Biomedical Engineering Reference
In-Depth Information
14.2
The diffusion equation
The differential equation that describes the one-dimensional diffusion problem is
given by
c
du
dx
d
dx
+
f
=
0,
(14.1)
where
u
(
x
) is the unknown function,
c
(
x
)
0 a given material characteristic func-
tion and
f
(
x
) a given source term. This differential equation is defined on a one-
dimensional domain
>
b
while the
boundary, which is formally denoted by
, is located at
x
=
a
and
x
=
b
.
Eq. (
14.1
) is an adapted form of the diffusion equation Eq. (
13.43
), introduced
in the previous chapter. Different symbols for the unknown (
u
instead of
ρ
) and
coefficient (
c
instead of
D
) are used to emphasize the general character of the
equation, applicable to different kinds of problems (see below). Furthermore, the
coefficient
c
can be a function of
x
and a source term
f
(
x
) is introduced.
Two types of boundary conditions can be discerned. Firstly, the essential bound-
ary condition, which must be specified in terms of
u
. For the derivations that
follow the boundary at
x
that spans the
x
-axis between
x
=
a
and
x
=
=
a
is chosen, to specify this type of boundary
condition:
u
=
U
at
u
,
(14.2)
where
u
denotes the boundary of the domain
at
x
=
a
. Secondly, a natural
boundary condition may be specified. Here the boundary at
x
=
b
is chosen to
specify the flux
cdu
/
dx
:
c
du
dx
=
P
at
p
,
(14.3)
where
p
denotes the boundary at
x
=
b
. For diffusion problems, an essential
boundary condition must be specified to have a well-posed boundary value prob-
lem. This is not necessarily the case for natural boundary conditions; they may be
absent.
Example 14.1
The diffusion equation describes a large range of problems in biomechanics and
is applicable in many different areas. In the way it was introduced in Section
13.4
the unknown
u
represents the concentration of some matter, for example: oxy-
gen in blood, proteins or other molecules in an extracellular matrix or inside a
cell, medication in blood or tissue. In that case the term
f
can be either a source
term (where matter is produced) or a sink term (where matter is consumed).
