Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Divergence Theorem and Incompressible Fluids

  1. Apr 27, 2008 #1
    1. The problem statement, all variables and given/known data

    Hi, I'm trying to follow the proof for the statement
    \nabla . u = 0

    I'm basing it off this paper:

    (page 7, 8)

    In case thats not accessable (I'm in university just now, and I'm not sure if thats a subscriber only paper) I'll write what I've got.

    so they start off with defining a fluid volume [tex]\Omega[/tex], and it's boundary surface as [tex]\partial \Omega[/tex], then defining the rate of change around this volume as

    [tex] \frac{d}{dt} Volume(\Omega ) = \int \int_{\partial \Omega} u.n [/tex]

    The volume should stay constant, thus

    [tex] \int \int_{\partial \Omega} u.n = 0 [/tex]

    from this step they mention the divergence theorem, then jump to

    [tex] \int \int \int_{\Omega} \nabla .u = 0 [/tex]

    It's this last jump I don't follow. From http://mathworld.wolfram.com/DivergenceTheorem.html" [Broken], I figured the divergence theorem changed to fit this problem would be..

    [tex] \int_{\Omega } (\nabla .u) d\Omega = \int_{\partial \Omega} u.da [/tex]

    they've dropped [tex] d\Omega [/tex] and gained two integrals, and I don't follow how they did this.
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Apr 27, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I'll admit that their notation isn't really intuitive. Basically, they chose to omit the variables of integration since integration of the surface and volume is implied by the double and triple integrals. However, you should note that what you have,
    Is identical to what they have,
    As I said previously, they have simply chosen not to write [itex]d\Omega[/itex] and [itex]dA[/itex], which is acceptable but can be confusing.
    Last edited: Apr 27, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook