Biomedical Engineering Reference
In-Depth Information
2
1.5
C2
1
0.5
C1
0
−0.5
−1
−1
0
1
2
3
4
5
6
7
8
9
x
Figure 16.8
Computational domain of the convection-diffusion problem.
with a thick line, the drug concentration is prescribed, say
u
=
1. The drug diffuses
into the liquid with a diffusion constant
c
, but is also convected by the fluid. The
aim is to compute the concentration profile in the two-dimensional channel for a
number of fluid velocities. The computational domain is indicated by the dashed
line, and is further outlined in Fig.
16.8
. Because of symmetry only the top half of
the vessel is modelled.
For stationary flow conditions, the velocity field
v
is described by means of a
parabolic profile (Poisseuille flow) according to
v
=
a
(1
−
y
2
)
e
x
.
As mentioned before, along boundary C1 the fluid flows into the domain with
a concentration
u
1
is prescribed. Along the remaining parts of the boundary the natural boundary
condition:
=
0, while along boundary C2 the concentration
u
=
n
·
c
∇
u
=
0,
is imposed. This means that the top wall is impenetrable for the drug, while this
condition must also be enforced along the symmetry line
y
=
0. Specification
of this condition on the outflow boundary is somewhat disputable, but difficult to
avoid, because only a small part of the circulation system is modelled. By choos-
ing the outflow boundary far away from the source of the drug the influence of
this boundary condition is small.
The corresponding mesh is shown in Fig.
16.9
. The problem is discretized using
bi-quadratic elements.
The steady convection diffusion problem is solved, for
c
=
1 and a sequence
0,1,10,25,100 of parameter
a
. Clearly, with increasing
a
the velocity in the
x
-
direction increases proportionally, hence convection becomes increasingly impor-
tant. For increasing
a
contours of constant
u
are depicted in Fig.
16.10
. In all
