Graphics Reference
In-Depth Information
-1-
The Equations of Fluids
Fluids surround us, from the air we breathe to the oceans covering two
thirds of the Earth, and are at the heart of some of the most beautiful and
impressive phenomena we know. From water splashing to fire and smoke
swirling, fluids have become an important part of computer graphics. This
topic aims to cover the basics of simulating these effects for animation. So
let's jump right in with the fundamental equations governing their motion.
Most fluid flow of interest in animation is governed by the famous
in-
compressible Navier-Stokes equations
, a set of partial different equations
that are supposed to hold throughout the fluid. The equations are usually
written as
∂u
∂t
+
u
u
+
1
·∇
ρ
∇
p
=
g
+
ν
∇·∇
u,
(1.1)
∇·
u
=0
.
(1.2)
These may appear pretty complicated at first glance! We'll soon break them
down into easy-to-understand parts (and in Appendix B provide a more
rigorous explanation), but first let's begin by defining what each symbol
means.
1.1 Symbols
The letter
u
is traditionally used in fluid mechanics for the velocity of the
fluid. Why not
v
? It's hard to say, but it fits another useful convention
to call the three components of 3D velocity (
u, v, w
), just as the three
components of position
x
are often taken to be (
x, y, z
).
The Greek letter
ρ
stands for the density of the fluid. For water, this is
roughly 1000 kg
/
m
3
, and for air this is roughly 1
.
3kg
/
m
3
,aratioofabout
700 : 1.
The letter
p
stands for
pressure
, the force per unit area that the fluid
exerts on anything.
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