Hold two sheets of paper a few centimetres apart and blow between them. Instinct says they should fly apart. They snap together instead. The fast-moving air you injected between them carries lower static pressure than the still air outside, and the outside pressure pushes the sheets in. That small parlor trick is the Bernoulli equation made visible, and the same physics meters fuel flow, lifts wings, and explains why a chimney draws harder on a windy day.
This article sets out where the Bernoulli equation comes from, works a venturi example end to end, and flags the assumptions that quietly decide whether the equation applies at all.
Why this calculation matters
The Bernoulli equation is the working engineer's first tool for relating pressure, velocity, and height in a moving fluid. It underpins flow measurement: venturi meters, orifice plates, and Pitot tubes all convert a measured pressure difference into a velocity or a flow rate using nothing more than Bernoulli plus continuity. It explains aerodynamic lift, the operation of carburettors and aspirators, cavitation risk at pump inlets, and the pressure recovery in a diffuser.
Its value is that it is an energy statement you can apply without solving the full equations of motion. If you can identify two points on a streamline and know the conditions at one of them, Bernoulli often gives you the other. That makes it the fast estimate that frames a problem before any detailed analysis begins — and frequently the only calculation a design actually needs.
The core formula
Bernoulli's equation is a statement of mechanical energy conservation along a streamline for steady, incompressible, frictionless flow. Per unit volume of fluid:
p + 0.5 * rho * V^2 + rho * g * z = constant
The three terms are the static pressure p, the dynamic pressure 0.5*rho*V^2 (the kinetic energy of the flow), and the hydrostatic term rho*g*z (the potential energy due to elevation). Their sum, the total pressure, stays constant from one point on a streamline to the next.
Written between two points, point 1 and point 2:
p1 + 0.5*rho*V1^2 + rho*g*z1 = p2 + 0.5*rho*V2^2 + rho*g*z2
The single most important consequence: along a streamline, where the velocity goes up, the static pressure must come down, and vice versa. Energy is simply being shuffled between the pressure term and the kinetic term.
Bernoulli rarely works alone. For flow through a duct that changes area, it is paired with the continuity equation — conservation of mass:
A1 * V1 = A2 * V2
Continuity tells you how much the velocity changes when the cross-section changes; Bernoulli then tells you what that does to the pressure. Together they solve the venturi.
A worked example
Consider water flowing through a horizontal venturi. The fluid density is rho = 998 kg/m^3. In the wide approach section the velocity is V1 = 2 m/s. The throat is constricted to one-quarter of the upstream area. The venturi is horizontal, so there is no change in elevation.
Step 1 — find the throat velocity from continuity. Mass conservation requires A1*V1 = A2*V2. With A2 equal to one-quarter of A1:
V2 = V1 * (A1 / A2) = 2 * 4 = 8 m/s
The flow speeds up by a factor of four through the throat, exactly as the area shrinks by a factor of four.
Step 2 — apply Bernoulli to find the pressure drop. With z1 = z2, the elevation terms cancel and the equation rearranges to give the static pressure drop:
dp = p1 - p2 = 0.5 * rho * (V2^2 - V1^2)
dp = 0.5 * 998 * (8^2 - 2^2)
dp = 0.5 * 998 * (64 - 4)
dp = 0.5 * 998 * 60
dp = 29,940 Pa
Step 3 — interpret the result. The static pressure falls by about 29.9 kPa between the wide section and the throat. The fast-moving throat flow carries the lower static pressure — the kinetic energy gained by speeding up was paid for out of the pressure term. This is precisely the principle a venturi flow meter exploits: measure that pressure drop, and you can back out the flow rate.
Notice how steep the trade is. The velocity quadrupled, but because the dynamic pressure depends on velocity squared, the pressure drop scales with the difference of the squares — which is why a modest-looking constriction produces a substantial pressure change.
Common mistakes
Applying Bernoulli across a pump, fan, or turbine. These machines add or extract energy, so the total pressure is not constant across them. Bernoulli holds only between points with no work crossing the boundary. Across a pump you need the energy equation with a pump-head term.
Ignoring friction in long runs. The frictionless assumption is reasonable across a short, smooth contraction like a venturi throat. Over tens of metres of pipe, viscous losses are significant and a head-loss term must be added, or Bernoulli will overpredict the downstream pressure.
Jumping between different streamlines. The classic form of Bernoulli connects two points on the same streamline. Comparing a point in a fast jet with a point in stagnant surrounding fluid is valid only under extra conditions, such as irrotational flow.
Forgetting compressibility. For gases moving faster than roughly 30 percent of the speed of sound, density changes are no longer negligible and the incompressible Bernoulli equation breaks down. A compressible form is needed.
Confusing static, dynamic, and total pressure. A wall tap senses static pressure; a Pitot tube facing the flow senses total pressure; their difference is the dynamic pressure. Mislabelling which one a gauge reads is a common source of error in flow measurement.
Try the interactive NovaSolver calculator
Working the venturi by hand once is instructive, but seeing the pressure and velocity respond as you reshape the duct builds real intuition. The Bernoulli's Theorem — Pipe Flow Visualizer on NovaSolver lets you set the inlet and outlet diameters, the inlet pressure and velocity, the fluid, and an elevation change, then returns the outlet velocity, outlet pressure, flow rate, dynamic pressure, and total head — with a live picture of the flow narrowing through the throat.
Related calculators
- Reynolds Number Calculator — check whether your flow is laminar or turbulent before trusting a frictionless estimate.
- Pipe Pressure Drop Calculator — adds the friction losses that Bernoulli alone leaves out of long pipe runs.
- Fluid Bernoulli calculator — another take on the energy balance for quick point-to-point checks.
The full set lives in the fluid dynamics tools hub.
Closing note
The Bernoulli equation is deceptively small for how much it explains. It is just an energy balance — static pressure plus dynamic pressure plus elevation, held constant along a streamline — yet it accounts for venturi meters, lift, draft, and the counterintuitive pull between two sheets of paper. The discipline is in the assumptions: steady, incompressible, frictionless, single streamline, no work added. Respect those, pair it with continuity, and Bernoulli will carry you a remarkably long way before you ever need a heavier tool.
