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Homework Help: Euler equations for ideal fluids, approximations

  1. Feb 27, 2014 #1
    1. The problem statement, all variables and given/known data


    The Euler equations for ideal compressible flow are given by
    [tex]
    \partial_t v + (v\cdot \nabla)v = g-\frac{1}{\rho}\nabla p \\
    \partial_t \rho + \nabla \cdot(\rho v) = 0
    [/tex]
    In my book these are written in terms of the small-value expansions [itex]\rho = \rho_0 + \delta \rho[/itex], [itex]p = p_0 + \delta p[/itex] and the equations become
    [tex]
    \partial_t v = -\frac{1}{\rho_0}\nabla \delta \rho \\
    \partial_t (\delta \rho) = -\rho_0 \nabla \cdot v
    [/tex]

    In the second equation, I don't understand why the RHS becomes [itex]\rho_0 \nabla \cdot v[/itex] instead of [itex](\rho_0+\delta \rho) \nabla \cdot v[/itex]?

    Thanks in advance.
     
  2. jcsd
  3. Feb 27, 2014 #2
    [itex]\delta \rho \nabla \cdot v[/itex] is negligible compared to [itex]\rho_0 \nabla \cdot v[/itex] because [itex]\delta \rho << \rho_0[/itex]
     
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