Skip to content

Bernoulli's principle

Air and fluids · level 6, lesson 3 of 6

A question to start

Hold two sheets of paper a few centimetres apart and blow between them. You might expect them to fly apart. Instead, they move together. Why?

The idea

As a fluid speeds up into a narrow part of a pipe, something must push it forward. That push comes from a pressure difference: the pressure behind it is higher than the pressure ahead. So where the fluid goes fast, its pressure is lower.

For a fluid that does not squash, flowing smoothly at one height, Bernoulli's principle says

\[p + \tfrac12 \rho v^2 = \text{constant along the flow}.\]

The term \(\tfrac12 \rho v^2\) is the dynamic pressure, the kinetic energy of each cubic metre of fluid. When \(v\) goes up, \(p\) comes down by the same amount.

still air:higher pstill air:higher pfast air: low p
Pressure drop at 20 m/s, \(\tfrac12\rho v^2\) Mars Earth
Air density 0.016 kg/m³ 1.217 kg/m³
Drop 3.2 Pa 243 Pa

Worked examples

1. Earth air at 20 m/s. \(\tfrac12 \times 1.217 \times 20^2 = 243\) Pa below the still air around it.

2. Mars air at 20 m/s. \(\tfrac12 \times 0.016 \times 400 = 3.2\) Pa. The same speed makes about 75 times less pressure difference.

3. Water in a narrowing pipe. Water at 2 m/s speeds up to 8 m/s. The pressure drops by \(\tfrac12 \times 1000 \times (8^2 - 2^2) = 30{,}000\) Pa.

Try it

  1. Where a fluid moves faster, is its pressure higher or lower?
  2. Find the pressure drop in Earth air moving at 10 m/s.
  3. Water flows at 1 m/s at 200,000 Pa, then speeds up to 5 m/s at the same height. Find its new pressure.
  4. Why do the sheets of paper move together when you blow between them?
Answers
  1. Lower.
  2. \(\tfrac12 \times 1.217 \times 100 \approx 61\) Pa.
  3. \(200{,}000 - \tfrac12 \times 1000 \times (25 - 1) = 188{,}000\) Pa.
  4. The fast air between them is at lower pressure, so the still air outside pushes them together.

Common mistake

"Fast air pushes harder." Fast air pushes harder on something in its way, head on. But its pressure sideways, the push on the walls it flows along, is lower.

Where it leads