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According to the Bernoulli's equation, if velocity decreases, then pressure increases.

I am trying to understand the Bernoulli's effect based on a situation.

Suppose we have a stream of water. Let's assume it is an ideal fluid. Imagine the water flows out from a wider pipe to a narrower pipe. Since the area decreases, according to the Continuity equation, velocity of water molecules increase. This causes an decrease in pressure.

I don't understand the last part. If water molecules' velocity increase, then their kinetic energy also increases. Wouldn't this causes more collision between pipe's wall and water molecules, thus giving higher pressure?

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Pressure is momentum transfer due to molecular collisions once you have subtracted out their average motion. So decrease in pressure due to increase in average speed may be construed as transfer of kinetic energy from random molecular motion to mean motion. This means that random molecular motion (by which I only mean molecular motion with average subtracted out) now contains less energy, less momentum, and thus results in lower pressure reading.

Recall how pressure is measured in a pipe, for example. Pressure gauge is fitted on the wall such that flow does not directly impinge on it; otherwise you would be measuring total energy which manifests itself as a pressure head (called stagnation pressure), and in an ideal fluid (in which there is no viscous dissipation) this latter pressure would be constant everywhere.

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  • $\begingroup$ If energy is transfered to average motion then the temperature of the fluid (related to kinetic energy of random motion) must drop. But this doesn't happen. So how increase in velocity increases pressure? $\endgroup$ – Antonios Sarikas Oct 21 '20 at 13:59
  • $\begingroup$ @AntoniosSarikas The temperature must change in accordance with the fluid's thermodynamic equation of state. However for usual changes in pressure in a flow the corresponding change in temperature may be negligible. $\endgroup$ – Deep Oct 22 '20 at 12:58
  • $\begingroup$ But if the temperature is the same then the walls "feel" the same force as the same amount of momentum (on average) is transferred. $\endgroup$ – Antonios Sarikas Oct 22 '20 at 13:30
  • $\begingroup$ @AntoniosSarikas Temperature will not be the same. It will change but the change will be negligible. $\endgroup$ – Deep Oct 22 '20 at 13:37
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    $\begingroup$ You keep repeating that "temperature of the fluid will remain constant" while I keep repeating that " temperature of the fluid will NOT remain constant" but that the change will usually be negligible. $\endgroup$ – Deep Dec 5 '20 at 14:36

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