Streamline flow and Bernoulli's principle

When a fluid flows gently, each particle passing a point follows the same path as the ones before it. These paths are streamlines, and the flow is streamline (steady) flow. Two streamlines can never cross, because a particle at the crossing point would have two directions at once. If the flow is too fast or rough, the paths become jumbled and the flow is turbulent.

We study an ideal fluid: one that is incompressible, has no viscosity, flows steadily and does not swirl. Two rules then hold along a tube of flow:

• Equation of continuity (mass is conserved): A₁v₁ = A₂v₂. Where the pipe is narrow, the fluid moves faster; that is why putting your thumb over a garden hose makes the water shoot out. If a pipe narrows from 4 cm² to 1 cm², the water speeds up four times.
• Bernoulli's principle (energy is conserved): along a streamline, P + ½ρv² + ρgh = constant.

The three terms are energies per unit volume: pressure energy, kinetic energy and potential energy. Bernoulli's principle says what one gains, the others lose. Its key message: where the fluid moves faster, its pressure is lower (at the same height).

Torricelli's law: water pours out of a hole a depth h below the open surface at v = √(2gh), the same speed a stone would reach falling freely through h. From a hole 5 m below the surface, v = √(2 × 9.8 × 5) ≈ 9.9 m/s.

Bernoulli's principle at work:

• A Venturi meter measures how fast a liquid flows: the pressure drop at its narrow neck tells the speed.
• A sprayer or perfume atomiser blows fast air across the top of a tube; the low pressure there draws the liquid up into a fine spray.
• Aeroplane wings (aerofoils) are shaped so that air goes faster over the top: lower pressure above, higher below, giving lift.
• A spinning ball drags air round with it, so the air moves faster on one side. The pressure difference makes the ball swing in the air: the Magnus effect.

It even explains one kind of heart attack. When fatty deposits narrow an artery, the blood must flow faster through the gap, its pressure drops, and the outside pressure can squeeze the artery shut for a moment.