Bernoulli equation
The Bernoulli equation is energy conservation for a flowing fluid. Along a streamline the sum of pressure energy, kinetic energy, and potential energy stays constant, which is why a fluid speeds up where pressure drops.
Variables
| P | Static pressure | Pa |
| ρ | Fluid density | kg/m³ |
| v | Flow velocity | m/s |
| g | Gravity | 9.81 m/s² |
| h | Elevation | m |
Rearranged
Worked example
Water (ρ = 1000) speeds up from 2 m/s to 6 m/s along a level pipe.
The pressure drop is ½ × 1000 × (6² − 2²) = ½ × 1000 × 32.
Each term is an energy per unit volume: pressure, motion, and height. As one rises another must fall, so a constriction that speeds the flow lowers its pressure. The ideal form assumes no friction; real pipes lose energy, handled separately as pressure drop.
Following the flow downstream?
See the Flow Rate Formula and the Pressure Drop Formula.
Energy conservation in flow
Bernoulli is the fluid version of conservation of energy. The three terms, pressure, kinetic, and potential, can trade among themselves but their sum is fixed along a streamline. Raise the fluid and pressure or speed must give way; speed it up and pressure drops. It holds for steady, incompressible, frictionless flow.
The venturi effect
Squeeze flow through a narrower section and continuity forces it to speed up; Bernoulli then says its pressure must fall. This venturi effect explains carburetors, atomizers, and the lift on a wing, and it is the basis of venturi and orifice flow meters, which read flow rate from the pressure difference across a constriction.
FAQ
What is the Bernoulli equation?
It states that pressure plus half rho v squared plus rho g h is constant along a streamline, expressing energy conservation for an ideal flowing fluid.
Why does pressure drop when fluid speeds up?
Because the total energy is fixed. If the kinetic term rises as the fluid accelerates, the pressure term must fall to keep the sum constant.
Does Bernoulli account for friction?
No. The basic equation assumes frictionless flow. Real pipe losses are added separately as a pressure drop or head loss term.