Graphics Reference
In-Depth Information
don't, but that's not really true: otherwise you wouldn't be able to hear
underwater!
However, the crucial thing is that usually fluids don't change their vol-
ume very much. It's next to impossible, even with an incredibly powerful
pump, to change the volume of water much at all. Even air won't change
its volume much unless you stick it in a pump, or are dealing with really
extreme situations like sonic booms and blast waves. The study of how
fluids behave in these situations is generally called
compressible flow
.It's
complicated and expensive to simulate, and apart from acoustics doesn't
enter that visibly into everyday life—and even sound waves are such tiny
perturbations in the volume and have so small of an effect on how fluids
move at a macroscopic level (water sloshing, smoke billowing, etc.) that
they're practically irrelevant for animation.
What this means is that in animation we can generally treat all fluids,
both liquids and gases, as
incompressible
, which means their volume doesn't
change.
2
What does this mean mathematically? There's a more rigorous
explanation in Appendix B again, but we can sketch out a quick argument
now.
Pick an arbitrary chunk of fluid to look at for some instant in time. We'll
call this volume Ω and its boundary surface
∂
Ω (these are the traditional
symbols). We can measure how fast the volume of this chunk of fluid is
changing by integrating the normal component of its velocity around the
boundary:
dt
volume(Ω) =
d
u
·
n.
∂
Ω
For an incompressible fluid, the volume had better stay constant, i.e., this
rate of change is zero:
u
·
n
=0
.
∂
Ω
Now we can use the divergence theorem to change this to a volume integral.
Basically, this is a multi-dimensional version of the Fundamental Theorem
of Calculus: if you integrate the derivative of a function, you get the original
function evaluated at the bounds of your integration (see Appendix A for
a review if you need to brush up on your vector calculus). In this case, we
2
Even if we need to somehow animate sonic booms and blast waves, they're basically
invisible and extremely fast moving, thus most audiences have no idea really how they
behave. It's probably a much better idea, from an artistic/perceptual/economic view-
point, to hack together something that looks cool than try to simulate them accurately.
