Bernoulli's Principle
Faster-moving fluid has lower pressure (energy conservation in flow).
Formula
Variables
Example
Fast air over a wing has lower pressure than the slower air beneath — producing lift.
How it works
Bernoulli’s principle says that for steady flow of an ideal fluid, the sum of pressure energy, kinetic energy, and gravitational potential energy per unit volume stays constant along a streamline. So if a fluid speeds up, its kinetic term increases and the pressure term must usually drop, provided the height does not change much. It comes from conservation of energy applied to fluid motion, and it works best when the fluid is incompressible, non-viscous, and the flow is smooth rather than turbulent.
When to use it
Use it when analyzing flowing liquids or gases in situations like pipes, nozzles, venturi meters, siphons, and airflow around objects. It is especially useful when comparing pressure and speed at two points in the same streamline and when height differences matter.
Worked example, step by step
Water flows through a horizontal pipe. At point 1, the pressure is P1 = 180 kPa and the speed is v1 = 2.0 m/s. At point 2, the speed increases to v2 = 6.0 m/s. Assume the water density is ρ = 1000 kg/m^3 and both points are at the same height, so h1 = h2. Use Bernoulli’s equation: P1 + 1/2ρv1^2 = P2 + 1/2ρv2^2 Solve for P2: P2 = P1 + 1/2ρ(v1^2 - v2^2) P2 = 180000 + 1/2(1000)(2.0^2 - 6.0^2) P2 = 180000 + 500(4 - 36) P2 = 180000 + 500(-32) P2 = 180000 - 16000 P2 = 164000 Pa Final answer: the pressure at point 2 is 1.64 × 10^5 Pa, or 164 kPa.
Common mistakes to avoid
- Using Bernoulli’s equation between two points in a flow with major friction losses or strong turbulence
- Forgetting that the height term ρgh matters when the fluid changes elevation
- Assuming higher speed always means lower pressure without checking that the points are in the same streamline and the flow conditions are appropriate
Did You Know?
Bernoulli's principle — faster flow, lower pressure — helps explain lift over an aircraft wing.
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