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Differential Equation with forcing/fluids question

  1. Sep 5, 2010 #1
    An issue has come up in my research: I think the problem can be phrased as such:

    Given a differential equation of the form
    [tex]Lu(t) = f(t)[/tex]
    Where the forcing function is of the form [tex]f(t)=\gamma e^{-t/\epsilon},\gamma[/tex] is a constant, and L is some linear second order operator.

    We want to see what happens to the particular solution in the limit as [tex]\epsilon \to 0[/tex] while keeping
    [tex]\int_{0}^{\infty}f(t')dt' = const[/tex]

    Physically, it seems as if the forcing function will turn into a boundary condition, although I am not sure how to make this rigorous. Is this always the case? When I go through and evaluate the limits for the particular solution I am finding results that do not make sense.

    In case a particular example helps, this is where this problem comes from.

    Consider the time independent linearized Navier Stokes equation for a viscous rotating fluid with forcing. The equation governing the dynamics is:
    [tex] u_{zz}-\frac{if}{\nu} u =-\frac{1}{\nu} A(z) [/tex]
    where [tex]A(z)=\gamma e^{-z/\epsilon}[/tex]
    I am getting results that do make sense physically for particular types of forcing functions A, so I would like to see what happens when A turns into a surface condition as opposed to a forcing function, since I know what to expect in that case.
  2. jcsd
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