Bernoulli's Equation
Along a streamline in steady, incompressible, inviscid flow, the sum of static pressure, dynamic pressure and hydrostatic pressure is constant. It is the mechanical-energy balance of fluid mechanics.
Variables
| Symbol | Meaning | Constraints / unit |
|---|---|---|
| P | Static pressure | Pa |
| ρ | Fluid density, constant | kg/m³ |
| v | Flow speed at the point | m/s |
| g | Gravitational acceleration | 9.81 m/s² |
| z | Elevation above a datum | m |
Assumptions — all four are required
- Steady flow — the velocity field does not change with time.
- Incompressible — density is constant (liquids; gases below roughly Mach 0.3).
- Inviscid — friction is negligible between the two points.
- Along a streamline — the two points lie on the same streamline (or the flow is irrotational).
Derivation sketch — work–energy on a fluid element
Consider a fluid element moving along a streamline. Net pressure work per unit volume over a displacement is ; the change in kinetic energy per unit volume is ; the change in potential energy per unit volume is . With no viscous losses, the work–energy theorem gives:
Integrating along the streamline yields the constant sum.
Worked example — why aeroplane wings feel lift
Air at moves at 70 m/s over the top of a wing and 60 m/s beneath it (same elevation). The pressure difference is:
Roughly 780 Pa of upward pressure difference — about 78 kg of lift per square metre of wing.
Beyond Bernoulli
Real pipe systems add a head-loss term (Darcy–Weisbach) and pump or turbine work. When viscosity matters everywhere — boundary layers, MHD flows, nanofluids — the full Navier–Stokes equations take over.