# 12.2 Bernoulli’s equation

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## Calculating pressure: pressure drops as a fluid speeds up

In [link] , we found that the speed of water in a hose increased from 1.96 m/s to 25.5 m/s going from the hose to the nozzle. Calculate the pressure in the hose, given that the absolute pressure in the nozzle is $1\text{.}\text{01}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}$ (atmospheric, as it must be) and assuming level, frictionless flow.

Strategy

Level flow means constant depth, so Bernoulli’s principle applies. We use the subscript 1 for values in the hose and 2 for those in the nozzle. We are thus asked to find ${P}_{1}$ .

Solution

Solving Bernoulli’s principle for ${P}_{1}$ yields

${P}_{1}={P}_{2}+\frac{1}{2}{\mathrm{\rho v}}_{2}^{2}-\frac{1}{2}{\mathrm{\rho v}}_{1}^{2}={P}_{2}+\frac{1}{2}\rho \left({v}_{2}^{2}-{v}_{1}^{2}\right)\text{.}$

Substituting known values,

$\begin{array}{lll}{P}_{1}& =& 1\text{.}\text{01}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\\ & & \text{}+\frac{1}{2}\left({\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}\right)\left[\left(\text{25.5 m/s}{\right)}^{2}-\left(\text{1.96 m/s}{\right)}^{2}\right]\\ & =& 4.24×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\text{.}\end{array}$

Discussion

This absolute pressure in the hose is greater than in the nozzle, as expected since $v$ is greater in the nozzle. The pressure ${P}_{2}$ in the nozzle must be atmospheric since it emerges into the atmosphere without other changes in conditions.

## 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 [link] .

## 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.

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