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Invariance of direction of vorticity

  1. Oct 30, 2016 #1
    1. The problem statement, all variables and given/known data

    The vorticity vector ##\vec{\omega} = \text{curl}\ \vec{v}##, defined as usual by ##\omega^{2}=i_{{\vec{\omega}}}\text{vol}^{3}##, is ##\textit{not}## usually invariant since the flow need not conserve the volume form.

    The mass form, ##\rho\ \text{vol}^{3}##, however, ##\textit{is}## conserved.

    From ##{\bf{\omega}}^{2}=i_{\vec{\omega} / \rho}\rho \text{vol}^{3}##, it can be shown that the vector ##\vec{\omega}/\rho## should be invariant; that is, ##\mathcal{L}_{\vec{v}+\partial / \partial t} (\omega^{2}/ \rho)=0##.



    How can you use ##\mathcal{L}_{\bf{X}}\circ i_{\bf{Y}}-i_{\bf{Y}}\circ\mathcal{L}_{\bf{X}}=i_{[{\bf{X}},{\bf{Y}}]}## to prove that the vector ##\vec{\omega}/\rho## is invariant?

    2. Relevant equations

    3. The attempt at a solution

    ##\mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(\frac{\omega^{2}}{\rho}\right)##

    ##= \mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(\frac{i_{(\vec{\omega}/\rho)}\ \rho\ \text{vol}^{3}}{\rho}\right)##

    ##= \frac{1}{\rho}\bigg(\mathcal{L}_{{\bf{v}+\partial / \partial t}}\left(i_{(\vec{\omega}/\rho)}\ \rho\ \text{vol}^{3}\right)\bigg)##

    ##= \frac{1}{\rho}\bigg(i_{(\vec{\omega}/\rho)}\left(\mathcal{L}_{{\bf{v}+\partial / \partial t}}\ \rho\ \text{vol}^{3}\right)+i_{[{\bf{v}+\partial / \partial t},(\vec{\omega} / \rho)]}\rho\ \text{vol}^{3}\bigg)##

    ##= \frac{1}{\rho}\bigg(0+i_{[{\bf{v}+\partial / \partial t},(\vec{\omega} / \rho)]}\rho\ \text{vol}^{3}\bigg).##

    How do you proceed next?
     
  2. jcsd
  3. Nov 5, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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