Friday, July 10, 2020

Vortex Rings / 3D Landau-Ginzburg

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Thursday, July 9, 2020

3D Landau Ginzburg / Staring into the Abyss

One step closer.

by MiMo
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So this is what the internal dynamics of the Dynamic Foam kind looks like. 
Pulsating bubbles driven by the fluid in the edges:

 There is a model that seems to resemble this (by MiMo):

Let's assume that the number of bubbles tends to infinity and and the state of the foam can be approximated by average pressure and flow intensity(with no direction, since the edges have arbitrary directions in average)
So we have scalar fields P(x,y,z,t) and I(x,y,z,t)
If we assume that the dynamic interaction between P and I resembles a wave (pressure induces flow -> flow induces pressure) and are out of phase by 90 degrees(sin / cos) we can describe the foam state like a complex field Psi
Psi(x,y,z,t) = a*P(x,y,z,t) + imaginary_unit*b*I(x,y,z,t). a and b are some arbitrary constants that define how strong pressure induces flow and flow pressure;  
The simplest equation for a wave like that is the Schrodinger Equation
imaginary_unit *speed_of_change_of_Psi = - (average_Psi_around_this_point - Psi)/surface_of_unit_sphere + some_function_of_Psi;
For a special function of Psi, that basically defines how pressure interacts with the flow in this current bubble, we can get quantized vortices, the smallest stable of which are exactly torus vortices(!)
Wyatt has already made a simulation of that equation in 2d and 3d.
The last thing left to do is just to rewrite the equation from average bubble flow and average bubble pressure for each bubble. And that's basically it. Its not that hard, but there may be some instabilities related to the fact that the foam is not completely uniform(as in the assumption of the averaged out foam from above).


L-G uses a ‘probe field’ as a trick to get these vortices: