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example 1

In the above example image Example 1, I understand the reason why the pressure will decrease when the velocity increases. The reason is given as follows by Richard Feynman:

The speed $v_2$ must certainly exceed $v_1$ to get the same amount of water through the narrower tube. So the water accelerates in going from the wide to the narrow part. The force that gives this acceleration comes from the drop in pressure.

But, I fail to understand how on increasing the velocity the pressure decreases, in every case.

For example, Richard Feynmann also gives the following Example 2:

Example 2:

Have you ever held two pieces of paper close together and tried to blow them apart? Try it! They come together. The reason, of course, is that the air has a higher speed going through the constricted space between the sheets than it does when it gets outside. The pressure between the sheets is lower than atmospheric pressure, so they come together rather than separating.

How do we understand this phenomenon in Example 2 just with the help of Newton's Laws, forces, energy conservation, etc. as explained in Example 1?

Can we explain every case with such a detail of forces as explained in the first Example 1?

In summary, I fail to understand this phenomenon intuitively.

Could someone explain it to me by taking 5 different examples in different contexts, so that we understand it intuitively, why it so happens?

Currently, in most cases it seems kind of magical that when velocity in a fluid increases, the pressure decreases.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Chris
    Dec 15, 2019 at 2:13
  • $\begingroup$ Bernoullis equation is basically an energy balance. When the velocity increases the kinetic energy increases so if the potential $$ \rho g h $$ stays the same then the pressure must drop $\endgroup$
    – ChemEng
    Dec 16, 2019 at 5:04

1 Answer 1

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The example where the sheets consistently approach each other shows that the air flow we usually produce by blowing from our mouths has subatmospheric pressure for a fair distance along its path.

But this cannot be easily explained with only Bernoulli's theorem, because the air from the blowing mouth has higher value of sum of the Bernoulli equation terms (in this case, kinetic energy per unit volume + pressure) than the air surrounding the sheets. We are putting new air of different flow in between the sheets, so we cannot equate the two Bernoullis equations and conclude that because speed is higher between the sheets, the pressure there has to be lower than the pressure outside. It may be lower, if the speed is high enough. Which it often is, but only up to some distance, as eventually the flow dissipates and its air pressure increases.

This means that the behaviour of the sheets should depend on how exactly the air is blown between them and where the sheets are held in place.

By the way, Bernoulli's equation, while not being enough to determine the motion of the sheets in this experiment even if it was entirely correct, actually isn't entirely correct for airflows, because air is quite compressible and has quite high effective viscosity. But the equation can often still be used to get rough estimates.

As the air elements move along their path, they slow down, take more curly paths and increase their pressure. If the conditions are right, the pressure in the later stages of the motion can be above the atmospheric pressure. If we put the sheets farther from the mouth in that place, the increased pressure should move them apart. I don't know if it is possible to demonstrate this with human blowing the air, but with strong a source of compressed of air, it seems possible.

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