Bernoulli Equation Calculator

Bernoulli's equation states that for an idealized (incompressible, frictionless) fluid flowing along a streamline, the sum of pressure energy, kinetic energy, and gravitational potential energy per unit volume stays constant: P + ½ρv² + ρgh = constant. It is one of the most widely used relations in fluid dynamics.

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, where P is pressure, ρ is fluid density, v is flow velocity, g is gravitational acceleration, and h is height, evaluated at two points along the same streamline.

The Venturi effect is the pressure drop that occurs when a fluid speeds up passing through a constriction, such as a narrowed section of pipe. Increasing v₂ relative to v₁ (with height unchanged) directly lowers the computed P₂, exactly reproducing the Venturi effect from the equation itself.

Torricelli's law gives the exit speed of fluid draining from a hole in an open tank: v = √(2gh), where h is the height of fluid above the hole. It is a special case of Bernoulli's equation where both the tank surface and the exit hole are open to the same atmospheric pressure, so the pressure terms cancel.

The classic form assumes the flow is incompressible (constant density), inviscid (no friction/viscous losses), steady, and evaluated along a single streamline with no external energy added (like a pump) or removed (like a turbine) between the two points.

Because total energy per unit volume must stay constant, an increase in kinetic energy (from higher velocity) must be balanced by a decrease in pressure energy, assuming height stays the same. This is the same principle that generates lift on an aircraft wing, where air moving faster over the curved upper surface has lower pressure than the slower air below.

Yes, as long as the flow speed is well below the speed of sound (so compressibility effects are small, generally below about Mach 0.3), Bernoulli's equation works well for gases like air, which is exactly why it explains aircraft lift and Venturi-effect air flow devices.

With both velocities zero, the equation reduces to simple hydrostatic pressure: P₂ = P₁ + ρg(h₁−h₂), the familiar 'pressure increases with depth' relationship for a fluid at rest.

Real fluids have viscosity (friction), which continuously converts some flow energy into heat, so real pressure and velocity relationships deviate from the ideal Bernoulli prediction, especially over long pipe runs or near solid boundaries. Engineers add empirical loss terms (like the Darcy-Weisbach friction factor) to account for this in practice.

Yes as a simplified illustration: entering the air density, and different speeds above and below a wing (with height difference typically negligible), shows the pressure difference that contributes to lift, though real wing aerodynamics also involves additional effects like circulation and viscous boundary layers.