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I was going through a proof of Pascal's law when I came across the following diagram.

FBD of a fluid element

The proof is premised on the assumption that all fluid elements of a fluid in static equilibrium will also be in static equilibrium. That is to say, if the fluid container is at rest (or moving with constant velocity), then not only will the fluid be in static equilibrium macroscopically but also every microscopic element of the fluid will also be in static equilibrium.

I do not see this as obvious. I can imagine a case where the fluid elements are accelerating in various directions but the net effect is such that the fluid appears stationary with respect to the container at the macroscopic scale. The "proof" seems more like a retrospective justification of the fact that the pressure on a fluid element is the same in all directions (in a non-accelerating fluid).

How then do we know Pascal's law to be valid? Is it purely empirical or can it be derived from Newton's laws?

Qmechanic
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Amogh
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1 Answers1

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I do not see this as obvious. I can imagine a case where the fluid elements are accelerating in various directions but the net effect is such that the fluid appears stationary with respect to the container at the macroscopic scale.

I suggest distinguishing three levels:

  • global equilibrium
  • macroscopic equilibrium
  • microscopic equilibrium

Miscorscopic here refers to molecules, which are indeed constantly moving chaotically. Macroscopic refers to fluid elements that are small (from the point of view of the derivation), but still contain huge numbers of molecules, so that we can apply continuous notions for systems of many molecules - temperature, pressure, etc. By global I mean the liquid as a whole. In other words, the "fluid elements" are macroscopic for the purposes of Pascal's law - sometimes we call them "physically small".

Now, we expect that in thermodynamic equilibrium there are no macroscopic motions in the liquid. If such motions existed, they would gradually vanish due to friction/viscocity and other effects, their energy resulting in heating the liquid. Such thermodynamic arguments are usually beyond the level of knowledge when Pascal law is introduced... but they are implicit in its derivation, as the Q. correctly points out.

Roger V.
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