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Magnetic drug targeting 203

where Re d 5 ðU 0 d=μ=ρÞ 5 ðU 0 d=νÞ is the Reynolds group based on d,and a new
2
group emerges, W mg;0 =ρU , which is the ratio of (potential) magnetic energy den-
0
sity to kinetic energy density group. The usage of d as scalehereisrelevantbecause
the gradient of the magnetic energy density (magnetic force) orthogonal to the
flow is important—the magnetic field acts to extract the MD from the vessel into
the ROI.
In fact, this energy group relates the contribution of the magnetic field to the MD
transport with respect to the kinematic energy of the flow (provided by the “pump”-
ing device) as leading term. Furthermore, the magnetic field may accelerate (upstream
the magnet region) or decelerate (downstream it) the flow, but it always attracts the
MNPs towards the magnet. As the magnetic body forces are the gradient of the mag-
netic energy density, this group may show off either

W mg;0 =d F mg;0;y W mg;0 =L L F mg;0;x
5 ; or 5 AR ð6:29Þ
2
2
ρU =d F mech ρU =d d F mech
0 0
where F mg;0;x and F mg;0;y are the scales for the streamwise and orthogonal to the stream,
respectively, magnetic body forces, F mech is the scale for the streamwise mechanical
body force, and AR is the geometric aspect ratio of the vessel.
2 , is essential to the MDT design, as it allows sizing the
Theenergygroup, W mg;0
ρU 0
magnetic field source (its energy), either PM or electromagnet, for a specific situa-
tion. It becomes then a problem of design to position and optimize the magnet
(Section 6.5) with respect to the ROI to achieve the required magnetomotive
force. The magnetic to mechanical forces group Eq. (6.29) may be derived from
the energy group to outline the order of magnitude of either the streamwise or the
orthogonal to the flow forces (in the above U is the scale for the streamwise veloc-
ity, but the scale for the orthogonal velocity may be used), and AR is the conver-
sion factor.
Two limiting cases may be noticed: (1) when the flow concurs with the magnetic
field lines the magnetic has no “extraction” effect and (2) if the magnetic field lines
are orthogonal to the flow, then the magnetic field has an drawing out effect. The
streamwise action of the magnetic field is discussed next.
Initially the fluid inside the vessel segment in the model has no magnetic proper-
ties, but it eventually changes as the magnetizable species (MD) fills the vessel. This
change in the magnetic susceptivity depends on mass concentration, and it is presented
here as χ(t) 5 c(t)/C b . This form creates couplings between the magnetic field, flow,
and species transfer such that Eqs. (6.21) (6.23) have to be solved simultaneously. A
segregated dynamic solver is recommended.
Fig. 6.27 shows the concentration profiles for several moments, and the flow field
at t 5 16 s. The computational domain is similar to that in Fig. 6.10, the PM has