The Bernoulli Principle: Trading Pressure for Speed

Energy conservation along a streamline

The Bernoulli equation is nothing more than the work-energy theorem applied to a moving fluid, tracked along a single streamline in steady, incompressible, frictionless flow. Add up pressure energy, kinetic energy and gravitational potential energy per unit volume, and the sum stays constant as the fluid moves:

P + 1/2 * rho * v^2 + rho * g * h = constant   // along a streamline
P1 + 1/2*rho*v1^2 = P2 + 1/2*rho*v2^2           // horizontal flow, h1 = h2

The equation is essentially trading pressure for speed: wherever the fluid speeds up, its pressure must drop, and wherever it slows down, pressure recovers. It is easy to misread as saying fast flow simply has low pressure in some independent sense — the correct reading is that pressure and speed are two forms of the same conserved energy budget, and one falls exactly as much as the other rises.

Continuity: why a narrower pipe means faster flow

Bernoulli's equation only tells you what happens to pressure once you already know the speed, and the speed itself is fixed by a separate, even simpler law: conservation of mass. For an incompressible fluid, whatever volume flows into a pipe each second must flow out each second, so the volumetric flow rate A·v is the same everywhere along the pipe:

A1 * v1 = A2 * v2       // continuity equation, incompressible flow

In a Venturi tube — a pipe with a narrow throat in the middle — the cross-sectional area A drops sharply at the constriction, so continuity demands the velocity v rises sharply to keep the product constant. Bernoulli then demands the pressure drop just as sharply at that same point. A pressure gauge at the throat of a Venturi tube reads lower than at the wide entrance and exit, purely because the fluid has to speed up to get through, and this pressure drop is the working principle of the Venturi flow meter, the carburettor, and medical and industrial devices that use a constriction to entrain a second fluid at low pressure.

Why the fluid itself matters

The continuity equation A1v1 = A2v2 is purely geometric and does not depend on which fluid is flowing — water, air, oil or mercury all speed up by the same ratio through the same constriction. Density rho only enters through the pressure term, 1/2·rho·v². For a given velocity change, a denser fluid like mercury (ρ ≈ 13,534 kg/m³) produces a far larger pressure drop than the same velocity change in air (ρ ≈ 1.2 kg/m³) or water (ρ ≈ 1,000 kg/m³) — roughly proportional to density, all else equal. This is why mercury manometers can register tiny pressure differences with a very short column, while measuring the same pressure difference in air would require an impractically large device.

Where the idealisation breaks

Bernoulli's equation as written assumes the flow is inviscid (no internal friction), incompressible and steady along a single streamline. Real pipe flow loses energy to viscous friction against the walls and internal turbulence, so pressure downstream of a constriction never fully recovers to its upstream value — an effect engineers account for with an empirical loss coefficient. At speeds approaching a substantial fraction of the speed of sound, air can no longer be treated as incompressible and the simple form of the equation breaks down, which is why high-speed aerodynamics needs the compressible generalisation instead. For the everyday range this simulation covers — liquids and slow-moving gas in a tube — the incompressible form is an excellent approximation.

Everyday Bernoulli

The same trade-off between speed and pressure explains why a shower curtain billows inward once the water is running (faster air inside the stream, lower pressure than the still air outside), why an airplane wing generates lift as air is forced to travel differently over its curved upper surface, why a spinning ball curves through the air (the Magnus effect combines this pressure difference with rotation), and why pinching a garden hose makes the water shoot out faster — the same continuity and pressure trade-off you can watch directly in the Venturi tube on this page.

Frequently asked questions

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