Bernoulli's equation

Engineering · Senior school and first-year engineering · about 10 min

When water speeds up through a narrow section of pipe, its pressure drops. Bernoulli's equation explains why: it is conservation of energy for a flowing fluid. It's behind Venturi meters, carburettors and part of the story of how wings make lift.

The key idea

Along a streamline in steady, incompressible flow with negligible friction:

P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}

$P$ is the pressure, $\tfrac{1}{2}\rho v^2$ the kinetic energy per unit volume, and $\rho g h$ the potential energy per unit volume. If one goes up, the others must come down.

Continuity: why the fluid speeds up

Fluid can't pile up inside a pipe, so the volume flow rate is the same at every section:

A_1 v_1 = A_2 v_2

Halve the diameter and the area drops to a quarter, so the speed goes up four times. Bernoulli then tells you what that does to the pressure.

Side view of a horizontal pipe cut open lengthways, with pale blue water inside. A wide section on the left narrows smoothly to a thin section on the right. A short dark blue arrow inside the wide section and a much longer dark blue arrow inside the narrow section, both pointing right. Thin dashed vertical lines mark section 1 in the wide part and section 2 in the narrow part. — The same flow has to pass through a smaller area, so it moves faster.

When you can use it

Bernoulli's equation assumes:

steady flow (not changing with time),
incompressible fluid (liquids, or gases at low speed),
negligible friction (no viscous losses),
points on the same streamline, with no pump or turbine between them.

Real pipes have friction, so engineers add a head-loss term; the ideal version is the starting point.

Worked example: water through a narrowing pipe

Water ( $\rho = 1000$ kg/m $^3$ ) flows through a horizontal pipe that narrows from $10$ cm to $5$ cm in diameter. In the wide section the speed is $2$ m/s and the pressure is $200$ kPa. Find the speed and pressure in the narrow section.

Side view of a horizontal pipe cut open lengthways, with pale blue water inside. A wide section on the left narrows smoothly to a section half as wide on the right. A dark blue arrow inside each section, pointing right. A vertical dimension across each section shows its diameter. — The pipe in this example.

Show the solution

Continuity. The area ratio is $\left(\frac{10}{5}\right)^2 = 4$ , so

v_2 = 2 \times 4 = 8 \text{ m/s}

Bernoulli, with $h_1 = h_2$ (horizontal pipe):

P_2 = P_1 + \tfrac{1}{2}\rho\,(v_1^2 - v_2^2) = 200\,000 + 500\,(4 - 64)

P_2 = 200\,000 - 30\,000 = 170 \text{ kPa}

The water speeds up four times and the pressure drops by $30$ kPa.

Common mistakes

Scaling speed with the diameter instead of the area

Speed scales with area, which goes with diameter squared: half the diameter means four times the speed.

Using it across a pump or a long rough pipe

A pump adds energy and friction removes it. Plain Bernoulli only holds where neither happens.

Mixing kPa and Pa

Work in pascals: $\tfrac{1}{2}\rho v^2$ with $\rho$ in kg/m $^3$ and $v$ in m/s comes out in Pa.

Check your understanding

1. In a horizontal pipe, where the fluid speeds up, the pressure...

2. A pipe's diameter is reduced to one third. The flow speed becomes...

3. Which of these breaks the assumptions of Bernoulli's equation?

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