Fluids in Motion and Bernoulli's Principle: Quick Revision

A brief note on Fluids in Motion and Bernoulli's Principle. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Steady flow: the velocity at each point does not change; streamlines never cross.
• Continuity: $A_1v_1 = A_2v_2$, so narrow means fast.
• Bernoulli: $p + \tfrac12\rho v^2 + \rho gh$ is the same along a streamline; faster or higher means lower pressure.
• Torricelli: a jet from a hole $h$ below the surface has $v = \sqrt{2gh}$.
• Uses: Venturi meters, sprays, wings, swerving balls, roofs in storms.

The facts at a glance

A worked example

A fire hose $6.0\ \text{cm}$ across delivers $12$ litres each second through a nozzle $2.0\ \text{cm}$ across.

1. In the hose. $v_1 = \dfrac{0.012\ \text{m}^3/\text{s}}{\pi(0.030\ \text{m})^2} = 4.24\ \text{m/s}$.
2. At the nozzle. A third of the diameter, nine times the speed: $38.2\ \text{m/s}$.
3. Pressure needed. $\tfrac12 \times 1000\ \text{kg/m}^3 \times ((38.2\ \text{m/s})^2 - (4.24\ \text{m/s})^2) = 7.2 \times 10^{5}\ \text{Pa}$ above the air's.
4. How high. Pointed up, the jet could rise $\dfrac{v^2}{2g} = 74\ \text{m}$, without air resistance.

Practice with new numbers

A program to try

The program works out the fire hose of the worked example.

Mistakes to avoid

• Diameter ratio for area ratio. Square it.
• $(v_2 - v_1)^2$ for $v_2^2 - v_1^2$. Square first, then subtract.
• Higher pressure where the fluid is fast. It is lower.
• The density in Torricelli's law. It cancels.
• The depth of the hole for the height of fall. The speed uses the depth below the surface; the fall uses the height above the ground.

Check yourself

1. Water at $1.0\ \text{m/s}$ enters a pipe section with a fifth of the area. How fast does it move there?

2. In a level pipe, water speeds up from $2.0\ \text{m/s}$ to $6.0\ \text{m/s}$. By how much does the pressure fall?

3. How fast does water leave a hole $0.80\ \text{m}$ below the surface? (Take $g = 10\ \text{m/s}^2$.)

4. Why do two sheets of paper held a little apart move together when you blow between them?

5. Water rises steadily $5.0\ \text{m}$ in a pipe of the same width. By how much does its pressure fall?