Graphics Reference
In-Depth Information
The letter
g
is the familiar acceleration due to gravity, usually
9
.
81
,
0) m
/
s
2
. Now is a good time to mention that in this topic we'll
take as a convention that the
y
-axis is pointing vertically upwards, and the
x
-and
z
-axes are horizontal. We should add that in animation, additional
control accelerations (to make the fluid behave in some desired way) might
be added on top of gravity—we'll lump all of these into the one symbol
g
. More generally, people call these
body forces
, because they are applied
throughout the whole body of fluid, not just on the surfaces.
The Greek letter
ν
is technically called the
kinematic viscosity
.Itmea-
sures how viscous the fluid is. Fluids like molasses have high viscosity,
and fluids like mercury have low viscosity: it measures how much the fluid
resists deforming while it flows (or more intuitively, how dicult it is to
stir).
(0
,
−
1.2 The Momentum Equation
The first differential equation (1.1), which is actually three in one wrapped
up as a vector equation, is called the
momentum equation
. This really is
good old Newton's equation
F
=
ma
in disguise. It tells us how the fluid
accelerates due to the forces acting on it. We'll try to break this down
before moving onto the second differential equation (1.2), which is called
the
incompressibility condition
.
Let's first imagine we are simulating a fluid using a particle system
(later in the topic we will actually use this as a practical method, but
for now let's just use it as a thought experiment). Each particle would
represent a little blob of fluid. It would have a mass
m
,avolume
V
,anda
velocity
u
. To integrate the system forward in time all we would need is to
figure out what the forces acting on each particle are:
F
=
ma
then tells
us how the particle accelerates, from which we get its motion. We'll write
the acceleration of the particle in slightly odd notation (which we'll later
relate to the momentum equation above):
Du
Dt
.
a
≡
The big
D
derivative notation is called the
material derivative
(more on
this later). Newton's law is now
m
Du
Dt
=
F.
