Last updated: 2026-07-01

Bernoulli's Equation

Bernoulli's equation links the pressure, speed, and height of a flowing fluid. It expresses conservation of energy along a streamline: as a fluid speeds up or climbs, its pressure must fall to keep the total energy per unit volume constant. The relation underlies the Venturi effect, pitot-tube airspeed measurement, and much of everyday hydraulics.

The equation

For steady, incompressible, inviscid flow along a single streamline, the sum of three terms is constant:

$P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}$

Written between two points, point 1 (upstream) and point 2 (downstream):

$P_1 + \tfrac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \tfrac{1}{2}\rho v_2^2 + \rho g h_2$

Each term is a pressure (energy per unit volume): $P$ is the static pressure, $\tfrac{1}{2}\rho v^2$ is the dynamic pressure, and $\rho g h$ is the gravitational (elevation) term. Solving for the downstream pressure gives the form this calculator uses:

$P_2 = P_1 + \tfrac{1}{2}\rho\left(v_1^2 - v_2^2\right) + \rho g\left(h_1 - h_2\right)$

Symbol table

Worked example

The Venturi effect

When a pipe narrows, the conservation of mass (continuity) requires $A_1 v_1 = A_2 v_2$, so the fluid must speed up in the narrow throat. Bernoulli's equation then forces the static pressure there to drop. This pressure reduction in a constriction is the Venturi effect. It is exploited by Venturi flow meters (which infer flow rate from the measured pressure drop), carburetors (which draw fuel into the low-pressure throat), and aspirators.

Lift intuition

Air moving faster over the curved top of a wing has lower static pressure than the slower air beneath it, and that difference contributes to lift. This is a useful intuition, but it is only part of the story: a full account of lift also involves the downward deflection of air and the circulation around the wing. Bernoulli's equation explains the pressure difference, not the complete origin of the force.

Assumptions and limits

Bernoulli's equation is exact only for flow that is steady, incompressible, and inviscid, evaluated along a single streamline with no pumps, turbines, or heat exchange between the two points. Real flows lose energy to friction and turbulence; over long pipes these losses are captured by adding a head-loss term. For gases above roughly Mach 0.3 the incompressible assumption breaks down and compressible relations are needed instead. Within its assumptions, however, the equation is a remarkably accurate and widely used tool.