Solve Bernoulli's equation for fluid pressure at a second point in a flow system, given pressure, velocity, and height at two points.
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Bernoulli's equation describes how pressure, velocity, and height trade off against each other along a streamline in steady, frictionless, incompressible fluid flow, expressing the principle that the sum of pressure energy, kinetic energy, and gravitational potential energy per unit volume remains constant between any two points in the same flow. This calculator solves the equation for pressure at a second point, given the pressure, velocity, and height at a first point along with the velocity and height at the second point, using standard gravitational acceleration (9.81 m/s²). The equation captures a genuinely counterintuitive result: where a fluid speeds up, its pressure must drop to conserve total energy, which is the same underlying principle behind phenomena as varied as airplane wing lift, a shower curtain being pulled inward by fast-moving water, and the reading on a pipe's pressure gauge changing as its diameter narrows. Engineers rely on this same relationship when sizing venturi meters and carburetors, both of which deliberately narrow a flow path to create a measurable pressure drop.
One of the more delightfully counterintuitive results in classical physics is that speeding a fluid up actually lowers its pressure, not raises it — the exact opposite of what intuition built from everyday experience with solid objects might suggest. Bernoulli's equation is the mathematical statement of exactly why this happens, and once understood, it explains an enormous range of everyday and engineering phenomena.
At its core, Bernoulli's equation is nothing more exotic than energy conservation applied to a moving fluid. Every unit volume of fluid carries three forms of energy as it flows: pressure energy (from the fluid's own internal pressure pushing on its surroundings), kinetic energy (from its velocity), and gravitational potential energy (from its height). Bernoulli's insight was recognizing that, for an idealized steady, frictionless, incompressible flow, the sum of these three energy forms stays constant along any single streamline, even as the individual pieces shift back and forth.
This constancy is exactly what forces the counterintuitive pressure-velocity tradeoff: if a fluid enters a narrower section of pipe, conservation of mass (the same amount of fluid has to pass through a smaller cross-section) forces its velocity to increase. Since total energy must stay the same and height typically doesn't change within a short pipe section, that increase in kinetic energy has to come from somewhere — and it comes directly out of the fluid's pressure energy, causing pressure to drop exactly where the fluid is moving fastest.
This exact mechanism explains a surprising range of phenomena once you know to look for it. An airplane wing is shaped so air flows faster over its curved upper surface than along its flatter underside, creating a pressure difference between the two surfaces that generates lift. A shower curtain gets pulled inward toward fast-moving water because the moving water's lower pressure creates a pressure difference against the still air on the other side. Even a simple Venturi meter, a common flow-measurement device, works by narrowing a pipe deliberately and measuring the resulting pressure drop to calculate flow rate.
The idealized assumptions behind the equation — steady flow, zero friction, and incompressible fluid — are never perfectly true in real engineering systems, but they're often close enough to give useful, practical answers, particularly for liquids moving at moderate speeds over short distances. For situations with significant friction (long pipe runs) or compressible flow (gases at high speed), engineers use extended versions of the equation that add correction terms for the energy Bernoulli's idealized version doesn't account for.
Bernoulli's equation is fundamentally a statement of energy conservation; if kinetic energy (tied to velocity) increases while total energy stays constant, pressure energy must decrease correspondingly to balance the equation.
Bernoulli's equation assumes steady flow (not changing over time), incompressible fluid (constant density), and negligible friction (no viscous energy losses) along a single streamline — real-world flows with significant friction need additional correction terms.
The equation technically applies to any fluid, but the incompressible assumption holds much better for liquids than for gases at high speed, where compressibility effects become significant and a modified form of the equation is needed.