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Homework Help: Why does div(v)=0 for a fluid means that the fluid is incompressible?

  1. Feb 7, 2006 #1
    why does div(v)=0 for a fluid means that the fluid is incompressible?
     
  2. jcsd
  3. Feb 7, 2006 #2

    Galileo

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    The fluid doesn't diverge or compress, so that the density in a certain fluid element always stays the same.

    Use the divergence theorem. Take an arbitrary volume region V of the fluid with surface S, then according to the divergence theorem:

    [tex]\int_{S}\vec v\cdot d\vec a=\int_{V}\vec \nabla \cdot \vec v dV[/tex]

    Do you know how to interpret this for your fluid?
     
  4. Feb 8, 2006 #3
    do you mean that the surface integral is the same as the volume integral?
    @@
     
  5. Feb 8, 2006 #4

    Galileo

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    No, the left integral is the fluid FLUX through the surface. The right one is a volume integral of the DIVERGENCE of the fluid.

    Have you heard of the divergence theorem before?
     
  6. Feb 8, 2006 #5
    i looked the theroem up!
    thank you!!!
    so in other words, (the amount of fluid leaving)- (the amount of fluid entering)=0=incompressible...
     
  7. Feb 8, 2006 #6

    Galileo

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    Right, that's basically it. Since div(v)=0 the flux through any closed surface is zero. The flux is a measure of how much fluid flows through the surface. Since the surface is closed there is no net accumulation of fluid in the volume.
     
  8. Feb 9, 2006 #7
    ^_^
    thank you very much!!!
     
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