# 14.6 Bernoulli’s equation  (Page 3/8)

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## Bernoulli’s principle

Suppose a fluid is moving but its depth is constant—that is, ${h}_{1}={h}_{2}$ . Under this condition, Bernoulli’s equation becomes

${p}_{1}+\frac{1}{2}\rho {v}_{1}^{2}={p}_{2}+\frac{1}{2}\rho {v}_{2}^{2}.$

Situations in which fluid flows at a constant depth are so common that this equation is often also called Bernoulli’s principle    , which is simply Bernoulli’s equation for fluids at constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path.) Bernoulli’s principle reinforces the fact that pressure drops as speed increases in a moving fluid: If ${v}_{2}$ is greater than ${v}_{1}$ in the equation, then ${p}_{2}$ must be less than ${p}_{1}$ for the equality to hold.

## Calculating pressure

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 $1.01\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ (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 ${p1}_{}$ .

## Solution

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

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

Substituting known values,

$\begin{array}{cc}\hfill {p}_{1}& =1.01\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}\text{+}\frac{1}{2}\left({10}^{3}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}\right)\left[{\text{(25.5 m/s)}}^{2}-{\text{(1.96 m/s)}}^{2}\right]\hfill \\ & =4.24\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}.\hfill \end{array}$

## Significance

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, because the water emerges into the atmosphere without other changes in conditions.

## Applications of bernoulli’s principle

Many devices and situations occur 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 as pumps to raise water to small heights, as is necessary for draining swamps, fields, or other low-lying areas. Some other devices that use the concept of entrainment are shown in [link] .

## Velocity measurement

[link] shows two devices that apply Bernoulli’s principle to measure fluid velocity. The manometer in part (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

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