Applications of Bernoulli's Theorem

Bernoulli's theorem, together with the equation of continuity (which follows from conservation of mass for an incompressible fluid, and states that $A_1 v_1 = A_2 v_2$ for any two cross-sections of a pipe of varying area, so that the fluid speeds up wherever the pipe narrows), together explain a wide range of everyday and instrumental phenomena.

The venturimeter. A venturimeter is an instrument used to measure the flow speed of a fluid moving through a pipe. It consists of a section of pipe that narrows smoothly to a constriction (the "throat") and then widens back out again, with a manometer (a pressure-measuring device, or simply a set of open vertical tubes) connected at the wide section and at the throat. By the equation of continuity, the fluid's speed must be higher at the narrow throat than at the wide section (since the same volume of fluid per second has to pass through a smaller area); by Bernoulli's theorem, applied at the same height so the $\rho g h$ terms cancel, this higher speed at the throat must correspond to a lower pressure there. The measured pressure difference between the wide section and the throat can therefore be used, together with the known areas of the two sections, to calculate the fluid's flow speed and hence the rate of flow through the pipe.

Dynamic lift on an aerofoil. An aircraft wing is shaped as an aerofoil: curved and comparatively longer on its upper surface, and flatter and comparatively shorter on its lower surface. As air flows past the wing, air travelling over the more curved upper surface is forced to move faster than air travelling under the flatter lower surface (to cover the greater distance along the curved upper surface in the same time as the air below it covers the shorter distance along the flatter lower surface). By Bernoulli's theorem, this faster-moving air above the wing is at a lower pressure than the slower-moving air below it. This pressure difference -- higher pressure pushing up from below, lower pressure pulling less strongly down from above -- produces a net upward force on the wing called dynamic lift, which is what keeps the aircraft airborne.

The atomizer (perfume spray). In an atomizer, a rubber bulb is squeezed to blow a fast stream of air horizontally across the open top of a narrow vertical tube, the lower end of which dips into a reservoir of liquid (perfume, or a similar fluid). The fast-moving air stream across the top of the tube is, by Bernoulli's theorem, at a lower pressure than the still air (and the liquid) below in the reservoir. This pressure difference pushes the liquid up the narrow tube, where it meets the fast air stream at the top and is broken up into a fine spray, or mist, of tiny droplets carried away by the air. …