A fluid dynamics principle stating that for steady, incompressible, nonviscous flow along a streamline, an increase in flow speed corresponds to a decrease in pressure (or potential energy per unit volume), conserving total mechanical energy along the streamline.

Principle

Along a streamline in the specified ideal conditions, total head (pressure head + velocity head + elevation head) remains constant, so kinetic energy increase implies static pressure decrease.

Demonstration

For flow through a horizontal constriction in a smooth pipe, velocity increases at the throat while static pressure measured by a manometer drops compared to the wider section, consistent with p + ½ρv^2 = constant.

Misapplication

Applying Bernoulli between two points not on the same streamline, in strongly viscous flows, compressible flows with shocks, or where energy is added/removed (pumps, turbines) leads to incorrect conclusions about pressure–velocity relations.

Consequence

Explains lift qualitatively on wings (pressure difference due to varying flow speed), predicts pressure changes in pipes and venturi meters, and provides a basis for engineering flow approximations when assumptions are met.

Reversal

The inverse claim—that higher speed always produces higher static pressure—holds in certain forced or compressible regimes (e.g., stagnation points) but contradicts Bernoulli's relation under its stated assumptions.

Boundary

Valid for steady, incompressible, inviscid flows along streamlines; excludes viscous boundary layers, turbulent mixing, compressibility at high Mach numbers, and flows with heat transfer or body forces that do work on the fluid.

Semantic Tension

Often conflated with a simplistic 'faster flow implies lower pressure' slogan; tension exists with dynamic lift explanations invoking circulation and with conservation-of-momentum analyses giving complementary perspectives.

Synthesis

Bernoulli's principle is an energy conservation statement for ideal fluid flow along a streamline: within its assumptions it links speed and pressure changes, but correct application requires verifying steady, incompressible, and inviscid conditions and streamline connectivity.