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Fluid dynamics

  1. Mar 4, 2008 #1
    1. The problem statement, all variables and given/known data
    Starting from the equations of conservation of mass and momentum for an inviscid compressible gas,
    [tex]\frac{\partial \rho}{\partial t} + \bigtriangledown . (\rho {\bf u}) = 0[/tex]

    [tex]\rho ( \frac{\partial {\bf u}}{\partial t} + ({\bf u}. \bigtriangledown){\bf u}) = \bigtriangledown p + {\bf F}[/tex]

    derive for a fixed volume V enclosed by a surface S:

    [tex]\frac{d}{dt} \int_V \frac{1}{2} \rho u^2 dV + \int_S \frac{1}{2} \rho u^2 {\bf u.n} dA = - \int_S p {\bf u.n} dA + \int_V (p\bigtriangledown. {\bf u} + {\bf F.u}) dV[/tex].

    2. Relevant equations
    Divergence theorem:

    [tex]\int_V \bigtriangledown . {\bf u} dV = \int_S {\bf u.n} dA[/tex]
    where A is the surface enclosing V


    3. The attempt at a solution
    I started from the first equation, multiplied this through by [tex]\frac{u^2}{2}[/tex], applied the divergence theorem and looked at what the full derivative given in the question is in terms of the chain rule, and ended up with

    [tex]\frac{d}{dt} \int_V \frac{1}{2} \rho u^2 dV + \int_S \frac{1}{2} \rho u^2{\bf u.n} dA = \int_V \rho {\bf u} . \frac{\partial {\bf u}}{\partial t} dV[/tex]

    Which looks like it might be right. However I am a bit wary though because I don't know if I should be considering the del operator acting on the [tex]\frac{u^2}{2}[/tex] that I multiplied through by. Anyway, assuming this is right (I have the alternative expression if it's not, and that also looks like it could be right) I tried to proceed by dotting the second fluid equation with [tex]{\bf u}[/tex], and things were looking quite promising but it didn't quite come out. I seemed to be having particular trouble with the [tex]p \bigtriangledown.{\bf u}[/tex] term.

    Am I going along the right lines here? Any hints on how I can finish this off?

    Thanks.

    Edit - Sorry about my lack of vectors, I can't seem to find a command that works on this forum. u and n are vectors through, and u^2 = u.u Also, I guess I should have mentioned (although the question doesn't actually specifiy) that rho, u and p are function sof position and time.
     
  2. jcsd
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