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Narrow pipes, fast flow

Air and fluids · level 6, lesson 2 of 6

A question to start

Put your thumb over the end of a garden hose and the water squirts much faster and farther. You have not turned the tap up. Where does the extra speed come from?

The idea

Water hardly squashes. So whatever flows into a pipe each second must flow out of it each second: the flow rate is the same all along. This is the continuity equation:

\[A_1 v_1 = A_2 v_2.\]

Where the pipe narrows, the fluid must go faster. Halve the area, and the speed doubles.

wide: slownarrow: fast

The same rule works for air when its speed is well below the speed of sound, because then the air barely squashes either.

Worked examples

1. The thumb. The hose end has area \(2.0\times10^{-4}\) m², with water at 3 m/s. Your thumb leaves a gap of \(0.5\times10^{-4}\) m². Speed through the gap: \(3 \times 2.0/0.5 = 12\) m/s.

2. A round pipe. Speed goes with \(1/\text{area}\), and area goes with the diameter squared. A pipe half as wide has a quarter of the area, so the speed is four times as big.

3. A river. Where a river narrows between rocks, it runs fast. Where it widens into a pool, it slows down.

Try it

  1. Why must the flow rate be the same all along a pipe?
  2. Water flows at 2 m/s in a pipe of area \(6\times10^{-4}\) m². What is its speed where the area is \(1.5\times10^{-4}\) m²?
  3. A pipe narrows from 4 cm across to 1 cm across. How many times faster is the water in the narrow part?
  4. Air at 2 m/s in a 0.06 m² duct passes into a 0.02 m² vent. How fast does it come out?
Answers
  1. Water hardly squashes, so no more can go in than comes out each second.
  2. \(2 \times 6/1.5 = 8\) m/s.
  3. The area is 16 times smaller, so the speed is 16 times bigger.
  4. \(2 \times 0.06/0.02 = 6\) m/s.

Common mistake

"Half the diameter, twice the speed." Speed depends on the area, which goes with the diameter squared. Half the diameter gives four times the speed.

Where it leads