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Another important situation is one in which the fluid moves but its depth is constant—that is, . Under that condition, Bernoulli's equation becomes
Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli's principle . It is 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.) 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 is greater than in the equation, then must be less than for the equality to hold.
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 (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 .
Solution
Solving Bernoulli's principle for yields
Substituting known values,
Discussion
This absolute pressure in the hose is greater than in the nozzle, as expected since is greater in the nozzle. The pressure in the nozzle must be atmospheric since it emerges into the atmosphere without other changes in conditions.
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.
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] .
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