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

Your Name]In summary, the problem at hand involves the self-gravitational instability of an incompressible fluid disk with a step discontinuity in its density distribution. To solve the equation for \delta\psi(x,z), we can use the method of separation of variables and the Fourier transform to convert the step functions into sinusoidal functions. This will allow us to solve the equation using standard methods for partial differential equations.
  • #1
omyojj
37
0
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:
Physics news on Phys.org
  • #2


Thank you for sharing your problem with us. I am interested in exploring the self-gravitational instability of incompressible fluids and its application to your specific situation.

Firstly, I would like to clarify that the density distribution you have described is not physically realistic. Incompressible fluids do not have a step discontinuity in their density distribution, as this would violate the fundamental principle of incompressibility. However, for the sake of solving your problem, we can consider a simplified scenario in which the density distribution has a small, continuous change around the boundaries.

Now, to solve the equation for \delta\psi(x,z), we can use the method of separation of variables. This involves assuming a solution of the form \delta\psi(x,z) = X(x)Z(z) and plugging it into the equation. We can then solve for X(x) and Z(z) separately, and combine the solutions to get the final solution for \delta\psi(x,z).

In this particular case, the right-hand side of the equation involves step functions of sinusoidal behavior in the x-direction. This can be taken care of by using the Fourier transform, which converts the step functions into a series of sinusoidal functions. The resulting equation can then be solved using standard methods of solving partial differential equations.

I hope this helps you to get started on solving your problem. If you need any further assistance, please do not hesitate to reach out to me.
 

1. What is the Poisson equation for a fluid?

The Poisson equation for a fluid is a partial differential equation that describes the relationship between the fluid's pressure and its density. It is used to model the behavior of fluids in a variety of applications, including fluid dynamics and thermodynamics.

2. What is a periodic problem in relation to the Poisson equation for fluids?

A periodic problem in relation to the Poisson equation for fluids refers to a situation where the fluid's properties, such as pressure and density, repeat in a regular pattern. This can occur in systems that have a periodic boundary condition, such as a fluid in a closed container with a periodic shape.

3. How is the Poisson equation for fluids solved for a periodic problem?

The Poisson equation for fluids can be solved for a periodic problem using numerical methods, such as finite difference or finite element methods. These methods involve discretizing the problem into smaller elements and solving the equations for each element, which can then be combined to obtain a solution for the entire system.

4. Is there an easier way to solve the Poisson equation for fluids in a periodic problem?

Yes, there are other methods that can be used to solve the Poisson equation for fluids in a periodic problem. One such method is the spectral method, which uses Fourier series to represent the solution in terms of trigonometric functions. This method can be more efficient than numerical methods, but it is limited to problems with simple geometries.

5. What are the applications of solving the Poisson equation for fluids in a periodic problem?

The Poisson equation for fluids in a periodic problem has many applications, including studying fluid flow in porous media, modeling heat transfer in periodic systems, and simulating the behavior of fluids in complex geometries. It is also used in the design and analysis of various engineering systems, such as turbines, pumps, and heat exchangers.

Similar threads

  • Classical Physics
Replies
9
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
300
  • Differential Equations
Replies
13
Views
2K
Replies
3
Views
681
  • Classical Physics
Replies
4
Views
1K
Replies
20
Views
5K
Replies
1
Views
578
  • Advanced Physics Homework Help
Replies
7
Views
2K
  • Classical Physics
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
10
Views
2K
Back
Top