Bernoulla

Meet Bernoulla, water rushing through a narrow pipe: where she speeds up, her pressure drops.

Daniel Bernoulli published the idea in 1738. For a steady flow of an incompressible fluid with no friction, the sum p + ½ρv² + ρgh is the same at every point along a streamline. It is energy conservation per unit volume: pressure energy, kinetic energy and gravitational energy can trade off, but their total stays fixed.

Where a horizontal pipe narrows, the same volume each second must pass through a smaller area, so the water speeds up. Bernoulli's equation then says its pressure must drop. That is the Venturi effect, used in Venturi meters to measure flow rates and in carburettors and atomisers.

Bernoulli's principle also explains why a shower curtain is drawn inward and helps account for pressure differences on aircraft wings, although full lift needs more of fluid dynamics than this one equation.

Honest limits: real fluids have viscosity, real flows can be turbulent, and gases compress at high speeds. The equation is an idealisation that works well for many everyday flows.