This technical report is an overview of the Navier-Stokes equations for simulating fluids. Navier-Stokes equations are 2 equations used to describe the behavior of fluids. They are.  
  del . u = 0 <- This means that if we take an infintesmall sphere around any point in the liquid, the amount of fluild leaving the sphere is exactly equal to the amount entering it. This is bascially because fluids can't be compressed

  du / dt = -(u . del) u - 1/rho del p + mu del^2 u + f

This equation gives the change in speed at any point in the fluid. This is the equation we really need (The first is more of a constraint which is not very useful to know the speed at any point)
  
The first term says that speed tends to propagate into the liquid.   
The second term says that liquids move from points of high pressure to points of low pressure.  
The third term says that the more viscous a fluid is, the smoother its movement gets.  
Finally, the fourth term is the external force.  
  
The rest of this report is basically a revision for Stam's Stable fluids paper. This report just gives a hand-wavy description of how Stam did everything.  This includes...  
  
 1. Using Helmholtz-Hodge decomposition to decompose the fluid into a gradient field and a divergence free field. This has the attractive property of making the illegal divergence disappear.  
  
 2. Using the method of characteristics to achieve unconditionally stable fluids (Tracing the fluid backwards instead of forward)  

  

I am still not very comfortable with NS-equations. The major problem I have is that I don't understand how the first equation still applies at the boundaries of the liquid? At the boundaries, the amount of fluid leaving can be higher than the amount entering or vice versa, and since we are SPECIFICALLY interested in simulating the boundary of the liquid, I don't see how the method still works fine!