Biomedical Engineering Reference
In-Depth Information
Concentration repartition is calculated by solving the mass conservation equa-
tion for the particles in the liquid, taking into account convection due to the flow
and to the magnetic forces [17]
∂
∂
c
� �
�
�
= Ñ
.(
D c
Ñ -
c V
(
+
uF
))
=
D c V
D -
Ñ - Ñ
c
.(
uF
c
)
(9.28)
f
mag
f
mag
t
In (9.28)
D
is the diffusion coefficient (unit SI: m
2
is defined by Einstein's equation
k T
B
D
=
(9.29)
6
πη
r
h
where
k
B
is the Boltzmann constant,
T
the Kelvin temperature. The mobility
u
is
defined by
1
u
=
(9.30)
6
πη
r
h
Note that
u
is the inverse of the drag coefficient
C
D
. In (9.28) the first term at the
right-hand side is the diffusion term that takes into account the Brownian motion,
the second term is the advection due to the motion of the carrier fluid, and the third
term is the convection due to the magnetic forces.
As for the calculation of particle trajectories, (9.28) requires the previous
calculation of the velocity field of the carrier liquid and the magnetic force
field.
Generally, two types of numerical approach can be done to solve (9.28): finite
elements method or finite differences/volumes method. In microfluidics, the bound-
aries of the domain have a very important impact and the finite elements method
is well adapted for such problems. However, in a simple geometry (rectangular or
axisymmetric), the finite differences method is very easy to use and one can write its
own numerical program to solve such problems.
Note that the problem is a weak coupled problem. It must be solved in three
steps. First, compute the carrier fluid velocities, second, compute the magnetic force
field, and third, solve for the concentration distribution in the domain. Using the
same numerical frame to solve successively the three equations is the most straight-
forward solution.
Note that boundary conditions must be specify to solve (9.28). At channel inlet,
boundary conditions for the concentration are of the type
c = c
0
or
c =
0 depend-
ing on the location of particulate injection; and the condition
¶
¶
at the channel
outlet. The solid walls are impermeable to fluid and particle transport, so that the
corresponding boundary conditions are
c
n
0
�
�
� � �
�
�
�
J n
.
= -
Dgrad c n cv n c u F
.
+
.
+
.
n
= -
Dgrad c n c u F
.
+
=
0
(9.31)
mag
mag
�
is the flux of particles.
where
J
