 v_2

Symbol	Name	Unit
$A1$	area	m^2 (square metre)
$A2$	area	m^2 (square metre)
$v1$	velocity	m/s (metre per second)
$v2$	velocity	m/s (metre per second)

Common misconceptions

A narrower pipe slows the fluid down, because there is less room for it to move through.
Flow rate means the speed of the fluid, so the two words describe the same quantity.
Continuity requires the pipe to have the same cross-section everywhere for the fluid to keep flowing.

Resources

Articles

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.

Mechanics and heatLTFY.01.006