Navier-Stokes Equation

Here is a summarized note I took after reading some material relevant to fluid effect.

This is the compact version of Navier-Stokes Equation excerpted from Jos Stam's paper. For simulation in Computer Graphics, physical accuracy isn't the primary but the appearance and performance is. In order to get a faster simulation and realistic at the same time, here it assumes constant density, which means the fluid is incompressible. Basically, fluid effect in CG is mostly based on this assumption.

The first equation is the condition of incompressibility. With constant density, the net flow in and out of a region should sum to zero, which means the divergence of velocity should be zero, no sinks or sources.

The second equation is based on Newton's second law F=ma. Before going further, I spent some time trying to understand 2 different descriptions for fluid mechanics, Eulerian and Lagrangian respectively. I recommend the following video on YouTube, which does a great job on explaining these 2 ideas. It also provides a clear demonstration about how to deduce material derivative, which is quite important to help understand the second equation here.

http://www.youtube.com/watch?v=mdN8OOkx2ko&feature=related

It's obvious to see that the left hand side of the assignment, combining with the first term on the right hand side is the material derivative. It says that the change of a point moving within a velocity field is the sum of the change at the fixed position and the one due to the movement.  The first term on the right hand side is what we call "Advection" (imaging if we drip a pigment drop into a moving liquid, how the pigment will move within it).

OK. Back to F=ma. 

Here we represent the mass as density since unit size is used; then expand the material derivative to get the following:

Well... divide each side by density, we get equation 2 :)

It's apparent that the sum of (F)orces corresponds to the rest 3 terms in equation 2:  pressure, viscosity and other external forces.

Pressure is used for enforcing incompressibility. Since the gradient points to the greatest rate of increase and fluid should flow from high pressure to low, negative sign is used for reversing.
Viscosity is used for simulating viscous fluid. Laplace operator is used here for getting smoothing result. External forces are something like gravity, buoyancy, etc.

So lots of research are around solving this equation, leading to various algorithms, such as SPH (Smoothed Particle Hydrodynamics), PIC/FLIP (Particles in Cell / Fluid Implicit Particle). It's kind of complicated and better has separate articles for these.