Biomedical Engineering Reference
In-Depth Information
Figure 6.3
Concentration balance in an elementary volume.
where
J
x
and
J
y
are the diffusion fluxes given by the Fick's law
¶
c
J
= -
D
x
¶
x
(6.2)
¶
c
J
= -
D
y
¶
y
In (6.2)
D
is the diffusion constant. Dividing (6.1) by D
x
D
y
and substituting
(6.2) yields
¶
c
¶
c
¶
c
(6.3)
+
u
+
v
= Ñ
.(
D c
Ñ
)
¶
t
¶
x
¶
y
or
¶
c
�
+
U c
.
Ñ = Ñ
.(
D c
Ñ
)
(6.4)
t
¶
where
U
is the vector (
u
,
v
,
w
). Recall that the material derivative notation is
D
¶
¶
¶
¶
¶
(6.5)
=
+
u
+
v
+
w
=
+ Ñ
U
.
Dt
¶
t
¶
x
¶
y
¶
z
¶
t
and then (6.3) can be cast under the form
Dc
= Ñ
.(
D c
Ñ
)
(6.6)
Dt
Assuming that
D
is constant, the advection-diffusion equation becomes
¶
c
�
+
U c D c
.
Ñ =
D
(6.7)
t
¶
