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How to interpret advection (v.del) v->v.(del v)

  1. Jul 7, 2010 #1
    I'm looking @ convective accerlation term in http://en.wikipedia.org/wiki/Navier_stokes_equation#Convective_acceleration. I don't understand the terminology. If v is a vector, it says that [tex](\mathbf{v}\cdot\nabla)\mathbf{v}[/tex] can be written as [tex]\mathbf{v}\cdot\nabla \mathbf{v}[/tex]. I thought that [tex]\nabla \mathbf{v}[/tex] is the transpose of the Jacobian matrix for [tex]\mathbf{v}[/tex]. As I'm not familiar with the terminology it almost looks like ([tex]\mathbf{v} \cdot \nabla \mathbf{v}[/tex] = vector . matrix), which can't be right. However, it appears [tex](v\cdot\nabla)v[/tex] is a vector. Can somebody shed some light on how [tex]v\cdot\nabla v[/tex] is a vector? If [tex](\mathbf{v}\cdot\nabla)\mathbf{v}=\mathbf{v}\cdot\nabla \mathbf{v}[/tex] & [tex]\nabla \mathbf{v} = (\mathbf{J}\mathbf{v})^T[/tex]. I'm sure I'm mutilating the terminology, if anybody could shed light on this, much appreciated.
     
    Last edited: Jul 7, 2010
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  3. Jul 7, 2010 #2

    arildno

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    In index notation, we have:
    [tex]\vec{v}\cdot\nabla=\sum_{j=1}^{3}v_{j}\frac{\partial}{\partial{x}_{j}}[/tex]
    where the indices ought to be fairly self-evident.
    Usually, we just omit the summation symbol, writing [tex]v_{j}\frac{\partial}{\partial{x}_{j}}[/tex] instead.

    Setting this alongside a vector [itex]v_{i}[/itex] then, we have:
    [tex](\vec{v}\cdot\nabla)\vec{v}=(v_{j}\frac{\partial}{\partial{x}_{j}})v_{i}=v_{j}\frac{\partial{v}_{i}}{\partial{x}_{j}}[/tex]

    The quantity [tex]\frac{\partial{v}_{i}}{\partial{x}_{j}}[/tex] is a matrix
     
  4. Jul 7, 2010 #3
    Thanks for your reply, I've seen that as well the (v.del) makes alot more sense. I know (v.del)v should equal v.del(v), but I'm curious how can I come to same result starting with v.del(v) which to me looks like a vector.matrix.
     
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