Bernoulli's equation describes how pressure, velocity, and elevation trade off against each other as a fluid flows along a streamline. It explains why a narrowing pipe speeds up flow and drops pressure, why airplane wings generate lift, and why water pressure at a faucet depends on the height of the tank feeding it.
What Bernoulli's equation says
Bernoulli's equation is a statement of conservation of energy for a flowing fluid: P + ½ρv² + ρgh stays constant along a streamline. The three terms — static pressure, dynamic pressure, and elevation (hydrostatic) pressure — can trade off against one another, but their sum at any point on the same streamline is the same as at any other point.
Because it's an energy-conservation statement, it only strictly holds for steady, incompressible, inviscid (frictionless) flow along a single streamline — real fluids always lose some energy to viscosity, so the equation is an idealization that works well when those losses are small.
Dynamic vs. static vs. elevation pressure
Static pressure (P) is what a pressure gauge moving with the fluid would read. Dynamic pressure (½ρv²) is the kinetic energy of the moving fluid packed into pressure units — it's what you'd measure if you brought the flow to a sudden stop. Elevation pressure (ρgh) is the familiar hydrostatic term: pressure increases with depth and decreases with height.
The Pressure Terms tab plots all three terms at both points side by side, which makes it easy to see, for example, that a velocity increase at point 2 is 'paid for' by a static-pressure decrease of the same size, while the elevation term stays fixed if both points are at the same height.
The Venturi effect
When a fluid moves from a wide section of pipe into a narrow one, the continuity equation (A₁v₁ = A₂v₂) forces its velocity to increase, since the same volume of fluid has to pass through a smaller opening in the same time. Bernoulli's equation then requires the static pressure to fall to compensate for that extra kinetic energy — this pressure drop is the Venturi effect.
The Venturi tab combines both equations directly: give it an inlet velocity and the two cross-sectional areas, and it computes the resulting throat velocity and pressure drop in one step.
Assumptions and when Bernoulli's equation doesn't apply
The equation assumes the fluid is incompressible (a good approximation for liquids, and for gases well below the speed of sound), inviscid (no internal friction), the flow is steady (not changing over time), and both points lie on the same streamline. It also assumes no external energy is added or removed between the two points — a pump or turbine in between would invalidate the simple form used here.
Real pipe flow always has some viscous loss, which Bernoulli's equation ignores — engineers add a 'head loss' correction term for friction in practical pipe-flow calculations. For high-speed gas flow approaching the speed of sound, compressibility effects also break the incompressible assumption.
Real-world applications
Carburetors and aspirators use the Venturi effect to draw in fuel or draw up a sample fluid, using a constriction to lower pressure at exactly the point where a side channel connects. Airplane wings generate lift partly because their curved upper surface causes air to speed up (and pressure to drop) relative to the flatter underside. Flow meters (Venturi meters, orifice plates) measure flow rate by reading the pressure drop across a known constriction. And Torricelli's law for tank draining — v = √(2gh) — is simply Bernoulli's equation applied to a large open tank with equal pressures at the surface and the exit.