Bernoulli's principle
Bernoulli's principle describes how pressure, speed and elevation are related in a flowing fluid. In its familiar horizontal-flow case, faster-moving fluid corresponds to lower static pressure along a streamline. It follows from energy conservation under idealized conditions and is used to understand effects such as flow through constrictions and pressure-based velocity measurement.
Statement and equation
Bernoulli's principle is a simplified case of Bernoulli's equation, which expresses conservation of mechanical energy for a fluid along a streamline. For steady, incompressible flow with negligible viscosity, the sum p + ½ρv² + ρgh remains constant along that streamline. Here p is static pressure, ρ is fluid density, v is flow speed, g is gravitational acceleration and h is elevation. Each term has units of energy per unit volume.
The pressure–speed relationship
At constant elevation, the gravitational term does not change, so an increase in flow speed corresponds to a decrease in static pressure if the stated assumptions apply. This is not a rule that pressure always falls wherever fluid speeds up: elevation differences, viscous losses, pumps, compressibility and the flow geometry can matter. The complete equation, not the simplified slogan, is needed when those factors are important.
Applications
In a narrowing section of a pipe, continuity of an incompressible flow requires fluid to move faster through the smaller cross-section. Bernoulli's equation then relates this higher speed to lower static pressure when elevation is unchanged and losses are negligible. The same relationship underlies Venturi meters and other pressure-difference methods for estimating flow speed. Real devices require calibration and may need corrections for viscosity and other non-ideal effects.
History and limits
Daniel Bernoulli published work on fluid motion in Hydrodynamica in 1738. The principle is a useful ideal-fluid model, not a complete account of every real flow. For example, Bernoulli's equation can help relate velocity and pressure around an airfoil, but it does not by itself explain the full production of aerodynamic lift, and the popular equal-transit-time explanation of wing lift is incorrect.
Key facts
- General relation
- p + ½ρv² + ρgh is constant along a streamline for steady, incompressible flow with negligible viscosity.
- Horizontal case
- At constant elevation and under the model's assumptions, higher flow speed corresponds to lower static pressure.
- Applications
- Venturi flow and pressure-difference velocity measurement use Bernoulli's equation.
- Important limitation
- The equation is an idealized energy relation; real flows may require accounting for viscosity, pumps, compressibility and other factors.
- Airfoil caution
- Bernoulli's equation contributes to understanding pressure and velocity around airfoils but does not validate the equal-transit-time lift explanation.
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References
- 1.14.6 Bernoulli’s Equation
OpenStax, Rice University · other · reliability: unverified · retrieved 30 October 2026
Supports: University physics text explaining Bernoulli's equation as energy conservation for incompressible, frictionless flow and discussing assumptions, pressure-speed relations, entrainment and velocity measurement.
- 2.Bernoulli’s theorem
Encyclopaedia Britannica · other · reliability: unverified · retrieved 30 October 2026
Supports: Reference article describing the relation among pressure, velocity and elevation for steady flow under idealized conditions, its energy-conservation basis, and the Venturi effect.
- 3.Incorrect Lift Theory
NASA Glenn Research Center · other · reliability: unverified · retrieved 30 October 2026
Supports: NASA's educational explanation identifies the equal-transit-time account of lift as incorrect while noting the valid pressure and velocity relationship described by Bernoulli's equation.
References are suggested by the research engine and may be incomplete. Unverified URLs are omitted rather than guessed.
Article history
- Version 130 October 2026 · EditorCreated by Zapia