Page 203 - Computational Modeling in Biomedical Engineering and Medical Physics

192 Computational Modeling in Biomedical Engineering and Medical Physics

The vessel volume is the only magnetizable medium. In the linear limit, its magne-
tization is
Þ γδH
M ff 5 γ arctan δHð ð6:11Þ
where γ [A/m] and δ [m/A] are empiric constants (Morega et al., 2015; S˘ av˘ astru,
2016). The magnetic circuit, magnetic flux and constitutive laws yield the mathemati-
cal model formulated in the magnetic vector potential, A,
21 Þ 5 J ;

e
0
r 3 μ r 3 A 2 M 2 σu 3 r 3 Að ð6:12Þ
e
where J is the external electrical current density (inside the coil only), the magnetic
field source, u the MAF velocity, σ the electrical conductivity of the MAF. Magnetic
insulation boundary condition closes the problem.
An order of magnitude analysis of Eq. (6.12) suggests the balance

A 0 M 0
; ; σμ U 0 LJ 0 BJ 0 :
μ L 2 L 0 ð6:13Þ
0
6
2
Using as reference quantities J 0 B 10 A/m (electrical current density scale), L B 0.1 m
(length scale), H 0 B J 0 L, A 0 B μ 0 LH 0 , M 0 B γδH 0 , U 0 B 0.1 m/s, rel. Eq. (6.13) yields
O 1ðÞ; O 1ðÞ; O 10 27 BO 1ðÞ; ð6:14Þ

where O( ) means “order of magnitude.” The transport term is much smaller than
the others, thus it can be discarded, leading to an important reduction in complexity
for reason that the magnetic field problem may be solved independently of the flow,
only once, in the beginning.
The arterial section of interest here is of “resistance” type (Feijóo, 2000) and MAF
is a Newtonian fluid. The quasisteady, incompressible, laminar, flow is modeled by
Eqs. (6.1) and (6.2), where f 5 f mg 5 μ MUrÞH is the magnetic body force. The
0 ð
boundary conditions are zero at the walls, uniform periodic velocity profile at
the entrance (Fig. 6.16) and uniform periodic pressure for the exit (Fig. 6.17). The
morphologies and the significance of these profiles are described in Morega et al.
(2012). For convenience, the leading pulse is set to 60 bpm.
The electromagnet is coreless, essentially a planar coil made of concentric, with
square cross-section turns, which can be powered independently. Fig. 6.18 (Morega
et al., 2015) shows the horizontal (left) and vertical (right) magnetic forces at the upper
side of the vessel wall that is closer to coil, for several powering schemes.
The grey blocks designate the currents entering the plane and the black ones show
the currents exiting the plane. In general, the extraction component exhibits maximum
values about the coil axis, whereas the streamwise component has maxima by the coil