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A Continuity equation in Lagrangian coordinates

  1. Jul 6, 2016 #1
    From the Eulerian form of the continuity equation, where x is the Eulerian coordinate:

    [itex] \frac {\partial \rho}{ \partial t } + u \frac {\partial \rho}{\partial x} + \rho \frac { \partial u}{\partial x} = 0[/itex]

    The incremental change in mass is, where m is the Lagrangian coordinate:

    [itex] dm = \rho dx[/itex]

    The specific volume is:

    [itex] V = \frac{1}{\rho}[/itex]

    How does one get the final form of the continuity equation in Lagrangian coordinates as follows:

    [itex]\frac{\partial V}{\partial t} = \frac{\partial u}{\partial m} [/itex]
     
  2. jcsd
  3. Jul 11, 2016 #2
    Thanks for the post! 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?
     
  4. Jul 12, 2016 #3
    $$\rho (t,\xi)\mathrm{det}\,\Big(\frac{\partial \xi}{\partial\hat \xi}\Big)=\hat\rho(\hat\xi),$$ here ##\hat \xi## are the Lagrangian coordinates in the initial moment ##t=0## and ##\xi=\xi(t,\hat\xi)## are the current Lagrangian coordinates; ##\hat\rho## is the initial desity and ##\rho (t,\xi)=\hat\rho(\xi(t, \hat\xi))## is the current density
     
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