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General pathline of a particle x at point epsilon at time ta

  1. Sep 5, 2016 #1
    1. The problem statement, all variables and given/known data

    Show that the path line of a particle at point x currently, and point ξ at time τ is given by

    ξ(τ) = x + (τ-t)Lx

    2. Relevant equations

    Pathline is solution to
    dx/dt = u
    x
    (t)|t=τ = X

    L
    is the velocity gradient and is a 2nd order tensor Lij = dui/dxj

    3. The attempt at a solution

    I am not really sure how to start, any hints or leads on how to begin proving this?
    I know how to obtain pathlines if the velocity field is given in terms of x,y,z,t (via integration) but how do I integrate the equations to obtain a general equation with a velocity gradient?

    Thanks
     
  2. jcsd
  3. Sep 10, 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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