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I am reading the Fast Fluid Dynamics Simulation on the GPU article and stuck with the following part:

We can use the Helmholtz-Hodge Decomposition Theorem to define a projection operator, p.jpg , that projects a vector field w onto its divergence-free component, u. If we apply p.jpg to Equation 7, we get:

enter image description here

Could you please help me understand how we can derive this equation with the Helmholtz-Hodge Decomposition Theorem here and what is the physical interpretation in the context of this article this formula has?

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I don't have time to spell out the explanation in detail, but this idea is sketched out in the exercise 6.7 on page 228 of our book:

https://goldbart.gatech.edu/PostScript/MS_PG_book/bookmaster.pdf

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  • $\begingroup$ Thank you, but this is more a task to solve rather than the explanation. It brings to live other questions directed to use Helmholtz-Hodge Decomposition Theorem, but doesn't help me understand my case. I have even more information here: Helmholtz decomposition, but is still hard for me to catch how this leads to the formula above and especially what is the physical interpretation of this. $\endgroup$ Sep 8, 2021 at 13:56
  • $\begingroup$ The basic idea is that the pressure does not dictate the motion of the fluid, but is instead a constraint which is dictated by the motion. The HH decomposition makes this clear. $\endgroup$
    – mike stone
    Sep 8, 2021 at 16:26

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