4.5.3: Bernoulli’s Principle
- Page ID
- 472547
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- Calculate with Bernoulli’s principle.
- List some applications of Bernoulli’s principle.
Although Pascal's principle is very useful in many applications, there are many more situations where it doesn't apply because the fluids are moving. There are a number of common examples of pressure dropping in rapidly-moving fluids. Shower curtains have a disagreeable habit of bulging into the shower stall when the shower is on. The high-velocity stream of water and air creates a region of lower pressure inside the shower, and standard atmospheric pressure on the other side. The pressure difference results in a net force inward pushing the curtain in. You may also have noticed that when passing a truck on the highway, your car tends to veer toward it. The reason is the same—the high velocity of the air between the car and the truck creates a region of lower pressure, and the vehicles are pushed together by greater pressure on the outside. (See Figure \(\PageIndex{1}\).) This effect was observed as far back as the mid-1800s, when it was found that trains passing in opposite directions tipped precariously toward one another.
![drawing of a truck and car driving adjacent to each other with vectors indicating the air flow past each of them and the pressure felt on either side of each vehicle.](https://chem.libretexts.org/@api/deki/files/469209/Figure_13_02_01a.jpg?revision=1&size=bestfit&width=225&height=208)
Hold the short edge of a sheet of paper parallel to your mouth with one hand on each side of your mouth. The page should slant downward over your hands. Blow over the top of the page. Describe what happens and explain the reason for this behavior.
The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation applies for any incompressible, frictionless fluid. The entire equation is beyond the scope of this course and requires an understanding of concepts introduced in chapter 5. However, for most of the applications we will want to consider, we can use a simplified version of this equation that assumes no change in elevation for the moving fluid. This is called Bernoulli's principle.
\[P_{1}+\frac{1}{2} \rho v_{1}^{2}=P_{2}+\frac{1}{2} \rho v_{2}^{2}. \nonumber \]
As we have just discussed, pressure drops as speed increases in a moving fluid. We can see this from Bernoulli’s principle. For example, 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.
Example \(\PageIndex{1}\): Calculating Pressure: Pressure Drops as a Fluid Speeds Up
In Example 8.6.2, 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 \(1.01 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\) (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
\[P_{1}=P_{2}+\frac{1}{2} \rho v_{2}^{2}-\frac{1}{2} \rho v_{1}^{2}=P_{2}+\frac{1}{2} \rho\left(v_{2}^{2}-v_{1}^{2}\right). \nonumber\]
Substituting known values,
\[\begin{aligned}
P_{1}=& 1.01 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2} \\
&+\frac{1}{2}\left(10^{3} \mathrm{~kg} / \mathrm{m}^{3}\right)\left[(25.5 \mathrm{~m} / \mathrm{s})^{2}-(1.96 \mathrm{~m} / \mathrm{s})^{2}\right] \\
=& 4.24 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}.
\end{aligned} \nonumber\]
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 Figure \(\PageIndex{2}\).
![Four entrainment devices, as described in the caption.](https://chem.libretexts.org/@api/deki/files/469205/Figure_13_02_02a.jpg?revision=1&size=bestfit&width=475&height=157)
Wings and Sails
The airplane wing is a beautiful example of Bernoulli’s principle in action. Figure
![A wing and a sail illustrating Bernoulli's principle. The air travels much slower across the straight side compared to the curved side, leading to a pressure difference.](https://chem.libretexts.org/@api/deki/files/469206/Figure_13_02_03.jpg?revision=1&size=bestfit&width=475&height=262)
MAKING CONNECTIONS: TAKE-HOME INVESTIGATION WITH TWO STRIPS OF PAPER
For a good illustration of Bernoulli’s principle, make two strips of paper, each about 15 cm long and 4 cm wide. Hold the small end of one strip up to your lips and let it drape over your finger. Blow across the paper. What happens? Now hold two strips of paper up to your lips, separated by your fingers. Blow between the strips. What happens?
Velocity measurement
Figure
\[P_{1}=P_{2}+\frac{1}{2} \rho v_{2}^{2}. \nonumber \]
Thus, pressure \(P_{2}\) over the second opening is reduced by \(\frac{1}{2} \rho v_{2}^{2}\), and so the fluid in the manometer rises by \(h\) on the side connected to the second opening, where
\[h \propto \frac{1}{2} \rho v_{2}^{2}. \nonumber \]
(Recall that the symbol \(\propto\) means “proportional to.”) Solving for \(v_{2}\), we see that
\[v_{2} \propto \sqrt{h}. \nonumber \]
Figure \(\PageIndex{4}\)(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.
![two drawings of a manometer, as described in the caption.](https://chem.libretexts.org/@api/deki/files/469204/Figure_13_02_04a.jpg?revision=1&size=bestfit&width=475&height=155)
Section Summary
- 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
\[P_{1}+\frac{1}{2} \rho v_{1}^{2}=P_{2}+\frac{1}{2} \rho v_{2}^{2}. \nonumber\]
- Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.
Glossary
- Bernoulli’s principle
- Bernoulli’s equation applied at constant depth: P1 + 1/2pv12 = P2 + 1/2pv22