Bernoulli's equation: along a streamline in steady, incompressible, frictionless flow, the sum of static pressure, dynamic pressure and gravitational term stays constant — P + ½ρv² + ρgh = constant. Faster flow means lower pressure, and rising fluid trades pressure for height. This free calculator solves for the outlet pressure P2 or the outlet speed v2, showing every step.
Bernoulli's equation is a statement of energy conservation for a moving fluid. Pick two points on the same streamline — an inlet (point 1) and an outlet (point 2) — and the equation says P1 + ½ρv1² + ρgh1 = P2 + ½ρv2² + ρgh2. Each term is an energy per unit volume: the static pressure P, the dynamic pressure ½ρv² carried by the motion, and the gravitational term ρgh. Their sum is fixed, so if one rises another must fall.
Rearranged for the outlet pressure, P2 = P1 + ½ρ(v1² - v2²) + ρg(h1 - h2); rearranged for the outlet speed, v2 = sqrt(v1² + 2(P1 - P2)/ρ + 2g(h1 - h2)). Choose the target in the calculator's Solve for menu, enter the pressures (Pa or kPa), the speeds (m/s), the two heights (m — negative for a drop) and the density (kg/m³), and read the answer with the worked steps. The gravitational constant used is g = 9.81 m/s².
Two behaviours are worth feeling directly. At constant height, speeding the fluid up drops its pressure — the effect behind a Venturi meter and an aerofoil's lift. And a liquid climbing to a greater height loses pressure by ρg times the rise, even with no change in speed. If you know the flow speed and want the resistive force instead, see the drag-force calculator, or look up a term in the physics glossary.
Water (ρ = 1000 kg/m³) flows along a horizontal pipe. At the inlet the pressure is P1 = 150,000 Pa and the speed is v1 = 2 m/s; where the pipe narrows the speed rises to v2 = 5 m/s, with h1 = h2 = 0. The outlet pressure is P2 = 150,000 + ½·1000·(2² - 5²) + 0 = 150,000 - 10,500 = 139,500 Pa — that is 139.5 kPa. The fluid sped up, so its pressure dropped by 10.5 kPa. Switching the calculator to solve for v2 with that same 139,500 Pa returns exactly 5.00 m/s, closing the loop.
Bernoulli's equation underlies how wings and propellers generate lift, how carburettors, atomisers and Venturi flow meters work, why a shower curtain billows inward, how a Pitot tube measures an aircraft's airspeed, and how pressure and flow trade off through pipes, nozzles and blood vessels. It is the first tool engineers reach for whenever a fluid speeds up, slows down or changes height.
Enter pressures in pascals (Pa) by default; you can switch either pressure field to kilopascals (kPa) with the unit selector, and the calculator converts for you (1 kPa = 1000 Pa). Speeds are in metres per second, heights in metres, and density in kg/m³. The outlet pressure result is shown in pascals with the kilopascal value alongside.
Only if the two points are at different heights. The term rho*g*(h1 - h2) accounts for the change in gravitational potential energy of the fluid. If the pipe is horizontal, set h1 = h2 (or leave both at 0) and the term vanishes. The single most common mistake is dropping the rho*g*h terms when h1 does not equal h2 — for a liquid going up or down a real height, they can dominate the pressure change.
When you solve for the outlet speed and the calculator returns no valid solution, the quantity under the square root came out negative: there is no real flow speed that satisfies those pressures and heights, so the combination is physically impossible. When you solve for pressure and get a negative outlet pressure, a real liquid would cavitate — its pressure cannot fall below zero absolute, and vapour bubbles form first — so the calculator flags it in the working.
Change the "Solve for" menu from P2 to v2. The calculator then uses the rearranged form v2 = sqrt(v1^2 + 2(P1 - P2)/rho + 2g(h1 - h2)). Enter both pressures, the inlet speed, both heights and the density, and it returns the outlet speed. Raising the outlet pressure P2 toward the inlet P1 (at equal heights) shrinks v2 back toward v1.
Yes, as a good approximation while the air flows well below the speed of sound — roughly under 100 m/s — where it behaves as effectively incompressible. That covers ventilation ducts, aerofoils at low speed, Venturi meters and atomisers. Above about a third of the speed of sound, compressibility matters and you need the compressible-flow form instead; use the density of air (about 1.2 kg/m³) rather than water.