Bernoulli Equation Calculator
Along a streamline, pressure, speed and height trade against one another in fixed proportion — Bernoulli’s principle. Enter conditions at one point of a flow, and this calculator returns the pressure at a second point with different speed or elevation.
Energy Conservation Wearing a Plumber's Hat
Bernoulli's equation is the work–energy theorem applied to a moving fluid: the sum P + ½ρv² + ρgh — pressure energy, kinetic energy and potential energy, each per unit volume — stays constant along a streamline for steady, incompressible, frictionless flow. Daniel Bernoulli published it in Hydrodynamica (1738), and its trade-offs run the engineered world: where a pipe narrows, the fluid speeds up (continuity) and its pressure drops (this equation) — the Venturi effect that meters flows, atomises paint, and empties carburettors into engines. The hydrostatic page of this hub is its v = 0 special case; the calculator here is the general statement, priced point-to-point, with the assumptions printed on its face because the equation's jurisdiction — no viscosity, no compressibility, one streamline — is exactly where beginners and brochures go wrong.
Worked Example: The Narrowing Pipe
The defaults run water (ρ = 1,000) from a section at 200,000 Pa and 2 m/s into a narrower section at 5 m/s, level: P₂ = 200,000 + ½ × 1,000 × (2² − 5²) = 200,000 − 10,500 = 189,500 Pa. Ten and a half kilopascals were spent buying speed — energy the fluid recovers if the pipe widens again downstream (a diffuser), or loses to turbulence if it doesn't. Raise point 2 by 3 m instead (dh = 3, speeds equal) and the height term collects its own toll: ρgΔh = 29,430 Pa — the hydrostatic page's arithmetic embedded in the flow's, exactly as the derivation promises. Pilots meet the same trade in air: at 1.225 kg/m³ the numbers shrink a thousandfold, which is why the pitot tube needs sensitive instruments and the aeroplane's lift bookkeeping needs care about which "Bernoulli explains flight" story is being told (see the FAQs).
The Pitot Tube: This Equation as an Instrument
Every airspeed indicator is a Bernoulli meter: the pitot faces the flow and reads the stagnation pressure (point 1: the incoming air's total), the static port reads the ambient P, and the difference — ½ρv² by this page's algebra with v₂ = 0 — yields the speed. Ships' logs, wind tunnels and Formula 1 aero rakes run the same subtraction. The instrument's honesty depends on the equation's assumptions, which at flight speeds in thin air hold well enough that a blocked pitot (the AF447 disaster's first domino) misleads the whole aircraft: three independent Bernoulli meters froze, and the crew, deprived of the equation's output, lost the plot the numbers had been narrating. The calculator's five inputs are the pitot's world, simplified to a desk. That an airliner's fate can hang on three heated tubes reading this equation is the strongest possible testimony to its reliability in its jurisdiction — and the strongest argument for the jurisdiction being printed, as here, wherever the equation is taught.
The Atomiser and the Perfume Bottle
The squeeze-bulb atomiser on a vintage dressing table is this page in cut glass: the bulb's jet races air across the top of the dip tube, the pressure there falls by the dynamic term, and the bottle's still surface — at full atmospheric pressure — pushes perfume up the tube into the stream, where it shreds into mist. Paint sprayers, carburettor jets and the Venturi scrubbers of industry are the same throat at larger scales. The calculator's worked example, read backwards (give it the fast jet's pressure deficit and ask what the still reservoir feels), is the atomiser's design sheet — Bernoulli's equation as a household appliance, two centuries before the equation had a website.
The Curveball's Confession
A spinning baseball carries its boundary layer around, speeding the flow on one side and slowing it on the other; the pressure field this equation audits is consequently lopsided, and the ball accelerates sideways — the Magnus effect, whose force the hub's sports pages could price from the same dynamic-pressure term. Bernoulli does not cause the asymmetry (viscosity and spin do), but it keeps its accounts: given the two sides' speeds, the lateral pressure difference is ½ρ(v₁² − v₂²) across the ball's width, which is the calculator's exact operation with air's density entered. The distinction between cause and accountancy is this page's recurring sermon — the equation is the flow's ledger, and ledgers do not care who spent the money.
Related Tools
More from the Physics Calculators hub: the Hydrostatic Pressure Calculator — this equation at rest — and the Mach Number Calculator, for the speeds where incompressibility itself resigns.
