Graphics Reference
In-Depth Information
-B-
Derivations
B.1 The Incompressible Euler Equations
The classic derivation of the incompressible Euler equations is based on
conservation of mass and momentum. Consider an arbitrary but fixed
region of space Ω, in the fluid. The mass of the fluid in Ω is
M
=
Ω
ρ,
and the total momentum of the fluid in Ω is
P
=
Ω
ρu.
The rate of change of
M
, as fluid flows in or out of Ω, is given by the
integral around the boundary of the speed at which mass is entering or
exiting, since mass cannot be created or destroyed inside Ω:
∂M
∂t
=
−
ρu
·
n.
∂
Ω
Here
n
is the outward-pointing normal. We can transform this into a
volume integral with the divergence theorem:
Ω
∇·
∂M
∂t
=
−
(
ρu
)
.
Expanding
M
and differentiating with respect to time (recalling that Ω is
fixed) gives
Ω
∇·
∂ρ
∂t
=
−
(
ρu
)
.
Ω
Since this is true for any region Ω, the integrands must match:
∂ρ
∂t
+
∇·
(
ρu
)=0
.
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