12.2 Bernoulli’s equation

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.

Conceptual questions