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Need help with fourier transformation to derive oseen tensor.

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Derive the Oseen tensor from the Navier-Stokes equation, Consider an incompressible fluid where the velocity field [tex]\vec{v}[/tex]([tex]\vec{r}[/tex],t) is given by
    [tex]\nabla[/tex].[tex]\vec{v}[/tex]=0 (1)
    Assume inertia forces are negligible.

    2. Relevant equations
    The Navier-Stokes eq becomes:
    -[tex]\nabla[/tex]P+[tex]\eta[/tex][tex]\Delta[/tex][tex]\vec{v}[/tex]+[tex]\vec{f}[/tex]=0 (2)
    With pressure P, [tex]\eta[/tex] the fluid viscosity and f the force acting on a unit volume.

    Now we define the Fourier transform as [tex]\vec{v}[/tex]k=[tex]\int[/tex][tex]\vec{v}[/tex]([tex]\vec{r}[/tex]) exp[i [tex]\vec{k}[/tex].[tex]\vec{r}[/tex]] d[tex]\vec{r}[/tex] and so on.
    Show that eq (1) and (2) can be rewritten as:
    [tex]\vec{k}[/tex].[tex]\vec{v}[/tex]k=0 and -[tex]\eta[/tex] [tex]\vec{k}[/tex][tex]^{2}[/tex] [tex]\vec{v}[/tex]k - i[tex]\vec{k}[/tex] Pk = -[tex]\vec{f}[/tex]k (3)

    3. The attempt at a solution
    In every book it is just said that fourier transforming eq (1) and (2) just leads to (3) but it is not explained. I've tried it but maybe someone can help me. Thanks in advance.
     
  2. jcsd
  3. Aug 26, 2011 #2

    kai_sikorski

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    Gold Member

    There is a nice derivation in the attached file.
     

    Attached Files:

  4. Aug 26, 2011 #3
    Haha, thank you!
    but I posted this question in 2009 and as I remember correctly was able to figure it out in the end and hand in my exercises.
    To save you some time, next time check the data of the post!
    Cheers
     
  5. Aug 26, 2011 #4

    kai_sikorski

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    I saw the date actually; just figured it might be useful to someone else at some point.
     
  6. Aug 26, 2011 #5
    You are right, didn't think about it. I never posted my answer, doubt I still have it.
    Good job!
     
  7. Feb 26, 2013 #6
    thanks it was useful to me!
     
  8. Feb 26, 2013 #7

    kai_sikorski

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    Gold Member

    Good to hear!
     
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