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Tensor gradient and scalar product

  1. Jun 5, 2010 #1
    Hi all,

    I need to evaluate the following equation :

    [tex] \mathbf{n} \cdot [\mathbf{\sigma} + \mathbf{a} \nabla\mathbf{\sigma}]\cdot\mathbf{n} [/tex]

    where [tex] \mathbf{n}[/tex] is the normal vector, [tex] \mathbf{a}[/tex] a vector, and [tex] \sigma [/tex] the stress tensor such that :

    [tex] \mathbf{\sigma} \cdot \mathbf{n} = -p\cdot\mathbf{n} + \mu [\nabla \mathbf{u} + (\nabla\mathbf{u})^T]\cdot \mathbf{n} [/tex]

    Actually, the first term (in the first equation) is not an issue , since it can be found in any serious book :smile: But I'm getting lost with the second one.

    I work much more in numerics than in maths, and my knowledge on the topic is very limited :redface: so I will be very grateful for any help, i.e.

    [tex] \mathbf{n} \cdot \mathbf{a} \nabla\mathbf{\sigma} \cdot\mathbf{n} [/tex]

    Any clue, simplification, explanation would be welcome .

    Thanks in advance
    Last edited: Jun 5, 2010
  2. jcsd
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