Navier-Stokes equation, about pressure... I'm a computer science student writting a dissertation about fluid simulation on real time applications. I'm trying to understand a few things regarding the pressure term:
1) When talking about the Helmholtz-Hodge decomposition, my books say that you can decompose any vector field into a divergence-free field, and a gradient of a scalar fied, and that scalar field "turns out" to be the pressure. How do they get to that conclusion?
(http://http.developer.nvidia.com/GPUGems/gpugems_ch38.html chapter 38.2.4 section "The Helmholtz-Hodge Decomposition")
2) In many books or articles when computing pressure people say to take the divergence of the momentum equation, that ends up in a poisson equation. I don't understand why they take the divergence of everything, probably due to my lack of understanding of how poisson equation works or something related.
(http://http.developer.nvidia.com/GPUGems/gpugems_ch38.html chapter 38.2.4 section "The Helmholtz-Hodge Decomposition - Second realization" , in this article they don't take the divergence of the momentum equation but it's similar and with the same purpose)
Keep in mind that I'm not a physics student so I kinda had to learn everything myself because in my university we don't really see much of this things, so probably my confusion is due to the lack of some basic concept that I haven't seen yet. Any help is appreciated, thanks.

here is a way to see (2). assuming the fluid to be incompressible , mass conservation tells you that $div\  u = 0.$  the newtons law for inviscid fluid is $\frac{du}{dt} = -\frac 1\rho\nabla P.$  take the divergence(dot product with $\nabla$) gives you the poisson equation $\Delta P = 0.$
