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B Pressure gradient causes conservative force?

  1. Oct 31, 2018 #1
    Is the force that accelerates afluid betwen two points of different pressure conservative?
     
  2. jcsd
  3. Oct 31, 2018 #2

    Charles Link

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    A very interesting question. Since the force per unit volume is ## \vec{f}_v=-\nabla P ## , and ## \nabla \times \nabla P=0 ##, (a vector identity that is commonly known from E&M theory), you might think that could be possible. In a real fluid however, there will also be viscous damping forces on the volume that are part of the total force equation that are often left out of the equation ## \vec{f}_v=-\nabla P ##. ## \\ ## So that the answer is the forces in a liquid are non-conservative, even though the part that comes from the minus of the pressure gradient could be considered to be conservative. ## \\ ## Note: This is my own analysis=others may agree or disagree=this is the first time I have encountered this question. ## \\ ## Additional note: You could think of these forces as acting on a small object of finite volume that is placed in the fluid. (And of course gravity will also act on the object, which is a conservative force).
     
    Last edited: Oct 31, 2018
  4. Oct 31, 2018 #3
    Thank you so much for your answer
    It is clear
     
  5. Oct 31, 2018 #4

    Charles Link

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    I added one more part above, about the force of gravity on an object in the fluid=be sure and see that. ## \\ ## Additional item: There is a vector identity, ## \int \nabla P \, d^3 x=\int P \, \hat{n} dA ##, (where ## \hat{n} ## is the outward pointing normal), and to have equilibrium, ## -\delta g \, \hat{z}+-\nabla P= 0 ##, (where ## \delta ## is the density (mass per unit volume)), so that you can actually derive Archimedes' principle from the pressure gradient force.
     
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