Biology Reference
In-Depth Information
Terminology classifies diffusion processes based on the structure of
the diffusion tensor:
•
If
D
is constant everywhere in
, diffusion is called “homogeneous”.
A
D
that varies in space defines “inhomogeneous diffusion”.
Ω
•
If
D
is proportional to the identity matrix,
D
D
, diffusion is
called “isotropic”; otherwise, “anisotropic”. Isotropic diffusion is
characterized by a flux whose magnitude does not depend on its
direction, and it can be described using a scalar diffusion constant
D
.
For isotropic, homogeneous diffusion, the diffusion equation
simplifies to
=
∂
ut
t
(,)
x
2
{
}
(15)
=∇
Du t
(,)
x
for
x
∈
ΩΩ
∂
,
0
<≤
tT
,
∂
∇
where
2
is the Laplace operator.
At
t
=
0, the concentration field is specified by an initial condition
ut
(,
x
==
0
)
u
()
x
x
∈
Ω
.
0
The model is completed by problem-specific boundary conditions pre-
scribing the behavior of
u
along
∂
. The most frequently used types of
boundary conditions are Neumann and Dirichlet conditions. A Neumann
boundary condition fixes the diffusive flux through the boundary to a pre-
scribed value
f
N
(
n
is the outer unit normal on the boundary):
Ω
u
∂
∂
=∇
ut
(,)
xn
⋅
=
f
(,)
x
t
for
x
∈
∂
Ω
,
0
<
tT
≤
;
N
n
whereas a Dirichlet condition prescribes the concentration
f
D
at the
boundary:
ut f
(,)
x
=
(,)
x
t
for
x
∈
∂
Ω
,
0
<
tT
≤
.
D
If the boundary function
f
is 0 everywhere on
∂
Ω
, the boundary
condition is called “homogeneous”.
