Applications of Bernoulli’s Principle

There are a number of devices and situations in which fluid flows at a constant height and, thus, can be analyzed with Bernoulli’s principle.

Entrainment

People have long put the Bernoulli principle to work by using reduced pressure in high-velocity fluids to move things about. With a higher pressure on the outside, the high-velocity fluid forces other fluids into the stream. This process is called entrainment. Entrainment devices have been in use since ancient times, particularly as pumps to raise water small heights, as in draining swamps, fields, or other low-lying areas. Some other devices that use the concept of entrainment are shown in the figure below.

Examples of entrainment devices that use increased fluid speed to create low pressures, which then entrain one fluid into another. (a) A Bunsen burner uses an adjustable gas nozzle, entraining air for proper combustion. (b) An atomizer uses a squeeze bulb to create a jet of air that entrains drops of perfume. Paint sprayers and carburetors use very similar techniques to move their respective liquids. (c) A common aspirator uses a high-speed stream of water to create a region of lower pressure. Aspirators may be used as suction pumps in dental and surgical situations or for draining a flooded basement or producing a reduced pressure in a vessel. (d) The chimney of a water heater is designed to entrain air into the pipe leading through the ceiling.

Wings and Sails

The airplane wing is a beautiful example of Bernoulli’s principle in action. Figure (a) below shows the characteristic shape of a wing. The wing is tilted upward at a small angle and the upper surface is longer, causing air to flow faster over it. The pressure on top of the wing is therefore reduced, creating a net upward force or lift. (Wings can also gain lift by pushing air downward, utilizing the conservation of momentum principle. The deflected air molecules result in an upward force on the wing — Newton’s third law.) Sails also have the characteristic shape of a wing. (See figure (b) below.) The pressure on the front side of the sail, \({P}_{\text{front}}\), is lower than the pressure on the back of the sail, \({P}_{\text{back}}\). This results in a forward force and even allows you to sail into the wind.

(a) The Bernoulli principle helps explain lift generated by a wing. (b) Sails use the same technique to generate part of their thrust.

Velocity measurement

The figure below shows two devices that measure fluid velocity based on Bernoulli’s principle. The manometer in figure (a) below is connected to two tubes that are small enough not to appreciably disturb the flow. The tube facing the oncoming fluid creates a dead spot having zero velocity (\({v}_{1}=0\)) in front of it, while fluid passing the other tube has velocity \({v}_{2}\). This means that Bernoulli’s principle as stated in \({P}_{1}+\cfrac{1}{2}{\mathrm{\rho v}}_{1}^{2}={P}_{2}+\cfrac{1}{2}{\mathrm{\rho v}}_{2}^{2}\) becomes

\({P}_{1}={P}_{2}+\cfrac{1}{2}{\mathrm{\rho v}}_{2}^{2}\text{.}\)

Thus pressure \({P}_{2}\) over the second opening is reduced by \(\cfrac{1}{2}{\mathrm{\rho v}}_{2}^{2}\), and so the fluid in the manometer rises by \(h\) on the side connected to the second opening, where

\(h\propto \cfrac{1}{2}{\mathrm{\rho v}}_{2}^{2}\text{.}\)

(Recall that the symbol \(\text{∝}\) means “proportional to.”) Solving for \({v}_{2}\), we see that

\({v}_{2}\propto \sqrt{h}\text{.}\)

Figure (b) below shows a version of this device that is in common use for measuring various fluid velocities; such devices are frequently used as air speed indicators in aircraft.

Measurement of fluid speed based on Bernoulli’s principle. (a) A manometer is connected to two tubes that are close together and small enough not to disturb the flow. Tube 1 is open at the end facing the flow. A dead spot having zero speed is created there. Tube 2 has an opening on the side, and so the fluid has a speed \(v\) across the opening; thus, pressure there drops. The difference in pressure at the manometer is \(\cfrac{1}{2}{\mathrm{\rho v}}_{2}^{2}\), and so \(h\) is proportional to \(\cfrac{1}{2}{\mathrm{\rho v}}_{2}^{2}\). (b) This type of velocity measuring device is a Prandtl tube, also known as a pitot tube.