12.2 Bernoulli’s equation  (Page 4/7)

Wings and sails

The airplane wing is a beautiful example of Bernoulli's principle in action. [link] (a) 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 [link] (b).) 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.

Velocity measurement

[link] shows two devices that measure fluid velocity based on Bernoulli's principle. The manometer in [link] (a) 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 $P_1+\frac{1}{2}{\mathrm{\rho v}}_1^2=P_2+\frac{1}{2}{\mathrm{\rho v}}_2^2$ becomes

Thus pressure $P_2$ over the second opening is reduced by $\frac{1}{2}{\mathrm{\rho v}}_2^2$ , and so the fluid in the manometer rises by $h$ on the side connected to the second opening, where

(Recall that the symbol $\text{∝}$ means “proportional to.”) Solving for $v_2$ , we see that

[link] (b) 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.

Summary

• Bernoulli's equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:
  $P_1+\frac{1}{2}{\mathrm{\rho v}}_1^2+\rho {\mathrm{gh}}_1=P_2+\frac{1}{2}{\mathrm{\rho v}}_2^2+\rho {\text{gh}}_2.$
• Bernoulli's principle is Bernoulli's equation applied to situations in which depth is constant. The terms involving depth (or height h ) subtract out, yielding
  $P_1+\frac{1}{2}{\mathrm{\rho v}}_1^2=P_2+\frac{1}{2}{\mathrm{\rho v}}_2^2.$
• Bernoulli's principle has many applications, including entrainment, wings and sails, and velocity measurement.