Possible wiffle-ball analytical solution

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icarus
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Possible wiffle-ball analytical solution

Post by icarus »

Hi all,

I think I have an analytical solution for the field of the domain interior to the so-called "wiffle-ball" quasi-spherical boundary, that forms within the Polywell reactor.

It demonstrates why the cusps losses of ions will become very small and ion/electron mixing also becomes very small when the spherical boundary is formed.

It may also yield some insights in to how the ion flow is focused in the very heart of the machine.

Is there someone who could work with me in plotting/visualising this solution as I haven't a great deal of time and that is not my strength? It shouldn't be too complicated, just an extension of something like Indrek's 1/48 symmetric model for the field for the exterior domain.

Thnx for any help.

tonybarry
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Post by tonybarry »

Hello icarus,

Welcome to Talk-Polywell. Analytical work is always welcome. May I ask how your dataset is structured so that visualising it might be undertaken? How do you presently visualise your data?

Currently Dr. Mike Rosen is using Blender as his visualising tool for a sim he has undertaken. This avenue may be of interest to you.

Regards,
Tony Barry

drmike
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Post by drmike »

Yeah, I just got some code working yesterday as a matter of fact! I'm trying to get E and B fields to show up as arrows in a 3D plot using *.obj file format. If you have time, you can check out my code here.
I"ve also got some basic math written up in the directory above one (see the *.pdf files).

I'm thinking that using the power of everyones 3D accelerators allows them to flip the 3D object around and get a much better idea of what things look like. I will try to expand on the 1/48 blocks to make them full spherical - should be straight forward. Getting a good visual requires seeing the whole thing, just stating the symmetry isn't good enough for most people.

Send me an e-mail by clicking on the "pm" button and we can talk off line.

Torulf2
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Post by Torulf2 »

If you are interesting in to have animations I can make it in Cinema4D for you. C4d can import OBJ format. I have some render in a previous tread.

drmike
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Post by drmike »

I'm no artist, so I won't try to push anything. What ever you think helps get the ideas across is probably better than what I can come up with. I couldn't even get the colors the way I wanted, but at least they change!

I do think animations will help, I just don't know what the best "gee whiz" thing to show is.

icarus
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Post by icarus »

Here's some replies:

tonybarry: I don't have a dataset generated but rather functional equations for the magneto-static field interior to the Magrid, I'll see if Blender will help.

torulf2: if I get something together I'll take you up on offer to generate animations, zooming in on the streamlines around stagnation points and stagn. lines on the wiffle-ball surface will be very instructive

drmike: what I have been working on is the formation and topology of the magnetic field interior to the wiffle-ball. From the hydrodynamic analogy for magneto-statics, it becomes obvious what flow-field will form in the interior of the magnetic field generated by the coils, if it indeed does form a quasi-spherical secondary flow field.

Imagine that the coils are vortex rings, i.e. 6 of them aligned as per the Polywell WB-6, etc and the steady flow would look exactly like this
http://www.mare.ee/indrek/ephi/wb1/wb1off_lic.jpg

Now if a secondary flow were to form in the very center, and it would necessarily be spherical and closed just from the symmetry of the situation, what would that flow look like to observe conditions of continuity and matching of streamlines across the spherical boundary?

One possible solution is using a "method of images" type approach, known as a Kelvin inversion (see sect. 2.8.3, chap. 2, Partial Differential Equations J. Kervorkian) for Laplace equation solution fields exterior and interior to circles or spheres. What this says is that the sources or dipoles (the vortex rings of our coils in this case) that are generating the flow exterior to the spherical surface are reflected (and inverted) in the spherical surface boundary to have virtual, image generators interior to the sphere. When we do this, the flow interior to the spherical surface exactly mirrors (in a spherical inversion sense) that flow exterior to the surface, ensuring that the surface is spherical also.

We would have a spherically inverted image Polywell of vortex rings at a distance ~ "r", (whereby the physical Polywell coils are at distance "1/r") inside the sphere that generates the interior field. The fields near the surface would be an exact mirror image about the surface, in the limit of an infinitesimally flat element of the spherical surface. For example, where the coil-center point cusps have field lines coming down towards the wiffle-ball surface they would stagnate on the surface and diverge about that point. In the interior, the image field lines would come up out of the center and stagnate on the inside of the sphere, at the same point, and around them diverge also, to mirror the divergence on the exterior. And vice-versa for regions of flow near the convergent lines upon the surface, that coincide with the line cusps between the coils.

