Bernoulli's Principle

Bernoulli's principle explains how pressure, speed, and energy are connected in a moving fluid such as air or water. It helps students understand why airplane wings generate lift, why a shower curtain can move inward, and why fluid speeds up in narrow pipes. This idea matters because it links motion and pressure in many real engineering systems. It is one of the core tools for analyzing fluid flow in physics.

For steady flow of an ideal fluid, Bernoulli's equation says that pressure energy, kinetic energy, and gravitational potential energy trade off along a streamline. When fluid speed increases, pressure often decreases if height stays about the same. Around an airfoil, air moving faster over the top surface is associated with lower pressure than the air below, contributing to lift. In real situations, viscosity, turbulence, and flow separation also matter, so Bernoulli's principle works best when used with careful physical reasoning.

Key Facts

Bernoulli's equation: P + 1/2 rho v^2 + rho g h = constant
If height is constant, then P + 1/2 rho v^2 = constant
Higher fluid speed usually means lower pressure along the same streamline
Continuity equation for incompressible flow: A1v1 = A2v2
Lift can be estimated from pressure difference: F = Delta P times A
Dynamic pressure is q = 1/2 rho v^2

Vocabulary

Bernoulli's principle: The principle that in steady ideal fluid flow, an increase in speed is associated with a decrease in pressure or potential energy along a streamline.
Pressure: Pressure is the force per unit area exerted by a fluid on a surface.
Streamline: A streamline is a path that is everywhere tangent to the fluid velocity at a given instant.
Continuity equation: The continuity equation states that for incompressible flow, the volume flow rate stays constant so A v remains the same.
Lift: Lift is the upward force on a wing caused by pressure differences and the deflection of airflow.

Common Mistakes to Avoid

Assuming Bernoulli's principle says fast air always causes low pressure everywhere, which is wrong because the relationship applies along a streamline and depends on the flow conditions.
Forgetting the height term rho g h, which is wrong because changes in elevation can affect pressure even when speed stays the same.
Using A1v1 = A2v2 for compressible or leaking flow without checking assumptions, which is wrong because continuity in that form requires steady incompressible flow with no fluid added or lost.
Saying lift comes only from equal transit time over and under the wing, which is wrong because air parcels do not need to meet at the trailing edge and lift depends on the full pressure and momentum pattern around the wing.

Practice Questions

1 Air flows horizontally through a pipe that narrows from area 0.040 m^2 to 0.010 m^2. If the speed in the wide section is 3.0 m/s, what is the speed in the narrow section?
2 Water moves at the same height in a pipe. At one point the speed is 2.0 m/s and the pressure is 1.80 x 10^5 Pa. At a second point the speed is 5.0 m/s. If the density of water is 1000 kg/m^3, what is the pressure at the second point?
3 A wing has faster airflow above it than below it. Explain why this can produce lift, and state one reason Bernoulli's principle alone does not give the full real world picture.