Bernoulli's equation

Bernoulli's equation is conservation of energy written for a moving fluid. Along a streamline the sum of three terms — the pressure, the kinetic energy per unit volume ½ρv², and the gravitational term ρgh — stays constant. None of them is conserved on its own; it is the total that holds, and the terms trade against each other.

The trade that surprises people is between speed and pressure. Where the fluid moves faster, the pressure must be lower, which sounds backwards until you look at what accelerates the fluid in the first place. A parcel entering a constriction speeds up, and the only thing that can push it forward is a pressure that is higher behind it than in front. High speed and low pressure go together because the drop in pressure is what produced the speed. This is why a shower curtain pulls inwards, why two ships running close abreast are drawn together, and why an aerofoil generates lift.

The conditions attached are not decoration. The derivation assumes steady flow, an incompressible fluid, and negligible viscosity, and real situations break these constantly. Viscous flow in a long narrow pipe loses pressure to friction and needs Poiseuille's law instead; turbulent flow has no well-defined streamlines to apply it along; fast gas flow compresses. Applied where its assumptions hold it is remarkably powerful, and applied where they do not it produces confident nonsense.