It is all very lengthy to describe in words but quite easy to visualise in the minds-eye from the equations and using simplified 2D to 3D extrapolation ... thus the need to generate plots/fields, etc.

It is a well-known analytic solution method (obviously only to a first-order) that I think has much merit for describing the magneto-static fields for a spherically-inverted secondary flow. If there is such a secondary field (flow) that is forming with the "wiffle-ball" there is only a limited number of correct topologies that can be forming. The Kelvin inversion is the first place to begin looking at what the field will be looking like, and will be quite correct very near to the spherical surface.

tombo
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Post by tombo »

Is the spherical surface just a mathematical construct to help solve the equations?
Or are you saying that the magnetic field itself is perfectly spherical where it contacts the plasma?
Does that imply that the field is concave toward the plasma?
If so then we have a stability problem.
-Tom Boydston-
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein

icarus
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Post by icarus »

tombo wrote:Is the spherical surface just a mathematical construct to help solve the equations?
Or are you saying that the magnetic field itself is perfectly spherical where it contacts the plasma?
Does that imply that the field is concave toward the plasma?
If so then we have a stability problem.
tombo: the whole solution is "just a mathematical construct", how well or not it reflects reality will be determined by measurements and yes it absolutely helps to solve the equations.

If the magneto-static field inverts (field reverses) to form the interior spherical secondary flow that I'm proposing then yes, the interface between the electron/ion regions will be quasi-spherical. I'm not sure which "plasma" you are referring to here, there is the mono-energetic region of electrons that may or may not behave like a plasma in the conventional sense and as far as I can estimate a region of mono-energetic ions mixed with electrons, all immersed in a strong electro-static field. A stretch to call the interior region a "plasma" in the usual sense ... maybe something like going towards a Bose-Einstein statistics makes sense for these regions ... White Dwarfs, who knows?

As to the stability of the whole system, all my training says first you need a dynamic model before you can analyse stability issues. Rules of thumb like "concave versus convex" tend to contain a myriad of assumptions.

Before we can build a dynamic model, we need a static model to perturb the system about, a known stationary solution, thus my spherical Kelvin inversion approach.

drmike
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Post by drmike »

It'd be interesting to read the math. Can you write it up in texmacs or use an equation editor in word or word perfect? I like texmacs myself, but I'm a linux nut :)

dweigert
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Post by dweigert »

OpenOffice.org has a Math/equation package..

tombo
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Post by tombo »

Icarus, I didn't mean to jump on you.
I really like it when serious lurkers come out of the woodwork.

What I meant was more along the lines of:
Does using a reflecting sphere as a mathematical method still work in light of the probability that the magnetic field is more like a spiky sphere?
I'm hoping that because it is a mathematical construct, it does not need to follow the physical field shape.
-Tom Boydston-
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein

kcdodd
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Post by kcdodd »

Art Carlson made this suggestion in another thread sounds similar? I think if you've got a solution it would be a huge benefit to many simulation attempts.
It is somewhere between possible and likely that you need to do some work to get into the high beta regime of interest. The easiest way I can think of to simulate that would be to put a superconducting ball into the middle of the configuration and to start the electrons from the surface of this. This may not be as hard as it sounds. I think if you go to circular coils, you can use image coils within the sphere to produce a magnetic configuration with zero normal field on the surface of the sphere.
Carter

icarus
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Post by icarus »

First cut of the solution, with Indrek's help.

http://www.mare.ee/indrek/ephi/tmp/test25.jpg

Image

kcdodd
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Post by kcdodd »

Looks good. How do you determine what values to use for the image?

edit:

I decided to attempt this as well just guess the value should solve for the sphere radius. I had it crop the fields inside the sphere since, well, there aren't supposed to be any there due to the electrons. In the pic some of the field lines actually terminate on the very surface of the sphere. That's bad but I am thinking that is just from the interpolation matlab has to do and it lost track of the field. I'm not sure though maybe the crop actually cropped a little too much, i hope not.

http://www.andromedaspace.com/files/highbball.png

Here's an electron trace just to test it. I had to do direct calculation (no interpolation) because of the whole cropping field line issue I mentioned. The electron pretty much just bounces around like it's supposed to.

http://www.andromedaspace.com/files/highbballrun.png
Carter

Indrek
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Post by Indrek »

Last night I wrote some Octave/Matlab code for a loop of current and the cube polywell:

http://www.mare.ee/indrek/ephi/octave/

The code is there so feel free to use it. I'd love to get some feedback as I'm a complete beginner at octave/matlab.

Here's a plot of the opposed coils:

Image

- Indrek

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