What is Bernoulli's principle?

Water pushed through a pipe that narrows has to speed up in the narrow part, the neck. The same litres pass every point each second, so where the pipe is half as wide across, with a quarter of the area, the water must go four times as fast. That is continuity: A₁v₁ = A₂v₂.

To speed the water up, something must push it forward. That push comes from the pressure behind it being higher than the pressure in front. So wherever the water is fastest, its pressure is lowest. The glass tubes standing on the pipe show it: the water in them rises less over the neck.

Bernoulli's principle puts numbers on it. Along the flow, P + ½ρv² + ρgh stays the same. P is the pressure, ½ρv² is the energy of motion in each cubic metre, and ρgh is the energy of height. A kilopascal is exactly one joule per litre, so the rule simply says each litre keeps its energy: it trades pressure for speed and back again.

The dashed line is that total. Each water column shows the pressure part, and the orange bar above it is the speed part. Together they always reach the line.

Blow fast air across the top of a thin tube standing in a liquid. The moving air has lower pressure than the still air resting on the liquid in the bottle. That still air pushes down harder, so the liquid climbs the tube. If it reaches the top, the air stream tears it into a fine mist.

How fast must the air go? The suction is about ½ρv². Lifting perfume 3 cm needs ρgh = 850 × 9.81 × 0.03 ≈ 250 Pa, so the air must reach about 20 m/s. A squeeze of an old-style perfume atomiser bulb does that easily. The same trick runs airbrushes, the old Flit gun for mosquitoes, and laboratory water aspirators.

A carburettor is the grown-up version. Air rushing into the engine passes a narrow venturi, where it speeds up and its pressure drops. A jet from the fuel bowl opens right there, so the engine sucks in petrol in step with the air it breathes. The harder the engine breathes, the more fuel it draws. Wilhelm Maybach's spray-nozzle carburettor of 1893 did this, and most cars and motorbikes used one until electronic fuel injection took over. See CarClear.

Run Bernoulli's rule backwards. A fluid sitting still at a high pressure has lots of pressure energy and no speed. Let it out through a small hole into the open air and the pressure falls to nothing, so all that energy becomes speed: ½ρv² = Δp, or v = √(2Δp ÷ ρ).

Gas hob. The regulator on an Indian LPG cylinder gives a gentle 29 millibars, only 3% above air pressure. But LPG gas is light, so through the tiny injector it still shoots out at about 50 m/s. That fast jet drags air into the mixing tube with it, so the flame gets air before it even burns. See ChimneyClear.

Dishwasher. The pump pushes water to the spray arms at about half a bar. Each nozzle turns that into a jet of about 10 m/s: pointed straight up it could climb 4 or 5 metres. See DishwasherClear.

RO purifier. The pump must keep about 5 bar on the membrane, yet most water leaves down the drain. A tiny flow restrictor on the reject line holds the pressure back by making that water squirt through a hole under a millimetre wide, at over 30 m/s. Fit a bigger hole and the pressure collapses. See ROClear.

A wing is tilted and curved so that it turns the air downwards. To bend the flow over its curved top, the pressure there must drop below the air around it, so the air on top speeds up. Underneath, the air is slowed and squeezed a little, so its pressure rises. Low above plus high below gives lift.

That is Bernoulli and Newton telling one story. Bernoulli links the fast air on top to its low pressure. Newton's third law says that if the wing pushes air down (the downwash), the air pushes the wing up. You need both halves, and neither is "the real one".

An airliner's wing holds up about 636 kg on every square metre at take-off. A ceiling fan blade is a small, slow wing turned on its side: it bends air down onto you. See FanClear.

Drafting. A vehicle drags a pocket of air along behind it. A cyclist in that pocket meets slower air, and drag grows with the square of the air speed, ½ρv². Tuck in 2.6 m behind a motorbike and your drag halves. In 2018 Denise Mueller-Korenek rode a bicycle at 296 km/h behind a dragster's fairing. See CycleClear.

A ball in a jet. Point a hair dryer up and a table-tennis ball hovers in it, even when you tilt it. The jet's own pressure is the same as the room's, so this is not simple Bernoulli. The jet bends round the ball (the Coandă effect), and air turned one way pushes the ball the other way, back into the stream.

Bernoulli's rule is exact only in an ideal world. It holds along one streamline, for flow that is steady, incompressible (the density stays put) and frictionless. Real fluids break each of these a little, and sometimes a lot.

Friction. Water rubbing on a pipe wall loses energy as heat, so pressure falls steadily along a long pipe even though the speed stays the same. The Darcy–Weisbach formula measures the loss: f × (L ÷ D) × ½ρv². Because the loss grows so fast as the pipe narrows, doubling a pipe's width cuts it by about 97% at the same flow. That's why a long, thin garden hose gives a feeble trickle.

Speed of sound. Air squashes. Below about a third of the speed of sound (Mach 0.3) its density changes by under 5% and Bernoulli works well. An airliner cruises near Mach 0.8, so its Pitot tube must use the full compressible formula, or it would misread its speed.

Myth-buster: "The air over the top of a wing must go faster so it can meet the air from the bottom at the back." No! Nothing makes the two meet. Release puffs of smoke together and the ones over the top arrive at the back well before their partners, and never meet them again. The "equal transit time" story gets the direction right and the reason wrong.

And the shower curtain that creeps in to cling to your legs? Hot air, falling spray and a spinning vortex of air all play a part. In 2001 David Schmidt won an Ig Nobel prize for simulating it on a computer and finding the vortex.