Bernoulli’s Principle

Bernoulli’s principle says that in a flowing fluid, faster flow means lower pressure. The fluid’s total energy — kinetic plus potential plus pressure — stays constant along a streamline. Where the fluid speeds up (through a constricted pipe, over an airfoil, around a curveball), pressure drops. Where it slows down, pressure rises. Daniel Bernoulli published the principle in 1738 in his book Hydrodynamica, and it remains the central equation in fluid dynamics. The principle explains why airplanes fly, why curveballs curve, why hurricane winds tear off roofs, and why a shower curtain billows inward when you turn on the water.

The Equation

For an ideal (incompressible, non-viscous) fluid flowing steadily along a streamline, Bernoulli’s equation is:

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

Each term is an energy density (energy per unit volume):

\$ P \$ — static pressure of the fluid
\$ \\frac{1}{2} \\rho v^2 \$ — kinetic energy per unit volume (dynamic pressure). \$ \\rho \$ is fluid density, \$ v \$ is flow speed.
\$ \\rho g h \$ — gravitational potential energy per unit volume. \$ g \$ is gravity, \$ h \$ is height above some reference level.

The sum is conserved along a streamline. So if speed goes up at constant height, pressure must drop to compensate. If height changes at constant speed, pressure adjusts. The total energy in the fluid stays put.

The Intuition Behind the Equation

Where does the kinetic-energy term come from? Apply the work-energy theorem to a fluid parcel. The net work done by pressure forces on each side of the parcel must equal its change in kinetic energy. Working out the algebra for a steady flow through a pipe of varying cross-section gives exactly Bernoulli’s equation.

Conservation of mass (the continuity equation) says the volumetric flow rate must be constant along a streamline: A₁v₁ = A₂v₂, where A is cross-sectional area and v is flow speed. So if the pipe narrows, the speed must increase. Bernoulli’s principle then says the pressure must decrease — because the kinetic energy increase had to come from somewhere, and the only energy source available is the pressure energy.

Worked Examples

Venturi tube

A pipe with a constricted section. Water enters the wide part at speed \$ v_1 \$ and pressure \$ P_1 \$. At the narrow part, conservation of mass forces speed to increase to \$ v_2 \$. Bernoulli’s principle gives the pressure drop:

$$ P_1 – P_2 = \frac{1}{2} \rho (v_2^2 – v_1^2) $$

This is exploited in carburetors, atomizers, and Venturi flow meters.

Airplane wing (simplified)

The classical (simplified) explanation: an airfoil is shaped so air flows faster over the top than the bottom. Bernoulli’s principle says faster flow = lower pressure, so the pressure on top is lower than on the bottom. The pressure difference times the wing area is the lift force.

This is a partial truth. Real aircraft lift involves the Coanda effect, downwash, and Newton’s third law (the wing pushes air down, and air pushes the wing up) just as much as Bernoulli. But the Bernoulli component is real and contributes substantially.

Tornado roof damage

In a hurricane or tornado, wind speeds outside a building can exceed 100 mph. The fast-moving air outside has much lower pressure than the (relatively still) air inside. The pressure difference pushes the roof upward and outward. This is one reason building codes in tornado country require strong roof-to-wall connections.

When Bernoulli’s Equation Doesn’t Apply

The standard Bernoulli equation makes four assumptions, all of which fail under some conditions:

Incompressible flow. Valid for water and slow-moving air. For air at speeds above about 100 m/s (Mach 0.3+), compressibility matters and you need the compressible Bernoulli equation.
Non-viscous flow. No friction. Real fluids have viscosity that dissipates energy. For slow flow through long thin pipes, viscous losses dominate and Poiseuille’s law applies instead.
Steady flow. Flow patterns don’t change with time. Turbulent or unsteady flow needs more sophisticated treatment.
Along a streamline. The conservation applies to fluid parcels following the same streamline. Comparing pressures between different streamlines requires additional conditions (irrotational flow).

For most introductory physics problems, the assumptions hold well enough. For rigorous engineering analysis, the Navier-Stokes equations (the full equations of fluid motion) take over.

Counterintuitive Demonstrations

Classic Bernoulli demonstrations that make great physics-class demos:

The blown ping-pong ball. A ping-pong ball levitates in a stream of air from a hair dryer. The fast air around the ball has lower pressure than the still surrounding air, which pushes the ball back into the stream from all sides.
The two-paper trick. Hold two pieces of paper hanging vertically about an inch apart. Blow between them. They move TOGETHER, not apart, because the fast moving air between them has lower pressure than the still air on the outside.
The shower curtain effect. When you turn on a shower, the curtain billows INWARD toward you. The fast-moving (and slightly warmer) air inside the shower has lower pressure than the still bathroom air.
The funnel and ping-pong ball. Place a ping-pong ball in a funnel and blow downward through the funnel. The ball stays put — even when you turn the funnel sideways or upside down — because air rushing around the ball creates lower pressure that holds it in place.

Frequently Asked Questions

What is Bernoulli’s principle in simple terms?

Bernoulli’s principle says that in a flowing fluid, faster flow means lower pressure, and slower flow means higher pressure. The fluid’s total mechanical energy (pressure energy + kinetic energy + gravitational potential energy) stays constant along a streamline. So when one component changes, another must compensate.

What is Bernoulli’s equation?

P + (1/2)ρv² + ρgh = constant along a streamline, for an ideal incompressible non-viscous fluid in steady flow. P is static pressure, ρ is fluid density, v is flow speed, g is gravity, h is height. Each term has units of energy per unit volume.

Does Bernoulli’s principle explain why airplanes fly?

Partially. The traditional simplified explanation is: airfoil shape forces air over the top to flow faster, which by Bernoulli’s principle means lower pressure on top, producing lift. This is real but incomplete. Real aircraft lift also involves the wing pushing air downward (Newton’s third law), the Coanda effect, and downwash. The Bernoulli contribution is one of several mechanisms working together.

Why does a shower curtain billow inward?

Because the water spray creates a fast-moving column of air inside the shower. The faster air has lower pressure than the still bathroom air outside the curtain. The pressure difference pushes the curtain inward. The effect is enhanced by slight warming of the shower air (which rises and creates a vacuum at the bottom that the bathroom air rushes to fill). Both are consistent with Bernoulli’s principle.

When does Bernoulli’s equation NOT apply?

When the fluid is compressible (high-speed air above Mach 0.3, where density changes), when viscosity matters (slow flow through narrow tubes, where friction dissipates energy), when the flow is unsteady (turbulent or pulsating), or when comparing streamlines that don’t share the same total energy. For rigorous engineering, the Navier-Stokes equations take over.

How is Bernoulli’s principle related to conservation of energy?

It IS conservation of energy, applied to a flowing fluid along a streamline. The three terms in Bernoulli’s equation (pressure, kinetic, gravitational) are three forms of energy per unit volume that can interconvert as fluid moves. The total stays constant because no energy is lost in an ideal fluid. Bernoulli’s principle is the work-energy theorem written in fluid-dynamics language.