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Verifying Fluid Dynamics Equation

  1. Sep 16, 2008 #1
    1. Verify that vjjvi=∂i(1/2*v2)-εijkvjωk,where ωkklmlvm


    2. Relevant equations



    3. The attempt at a solution
    I am confused on how to proceed with this problem up to this point I have decomposed ωk into Curl(v). Which leaves vj X Curl(v). Decomposing this leaves vmivm-vllvi. I am stuck on where to go from here. How do I get rid of the ]=∂i(1/2*v2) term? Any help is appreciated. Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 17, 2008 #2

    tiny-tim

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    Welcome to PF!



    Hi ICSunSpots! Welcome to PF! :smile:

    Hint: ∂i(1/2*v2) = vjivj, isn't it? :wink:
     
  4. Sep 17, 2008 #3
    i(1/2*v2) = vjivj


    Thank you for your reply. That would certainly make my solution thus far work.
    However,
    I am fairly new to index notation. Can you explain to me in vector notation what vjivj means? I understand the ]∂ivj term, but what does the vector vj in front of it do? Thanks.
     
  5. Sep 17, 2008 #4

    tiny-tim

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    I'm using the summation convention … repeated indices are added over all possible basis values.

    So v2 = vjvj,

    and so ∂iv2 = ∂i(vjvj) = (∂ivj)vj + vj(∂ivj) = 2vjivj :smile:
     
  6. Sep 17, 2008 #5
    Great! It's clear as day now. This index notation is really neat stuff, it just takes a little bit for it to be intuitive. Thank you for your help, I can now finish the problem.
     
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