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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?

Idonknow
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1 Answers1

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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.

Deep
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  • 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? – Antonios Sarikas Oct 21 '20 at 13:59
  • @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. – Deep Oct 22 '20 at 12:58
  • But if the temperature is the same then the walls "feel" the same force as the same amount of momentum (on average) is transferred. – Antonios Sarikas Oct 22 '20 at 13:30
  • @AntoniosSarikas Temperature will not be the same. It will change but the change will be negligible. – Deep Oct 22 '20 at 13:37
  • But aren't the surroundings hot enought to heat the system so its temperature won't change (which is a real scenario)? – Antonios Sarikas Oct 22 '20 at 13:44
  • @AntoniosSarikas I am not sure I understand you. May be you should post it as a detailed question. – Deep Oct 22 '20 at 14:19
  • The internal kinetic energy must be proportional to temperature, random motion as you stated (a moving box doesn't increase its temperature). If this energy is that is converted into average motion (moving box) then I understand that the pressure must drop accordingly. But I can't understand how will be the case for a fluid. A drop in random kinetic energy will result in a drop in temperature. But if the temperature remains constant that means that pressure remains constant, because random kinetic energy remains constant. So for a fluid temperature must drop. – Antonios Sarikas Dec 05 '20 at 13:34
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    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. – Deep Dec 05 '20 at 14:36
  • I understand what you said about the temperature drop. Lets assume that first fluid (system) is in thermal equilibrium with surroundings at temperature $T_i$. Now if its temperature drops (due to average motion) some heat will flow from surroundings to the system but the final temperature $T_f$ will be less that $T_i$ so the collisions will transfer less momentum. I apologise for keep repeating it. I thought (wrong) that the new temperature must be the same with the initial. – Antonios Sarikas Dec 18 '20 at 18:20
  • @AntoniosSarikas I recommend that you post your doubt as a question. – Deep Dec 19 '20 at 06:28
  • So essentially if you observed a moving fluid element or parcel of water in its center of mass frame you'd be observing the molecular motions that contribute to the fluid element's pressure on its surroundings? ( which include the rest of the fluid and the walls) that is a rather intuitive explanation. – MattGeo May 18 '22 at 11:09