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Material Derivative

  1. May 29, 2010 #1
    A fluid moves so that its velocity is [tex] \vec {u} \equiv (2xt,-yt,-zt) [/tex], written in rectangular Cartesian coordinates. Show that the surface F(x,y,z,t) = [tex] x^2exp (-2t^2)+(y^2+2z^2)exp(t^2)=constant [/tex] moves with the fluid (so that it always contains the same fluid particles; that is, DF/Dt=0)



    2. Relevant equations
    I got this from http://en.wikibooks.org/wiki/Marine..._Wave_Theory#Kinematic_Free_Surface_Condition

    [tex] \frac{DF(x,y,z,t)}{Dt}=w. [/tex] If the surface is defined by z=A(x,y,t), then [tex] w = \frac {\partial A}{\partial t} + u \frac {\partial A}{\partial x}+v \frac{\partial A}{\partial y} [/tex] but I'm confused. If A refers to the surface, my surface has z's in it. Also what does the F stand for in the wiki equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 29, 2010 #2
    I believe [itex]F[/itex] is also the surface. If you look at one of the prior sections, it says,


    I would also assume that if your surface has a component of [itex]z[/itex] in it, you cannot use that form of the derivative and must use the total derivative.
     
  4. May 29, 2010 #3
    Thanks very much
     
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