Biomedical Engineering Reference
In-Depth Information
This differential equation should be satisfied for all
x
in the considered domain
V
and for all times
t
. By specifying the initial conditions (prescribed
ρ
for all
x
in
V
at
t
0) and with one single boundary condition for each boundary point of
V
(either the density
=
D
(
∇
ρ
ρ
or the outward mass flux
ρ
v
·
n
=−
)
·
n
should be
prescribed), in principle a solution for
x
,
t
) can be calculated.
To illustrate some of the problems that arise, we confine ourselves to an attempt
to solve a simple one-dimensional problem. A domain is given by 0
ρ
(
L
.
Diffusion of a certain material (diffusion coefficient
D
) in the
x
-direction can take
place. For the density
≤
x
≤
ρ
=
ρ
(
x
,
t
) the following partial differential equation holds:
2
∂ρ
∂
D
∂
ρ
∂
x
2
−
=
0,
(13.43)
t
emphasizing that the spatial derivative
δρ/δ
t
is written here as the partial deriva-
tive of
to the time
t
(with constant
x
). Misunderstandings because of this will
not be introduced, because exclusively an Eulerian description will be used. The
differential equation in this example is completed with:
•
ρ
the initial condition:
ρ
=
0for0
≤
x
≤
L
and
t
=
0,
•
the boundary conditions:
ρ
=
ρ
0
(with
ρ
0
a constant) for
x
=
0and
t
>
0
∂ρ
∂
x
=
0 (no outflow of material) for
x
=
L
and
t
>
0.
Even for this very simple situation an exact solution is very difficult to determine.
A numerical approach (for example by means of the Finite Element Method) can
lead to a solution in a simple way. This is the topic of Chapter
14
. Here, it can be
stated that the solution for
ρ
(
x
,
t
)at
t
→∞
has to satisfy
ρ
(
x
,
t
)
=
ρ
0
for all
x
.A
large number of closed form solutions can be found in [
4
].
For filtration problems Darcy's law can be applied:
=−
κ
∇
ρ
v
p
,
(13.44)
with
the permeability. The mass balance, see Section
11.2
, can be written as
follows:
κ
δρ
δ
t
+ ∇·
(
ρ
v
)
=
0.
(13.45)
Further elaboration is limited to stationary filtration (time
t
does not play a role).
In that case
δρ
δ
∇·
(
ρ
v
)
=
0.
t
=
0 and so
