Biomedical Engineering Reference
In-Depth Information
16.6
Convection-diffusion equation
Assuming isotropic diffusion, the convection-diffusion equation is given by
∂
u
∂
t
+
∇
∇·
(
c
∇
v
·
u
=
u
)
+
f
,
(16.26)
with
v
the convective velocity. This equation should hold on the spatial domain
during a certain period of time, say
S
=
[0,
T
]. Initial boundary conditions must
be specified:
u
(
x
,
t
=
0)
=
u
ini
(
x
)in
,
(16.27)
as well as essential and natural boundary conditions:
u
=
U
at
u
(16.28)
n
·
c
∇
u
=
P
at
p
.
(16.29)
The weak form is obtained analogously to the procedure of Section
16.4
, giving
w
∂
u
u
)
d
∇
+ ∇
(
c
∇
∂
t
+
w
v
·
u
w
·
c
∇
=
wf d
+
w
n
·
ud
.
(16.30)
Spatial discretization is performed in a two-dimensional configuration by intro-
ducing
T
(
x
,
y
)
∼
e
,
w
h
|
e
=
∼
(16.31)
and
T
(
x
,
y
)
∼
e
.
u
h
|
e
=
∼
(16.32)
For a particular element
e
, the individual integrals of Eq. (
16.30
) can be
converted to:
w
∂
u
∂
e
M
e
d
∼
e
d
=
∼
,
(16.33)
t
dt
e
with:
T
d
M
e
=
∼∼
.
(16.34)
e
Further, by using
v
=
v
x
e
x
+
v
y
e
y
,
(16.35)
it can be written:
wv
· ∇
ud
=
∼
e
C
e
∼
e
,
(16.36)
e
