Biomedical Engineering Reference
In-Depth Information
Cell concentration equation
∂n
∂t
+
(
nV
∗
)+
(
nV
c
)=
D
c
∇
2
n
∇·
∇·
(14.3)
Boussinesq approximation
ρ
=
ρ
0
(1 +
β
(
n
−
n
0
))
(14.4)
14.2.1.2 Initial and Boundary Conditions
At
t
∗
=0
,
it is supposed that the initial concentration is uniform, that is,
n
(
X,Y,t
∗
)=
n
(
X,Y,
0) =
n
(14.5)
At the impermeable boundaries the condition of zero-normal fluid velocity
requires
X
=0
,L
:
Y
=0
,H
:
V
X
=0
V
Y
=0
(14.6)
while the condition of zero-concentration flux is (
e
is the unit normal vector)
e
=
n
+
n
(
V
∗
+
V
c
)
J
∗
•
−
D
c
∇
•
e
=0
X
=0
,L
:
J
X
=
−
∂n/∂X
=0
⇔
(14.7)
J
Y
=
nV
c
Y
=0
,H
:
−
D
c
∂n/∂Y
=0
14.2.2 Diffusion State
For
V
c
=(0
,V
c
), the system (14.1 through 14.4) under the initial and bound-
ary conditions (14.5 through 14.7) admits the following steady state solution:
n
V
c
H
D
c
exp
V
c
D
c
Y
V
∗
= 0
exp
V
c
H
D
c
and
n
=
(14.8)
−
1
which satisfies the conservation of microorganisms:
L
H
1
LH
n
(
X,Y,t
∗
)
dY
n
=
dX
(14.9)
0
0
Search WWH ::
Custom Search