Frequently Asked Questions
Does Bernoulli explain how wings lift?
Partly, and the brochure version ("faster air on top because it has farther to travel") is wrong in its premise — parcels do not reunite, and the top flow is faster than the equal-transit story claims. The honest account is a partnership: the wing deflects air downward (Newton's third law takes the reaction as lift), and the pressure field that accomplishes the deflection obeys Bernoulli along its streamlines — lower pressure above, higher below, consistent with the speed difference the circulation creates. The equation quantifies the pressure pattern faithfully; it does not, alone, explain why the pattern exists. This page computes the equation, and this FAQ declines the myth — the two services a physics hub owes its readers in equal measure.
What breaks the equation in real ducts?
Viscosity first: friction converts the constant's energy into heat along real pipes (the Darcy–Weisbach head-loss term is engineering's patch on Bernoulli), so real P₂ falls below the calculator's figure — increasingly with length, roughness and speed. Then compressibility (past about Mach 0.3 the density term moves), unsteadiness (start-up surges, water hammer), and streamline-crossing (the two points must share a streamline, or the constant differs between them). The calculator's answer is the ideal benchmark from which real designs subtract their losses — the standard relationship between a physics page and an engineering office.
Why do shower curtains billow inward?
The beloved demonstration, with an honest asterisk: the running shower's air moves, and by this equation its pressure sits lower than the still bathroom air's — a real contribution, though entrainment and the warm air's buoyancy share the blame in careful analyses. The same family of effect flattens roofs in storms (fast air above, still pressure below, the difference lifting) and passes a football's curveball (the Magnus force — a cousin via the pressure field around spin). The curtain is the friendliest of the Venturi clan, and the asterisk is the hub's standing rule: an effect can be real, Bernoulli-flavoured, and still not wholly Bernoulli's.
What is the Venturi meter measuring?
Flow rate, by this equation plus continuity: a known constriction forces the speed up by the area ratio, the pressure drop across it is measured, and the two equations solve for the discharge — no moving parts, a small permanent loss, and a century of service in waterworks and carburettors. Enter its geometry backward here: guess the throat speed, compute the pressure drop, compare with the gauge, iterate. The calculator's point-to-point form is the meter's entire theory, which is why process engineers keep the equation where this page keeps it — at the top of the fluids drawer.
How does a carburettor use it?
The venturi in the carburettor's throat speeds the intake air and drops its pressure by this equation's amount; the float bowl, vented to the fuller pressure, pushes fuel through a jet into the low-pressure throat, where it atomises into the stream. The metering is self-scaling — more air, more drop, more fuel — a pneumatic computer running Bernoulli's algebra at every throttle position, rendered obsolete only by electronic injection's greater precision. A generation of mechanics tuned engines by ear against this equation's behaviour; the calculator restores its arithmetic to view.
How does a siphon fit the equation?
Beautifully, as a three-point audit: the reservoir surface (still, atmospheric), the siphon's crest (raised, moving — pressure below atmospheric by both the speed and height terms, which is why a siphon taller than about 10 m of water fails as the crest pressure reaches vapour pressure and the column cavitates), and the outlet (lower, fastest, atmospheric again). The calculator prices points in pairs along exactly this path. The siphon's ancient mystery — water flowing uphill unaided — dissolves into the equation's bookkeeping: the downhill leg's weight pays the uphill leg's fare, with the crest pressure as the visible receipt.
Why do tall chimneys draw better?
Partly Bernoulli's cousin, buoyancy (hot gas is lighter, the stack effect — a height term of exactly this page's ρgh with two densities), and partly the wind's Bernoulli term across the cap. The calculator's height field, run with flue-gas densities, prices the static half of the draft; stoves and forges were tuned by it empirically for centuries.
Who was Bernoulli?
Daniel Bernoulli (1700–1782), of the Basel dynasty that staffed European mathematics for a century (his father Johann and uncle Jacob feuded famously; Daniel's own father competed with him bitterly over the Hydrodynamica's priority). The equation named for Daniel was cast into its modern energy form by successors (Euler's equations contain it), but the 1738 book's core insight — pressure and speed as interchangeable currencies of a flow — is his, and it remains the most practically consequential idea in fluid mechanics: this page, the pitot tube, and the waterworks of every city divide the inheritance.