Biomedical Engineering Reference
In-Depth Information
Again, if the concentration of magnetic beads is not too important, the external
magnetic field is not affected significantly by the presence of the magnetic beads.
In such a case, the magnetic field must be calculated before attempting the calcula-
tion of the trajectories. Under this assumption, the magnetic force on a particle of
volume
v
P
is given by
�
χ
æ
1
2
ö
2
F
=
µ
v
D Ñ
ç
H
(9.15)
÷
mag
p
0
è
ø
where D
c
is the difference of magnetic susceptibility between a particle and the
fluid,
H
the magnetic field. Equation (9.15) indicates that the magnetic force is
aligned with the direction of the gradient of the square of the magnetic field. Finally,
the gravity term is simply given by
�
F
=
g v
D
ρ
y
ˆ
(9.16)
grav
p
where
g
is the acceleration of gravity,
v
p
the volume of the particle,
y
the vertical
unit vector (oriented downwards) and D
r
the difference between the volumic mass
of the particle and that of the liquid. After substitution of (9.14), (9.15), and (9.16)
in (9.13), one obtains the equation for the particles velocity
�
� �
dV
æ
1
ö
p
2
m
=
µ
v
D Ñ
χ
H
-
6
πη
r V V
(
-
)
+
g v
D
ρ
y
ˆ
(9.17)
ç
÷
0
p
h
p
f
p
è
ø
d t
2
x
and
y
coordinates of the particle at a given time are linked to the velocity by the
relations
d x
=
V
p x
,
d t
(9.18)
d y
=
V
p y
,
d t
Equations (9.17) and (9.18) define the particle trajectory. It is very seldom that
they can be solved analytically, but as we will see, it is always interesting to spend
some time investigating if an analytical solution may exist—even for the price of
some simplification. An example will be given in Section 9.10. Most of the time, a
numerical approach is required. Different methods can be used such as Runge Kutta
or predictor-corrector. We indicate in Section 9.9 a very simple first-order predic-
tor-corrector method that is very efficient when the velocity of the carrier fluid is
sufficiently low.
9.5 Example of a Ferromagnetic Rod
A very didactic example of trajectories of magnetic microparticles is that of
the ferromagnetic rod. In this particular case, a closed form solution exists for
the magnetic field and the trajectories may be calculated easily [7, 10]. Thus,
it is a good test for the verification of numerical models. Moreover, it bears the
