Biomedical Engineering Reference
In-Depth Information
V
z
v
x
h
x
y
Figure 13.6
Flow between two mutually translating parallel plates.
Thus, for the velocity in
x
-direction a linear profile with respect to the
z
-coordinate
is found. The flow in this example is known as the 'Couette flow'.
In this section two very special cases (yet both of practical interest) of fluid
flows are treated that permit an analytical solution. For very few practical prob-
lems this is possible. The Finite Element Method as discussed in Chapters
14
to
18
enables us to construct approximate solutions for very complex flows, however
the specific algorithms to do that for viscous flows are beyond the scope of the
contents of this topic.
13.4
Diffusion and filtration
In Section
12.8
the constitutive equations for diffusion and filtration to describe
transport of material ('fluid') through a stationary porous medium, have been
discussed. By adding the relevant balance laws (see Chapter
11
) the problem
description can be further elaborated. This will be the subject of the present
section.
For diffusion Fick's law can be applied:
D
∇
ρ
ρ
v
=−
,
(13.40)
with D the diffusion coefficient.
The mass balance, see Section
11.2
, can be written as
δρ
δ
t
+ ∇·
(
ρ
v
)
=
0.
(13.41)
Elimination of the velocity
v
from the two equations above, leads to a linear partial
differential equation for the density
ρ
=
ρ
(
x
,
t
) according to:
δρ
δ
t
−
D
∇·
(
∇
ρ
)
=
0.
(13.42)
