Biomedical Engineering Reference
In-Depth Information
ext
r
2
-
2
H
=
H
cos
θ
(1
+ L
a
r
)
0
(9.25)
ext
2
-
2
H
=
H
sin ( 1
θ
- + L
a
r
)
0
θ
µ
-
µ
(
)
w
f
L =
with
and
μ
w
,
μ
f
are respectively the magnetic permeability of the rod
µ
+
µ
(
)
w
f
and the fluid. The magnetic force is then
æ
ö
é
2
ù
a
�
2
L
+
cos2
θ
a
ê
ú
ç
÷
2
2
F
= -
2
µ χ χ
(
-
)
v
H
L
(9.26)
r
m
0
p
f
p
ë
û
ç
÷
0
3
r
ç
÷
è
sin2
θ
ø
Note the similarity between the two expressions of the magnetic forces (9.24) and
(9.26). In both cases, in the vicinity of the rod, there are two attraction zones aligned
with the external field (for
q
close to zero or to
p
) and two repulsion zones in the
direction perpendicular to the external field (for
q
close to
p
/2 and -
p
/2). By taking
the ratio between radial and azimuthal components of the force in either expression
(9.24) or (9.26), it can be deduced that attraction forces are larger than repulsion
forces. We show the magnetic field force in the vicinity of the rod in Figure 9.10.
9.5.3 Trajectories (Carrier Fluid at Rest)
We investigate first the case where the fluid is at rest. Using the algorithm for the
calculation of trajectories described in Section 9.9, and the magnetic forces calcu-
lated in the preceding section, we find that the microparticles migrate from the two
repulsion zones towards the two attraction zones, as shown in Figure 9.11. Theo-
retically, the repulsion and attraction regions extend to infinity, but the magnetic
Figure 9.10
Magnetic force field around a cylindrical rod placed in a uniform magnetic field
(directed from left to right) showing to attraction zones and two repulsion zones.
