Graphics Reference
In-Depth Information
Putting it all together, here's how a blob of fluid moves:
m
Du
Dt
=
mg
−
V
∇
p
+
Vη
∇·∇
u.
Obviously we're making errors when we approximate a fluid with a
small finite number of particles. We will take the limit then as our number
of particles goes to infinity and the size of each blob goes to zero. This
poses a problem in our particle equation, because the mass
m
and volume
V
of the particle are then going to zero. We can fix this by dividing the
equation by the volume and then taking the limit. Remembering
m/V
is
just the fluid density
ρ
,weget
ρ
Du
Dt
=
ρg
−∇
p
+
η
∇·∇
u.
Looking familiar? We'll divide by the density and rearrange the terms a
bit to get
Du
Dt
+
1
p
=
g
+
η
ρ
∇
ρ
∇·∇
u.
To simplify things even a little more, we'll define the kinematic viscosity
as
ν
=
η/ρ
to get
Du
Dt
+
1
ρ
∇
p
=
g
+
ν
∇·∇
u.
We've almost made it back to the momentum equation! In fact this
form, using the material derivative
D/Dt
, is actually more important to us
in computer graphics and will guide us in solving the equation numerically.
But we still will want to understand what the material derivative is and
how it relates back to the traditional form of the momentum equation. For
that, we'll need to understand the difference between the
Lagrangian
and
Eulerian
viewpoints.
1.3
Lagrangian and Eulerian Viewpoints
When we think about a continuum (like a fluid or a deformable solid)
moving, there are two approaches to tracking this motion: the Lagrangian
viewpoint and the Eulerian viewpoint.
The Lagrangian approach, named after the French mathematician La-
grange, is what you're probably most familiar with. It treats the continuum
just like a particle system. Each point in the fluid or solid is labeled as a
separate particle, with a position
x
and a velocity
u
. You could even think
