1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solving fluid's Poisson equation for periodic problem or more easy way?

  1. Jul 21, 2010 #1
    The problem is about mathematics but it originates from the self-gravitational instability of incompressible fluid, so let me explain the situation first.

    I have an incompressible uniform fluid disk that is infinite in the x-y direction.

    The disk has a finite thickness [tex] 2a [/tex] along the z-direction. (-a<z<a)

    The space exterior to the disk is assumed to be filled with a rarefied medium that has constant pressure equal to the fluid's, which prevents the disk from dispersing.

    Thus, the initial density distribution has a step discontinuity and can be written as

    [tex] \rho(x,y,z) = \rho_0 ( \theta(z-a) - \theta(z+a) ) [/tex]

    where [tex] \theta(z) [/tex] is a step function.

    Now I want to apply small Lagrangian perturbation to the fluid of the form

    [tex] \xi_{x,z}(x,z) = \xi_{x,z}(z) e^{ikx + i\omega t} [/tex]

    where [tex] \xi_{x,z} [/tex] is the x,z-component of Lagrangian displacement vector.

    Perturbation has its wavenumber k along the x-direction, and I assumed time dependence.

    Also I consider only the perturbation that has even reflection symmetry for displacement, that is , [tex] \xi_{x,z}(z) = - \xi_{x,z}(-z) [/tex] ()

    (Sausage type: The rectangular shape of the slab changed slightly (though infinitesimally) so that it looks more like a cylinder now)

    Deep inside the disk, there would be no change in density because the fluid itself is incompressible.
    ([tex] \nabla \cdot {\mathbf{\xi}} = 0 [/tex])

    But near the boundary surfaces, discrete density changes in Eulerian density variable [tex] \delta\rho(x,z) [/tex] could occur if the difference b/w a and the height from the midplane(z) is smaller than the Lagrangian displacement vector at z=a.

    [tex] \delta\rho(x,z) = \rho_0 [ \theta(z - \xi_z(z=a)e^{ikx} - \theta(z - a) ] + \rho_0 [ \theta(z+a) - \theta(z - \xi_z(z=-a)e^{ikx}) ] [/tex]

    (Of course, Lagrangian density perturbation is everywhere zero, i.e., [tex] \Delta \rho = 0 [/tex] .)

    Now I want to introduce self-gravity at this point because I want to examine the strength of perturbed gravity that makes the system unstable to this small disturbances.

    [tex] \nabla^2 \delta\psi = 4\pi G \delta\rho [/tex]

    Can I solve the above equation for [tex]\delta\psi(x,z) [/tex] with right-hand side involving step functions of sinusoidal behavior in x-direction?

    Any hint or help would be much appreciated.

    Thank you.

    BTW, excuse my English..
    Last edited: Jul 21, 2010
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted