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Fluid Dynamics, Linearization Question

  1. May 9, 2008 #1
    1. The problem statement, all variables and given/known data
    Consider an ideal fluid large enough to experience its own gravitational attraction. If the fluid is initially at hydrostatic equilibrium with density [tex]\rho_{0} (r)[/tex] and pressure [tex]p_{0}(r)[/tex] , it can develop small amplitude pressure waves which may be analyzed as follows.

    Show that [tex]\delta p = p - p_{0}[/tex] and [tex]\delta\rho = \rho - \rho_{0}[/tex] obey the linearized equations of motion

    [tex]\frac{\partial\rho}{\partial t} = - \nabla . ( \rho_{0} v)[/tex]

    [tex]\rho_{0} \frac{\partial v}{\partial t} = -\nabla\delta p + \delta\rho g [/tex]

    2. Relevant equations

    The relevant equations are

    [tex]\rho\frac{\partial v}{\partial t} + \rho (v . \nabla) v = - \nabla p[/tex] + g
    [tex]\frac{\partial\rho}{\partial t} + \nabla (\rho .v) = 0 [/tex]

    3. The attempt at a solution

    For the solution I began writing
    [tex] p = p_{0} + \delta p[/tex] (1)
    [tex] \rho = \rho_{0} + \delta\rho[/tex] (2)

    and assumed that [tex]\delta\rho <<< \rho_{0}[/tex] (3)
    and [tex]\delta p <<< p_{0} [/tex] (4)

    I then substituted (3) and (4) into (1) and (2) and only kept the lowest order terms (since [tex]\delta\rho[/tex] is small compared to [tex]\rho[/tex] ).

    So then I get [tex]\rho_{0}\frac{\partial v}{\partial t} = - \nabla\delta p + \delta\rho g[/tex]
    and [tex]\frac{\partial \delta\rho}{\partial t} + \rho_{0}\nabla .v = 0 [/tex]

    If anyone could help from here that'd be much appreciated!
  2. jcsd
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