Graphics Reference
In-Depth Information
1.3.1 An Example
Hopefully to lay the issue to rest, let's work through an example in one
dimension. Instead of
q
we'll use
T
for temperature. We'll say that at one
instant in time, the temperature profile is
T
(
x
)=10
x
;
that is, it's freezing at the origin and gets warmer as we look further to the
right, to a temperature of 100 at
x
= 10. Now let's say there's a steady
wind of speed
c
blowing, i.e., the fluid velocity is
c
everywhere:
u
=
c.
We'll assume that the temperature of each particle of air isn't changing—
they're just moving. So the material derivative, measuring things in the
Lagrangian viewpoint says the change is zero:
DT
Dt
=0
.
If we expand this out, we have
∂T
∂t
+
∇
T
·
u
=0
,
∂T
∂t
+10
·
c
=0
∂T
∂t
⇒
=
−
10
c
;
that is, at a fixed point in space, the temperature is changing at a rate
of
10
c
. If the wind has stopped,
c
= 0, nothing changes. If the wind is
blowing to the right at speed
c
= 1, the temperature at a fixed point will
drop at a rate of
−
−
10. If the wind is blowing faster to the left at speed
c
=
2, the temperature at a fixed point will increase at a rate of 20. So
even though the Lagrangian derivative is zero, in this case the Eulerian
derivative can be anything depending on how fast and in what direction
the flow is moving.
−
1.3.2 Advecting Vector Quantities
One point of common confusion is what the material derivative means
when applied to vector quantities, like RGB colors, or most confusing of
all, the velocity field
u
itself. The simple answer is: treat each component
