Biomedical Engineering Reference
In-Depth Information
9.10.1 Trajectories
Due to the design of the MFFF device, with a vertical external force field and a
horizontal hydrodynamic drag, (9.17) can be decomposed on the
x
and
y
directions
and we obtain the following system for the particle velocity
dV
p x
,
= -
c V
(
-
V
)
1
,
,
p x
f x
d t
(9.33)
dV
p y
,
= -
c V
+
c
1
p y
,
2
d t
where
c
1
and
c
2
are given by
6
πη
r
c
=
1
m
1
æ
ö
2
v
D Ñ
ç
χ
H y
.
ˆ
÷
p
g v
D
ρ
è
ø
2
p
c
=
+
2
m
m
Equation (9.33) forms a noncoupled, first-order, differential system. To solve nu-
merically for such a system is classical; almost any mathematical software possesses
an algorithm to solve such a system. However, (9.33) admits a closed form solution
for the velocity if we assume that
c
1
and
c
2
are constant [21]; this is not very restric-
tive since
c
1
is constant in an homogeneous fluid, and
c
2
also if the magnetic gradi-
ent is uniform—which is the case when the dimension of the magnet is sufficiently
large compare to the dimension of the channel. The closed form solution is then
-
c t
-
c t
V
=
V
e
+
V
[1
-
e
]
1
1
p x
,
p x
, 0
fx
(9.34)
c
2
1
-
c t
-
c t
V
=
V
e
1
+
[1
-
e
1
]
p y
,
p y
, 0
c
where the subscript zero corresponds to the initial values at
t
= 0. We have thus de-
rived an analytical expression for the particle velocity that depends on the values of
the magnetic force, drag coefficient and gravity. In (9.34) the applied forces appear
through the ratio
c
2
/
c
1
1
æ
ö
2
v
D Ñ
χ
H y
.
ˆ
+
g v
D
ρ
ç
÷
p
p
c
c
è
ø
2
2
1
=
(9.35)
r
6
πη
It is seen in (9.35) that
c
2
/ c
1
represents the ratio between the applied external (ver-
tical) forces and the hydrodynamic (horizontal) drag force (
c
2
/ c
1
has the dimension
of a velocity). This ratio determines the trajectories. Two particles experiencing the
same ratio
c
2
/ c
1
and starting for the same point at inlet will follow the same trajec-
tory to some bias due to the Brownian motion.
We now advance a step further and search for a solution for the trajectory. We
have to solve the first-order differential system
