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Double inner product?

  1. Feb 2, 2010 #1
    Hi, I'm having trouble understanding how to perform the following calculation:

    [tex]
    u=(u,v,w)
    [/tex]

    [tex]
    (\nabla u + (\nabla u)^T) : \nabla u
    [/tex]

    I get the following by doing the dot product of the first term and then adding the dot product of the second term, but I'm pretty sure it isn't correct.

    [tex]
    2\left(\frac{\partial u}{\partial x}\right)^2
    + 2\left(\frac{\partial v}{\partial y}\right)^2
    + 2\left(\frac{\partial w}{\partial z}\right)^2
    [/tex]

    Could someone please shed some light on how the double inner product should work?
    Thanks
     
  2. jcsd
  3. Feb 2, 2010 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Your notation here doesn't make sense to me. If you are using "u" to represent a vector, don't use the same "u" to represent one of its components. If you are writing [itex]\nabla u[/itex] as, say, a row vector, then [itex[(\nabl u)^T[/itex] would be a column vector and you cannot add them. In general, a vector and its transpose are in different vector spaces and cannot be added. Finally, I don't know what ":" means. Was that supposed to be [itex]\cdot[/itex]?

     
  4. Feb 5, 2010 #3
    I think it is related to the definition in section 1.3.2 found here:
    http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html

    It is for a pair of rank 2 tensors, and is denoted by a :

    The \nabla u's used in the original post are interpreted as second rank tensors, and the double inner product is applied between the terms on each side of the :

    Haven't got time to check the calculation myself.

    Torquil
     
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