Fluid Mechanics

Narrow the Section and the Flow Speeds Up

Continuity A·v = const (mass conservation); narrower means faster inversely, Q = A·v

Send water through a pipe that is fat in some places and thin in others. Squeeze the throat yourself and find out why the current quickens where it narrows.

Drag the probe along the pipe and read the area A and the speed v. Narrower is faster, and the product A·v is the same everywhere.

Now pinch the throat yourself. Halve the area and the speed doubles: area and speed are locked in inverse proportion.

A and v move oppositely, yet their product is the same at both sections. This product A·v is the volume passing per unit time, the flow rate.

As an equation, A1 v1 = A2 v2, so Q = A·v stays constant. Drag v1 and v2 = v1·A1A2 follows. (Incompressible; for a gas, ρ·A·v.)

Pinch the hose end and the same water crowds through a thin gap, shooting out far. Nozzles, spray bottles, and syringes all work this way.

In PracticeTo sum up: in steady flow, as much must leave as enters, so the product of cross-sectional area and speed is constant everywhere along the pipe. A1 v1 = A2 v2, that is, the volume flow rate Q = A·v is conserved. So when the section narrows, the speed grows in inverse proportion: half the area means twice the speed. This holds for an incompressible fluid; for a compressible gas it is ρ·A·v that stays constant. The intuition that narrowing and acceleration are one and the same runs through nozzles, hoses, blood vessels, and a river's rapids. In the next lesson, Bernoulli, we view the same flow as energy and reveal why the pressure drops where the flow speeds up.

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