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

1. Jul 21, 2010

### omyojj

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 $$2a$$ 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

$$\rho(x,y,z) = \rho_0 ( \theta(z-a) - \theta(z+a) )$$

where $$\theta(z)$$ is a step function.

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

$$\xi_{x,z}(x,z) = \xi_{x,z}(z) e^{ikx + i\omega t}$$

where $$\xi_{x,z}$$ 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 , $$\xi_{x,z}(z) = - \xi_{x,z}(-z)$$ ()

(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.
($$\nabla \cdot {\mathbf{\xi}} = 0$$)

But near the boundary surfaces, discrete density changes in Eulerian density variable $$\delta\rho(x,z)$$ 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.

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

(Of course, Lagrangian density perturbation is everywhere zero, i.e., $$\Delta \rho = 0$$ .)

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.

$$\nabla^2 \delta\psi = 4\pi G \delta\rho$$

Can I solve the above equation for $$\delta\psi(x,z)$$ 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