Bernoulli Equation Calculator

Where Fluid Speeds Up, Pressure Drops

Bernoulli's equation is the reason a narrowed pipe carries fluid faster but at lower pressure, and the reason an airplane wing generates lift. It's a statement of energy conservation along a streamline: the sum of pressure energy, kinetic energy, and gravitational potential energy per unit volume stays constant as a fluid flows, as long as friction and turbulence are negligible. This calculator solves that relationship for whichever pressure or velocity term is unknown.

The Formula

P1 + ½ρv1² + ρgh1 = P2 + ½ρv2² + ρgh2

Here ρ is fluid density, g is standard gravity (9.81 m/s²), and the subscripts 1 and 2 refer to two points along the same streamline. Rearranged, the calculator solves directly for P2, P1, v2, or v1 depending on which three values you supply.

Where This Shows Up

Venturi meters and carburetors — a constriction speeds the fluid, dropping local pressure in a precisely predictable way, which is how these devices measure or meter flow.
Aerodynamic lift — airflow moving faster over a curved wing surface creates a pressure difference relative to the slower flow underneath.
Plumbing and HVAC design — pipe sizing decisions account for the pressure drop that comes with any change in cross-section or elevation.
Medical diagnostics — blood velocity through a narrowed artery can be related back to pressure changes using the same underlying principle.

A Worked Example

Consider water (ρ = 1,000 kg/m³) flowing through a horizontal pipe (h1 = h2, so the elevation terms cancel) at 200,000 Pa and 2 m/s, entering a narrower section where it speeds up to 6 m/s:

P2 = P1 + ½ρ(v1² − v2²)
P2 = 200,000 + 0.5 × 1,000 × (2² − 6²)
P2 = 184,000 Pa

The pressure drops by 16,000 Pa even though the pipe is horizontal — entirely a result of the increase in speed.

How to Use This Calculator

Choose what to solve for: P2, P1, v2, or v1.
Enter Fluid Density (kg/m³), and optionally the two elevations h1 and h2 in meters (defaults to 0 for a horizontal system).
Enter the remaining known pressures and velocities for the mode you selected.
Select Calculate to get the unknown value along with the full worked formula.

Related Calculations

For pipe flow behavior beyond pressure and velocity, see the Reynolds Number Calculator to check whether the flow is laminar or turbulent, or use the Pressure Calculator for simpler force-based or hydrostatic pressure problems.