Biomedical Engineering Reference
In-Depth Information
Figure 6.8
During some biochemical reactions, micro- and nanoparticles can be temporarily
immobilized at the wall until equilibrium is found. Depletion layers similar to concentration boundary
layers form in the vicinity of the solid wall.
The system (6.13) is a system of five scalar equations (in the three-dimensional
case). There are five unknowns (
u
,
v
,
w
,
P
,
c
), two fluid properties
r
and
m
, the diffu-
sion constant of the species
D
, and two external actions on the fluid: the body force
par unit volume
F
and the concentration source or sink per unit volume
S
.
Note that system (6.13) is only a weakly coupled system under the condition
that the concentration is sufficiently small to not affect the buffer fluid viscosity and
density. In the next section we treat the problem of the variation of the fluid proper-
ties with concentration.
Usually, in a microfluidics microsystem, the flow of the buffer fluid is permanent
(steady state) and only the concentration changes with time. In such a case, if we
assume that
r
,
m
, and
D
are constant, and that there are no body forces (gravity is
usually negligible in very small systems). In such a case, the system (6.13) collapses
to
�
Ñ =
.
U
0
� �
�
1
(6.14)
U U
.
Ñ = - Ñ + D
P
ν
U
ρ
�
¶
c
+
U c D c
.
Ñ =
D +
S
t
¶
And if the hypothesis of a creeping flow is valid (i.e., the Stokes approximation
is justified), the system (6.14) collapses to the linear system, under the condition
that the function
S
is well behaved
�
Ñ =
.
U
0
�
Ñ =
P
η
D
U
(6.15)
�
¶
c
+
U c D c
.
Ñ =
D +
S
¶
t
