Bernoulli's principle

Energy in a moving fluid

As water speeds up into a narrow pipe, something must push it forward. That push comes from a pressure difference: the pressure behind is higher than in front. So where a fluid moves faster, its pressure is lower.

Daniel Bernoulli put this as a law of energy. For an ideal fluid (no viscosity, no change of volume) in steady flow, along one streamline:

Each term is an energy per unit volume, in $\text{J/m}^3$, which is the pascal:

• $p$, the work the pressure can do;
• $\tfrac12\rho v^2$, the kinetic energy;
• $\rho gh$, the potential energy.

You should find that both speeding up and climbing lower the pressure, and that each metre of height costs about $10\ \text{kPa}$.

Two special cases

• A level pipe ($h$ the same): $p_1 - p_2 = \tfrac12\rho(v_2^2 - v_1^2)$.
• A fluid at rest ($v = 0$): $p_1 - p_2 = \rho g(h_2 - h_1)$, the pressure-depth law of the first course.

Watch out. Bernoulli's equation needs the difference of the squares, $v_2^2 - v_1^2$, never $(v_2 - v_1)^2$.

A worked example

Water in a main is at $2.8 \times 10^{5}\ \text{Pa}$, moving at $1.2\ \text{m/s}$ at ground level. The main rises $6.0\ \text{m}$ and its diameter halves. Find the pressure up there.

1. Speed. Half the diameter is a quarter of the area: $v_2 = 4 \times 1.2\ \text{m/s} = 4.8\ \text{m/s}$.
2. Kinetic term. $\tfrac12 \times 1000\ \text{kg/m}^3 \times (1.44\ \text{m}^2/\text{s}^2 - 23.04\ \text{m}^2/\text{s}^2) = -10\,800\ \text{Pa}$.
3. Height term. $1000\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2 \times 6.0\ \text{m} = 58\,800\ \text{Pa}$.
4. Pressure. $2.8 \times 10^{5}\ \text{Pa} - 10\,800\ \text{Pa} - 58\,800\ \text{Pa} = 2.10 \times 10^{5}\ \text{Pa}$.

Practise with new numbers

Try it in Python

The program works out the water main of the example.

Remember these

• $p + \tfrac12\rho v^2 + \rho gh$ is the same along a streamline.
• Faster or higher means lower pressure.
• Level pipe: $p_1 - p_2 = \tfrac12\rho(v_2^2 - v_1^2)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Water speeds up from $1.0\ \text{m/s}$ to $4.0\ \text{m/s}$ in a level pipe. How much does its pressure fall?

2. Which term of Bernoulli's equation is the kinetic energy per unit volume?

3. Show that Bernoulli's equation gives $p = p_0 + \rho gh$ for a liquid at rest.

4. Why do two boats moving side by side, close together, drift toward each other?