Page 476 - Handbook of Biomechatronics

470 Ahmet Fatih Tabak

momentum transfer to the fluid. If the Sp number is smaller than unity,
reciprocation of the elastic flagellum is converging to the limit of a rigid slen-
der object. As the Sp number becomes very large, the undulation is not
effective as the flagellum is virtually indistinguishable from the surrounding
liquid at such conditions. Therefore, there exists a distinct Sp number with
maximum hydrodynamic thrust for the selected natural micro-swimmer or
the tailored artificial flagellum. The Sp number is given as

L L
Sp ¼ ¼ 0:25 (7)
L v ð κ=c n ωÞ
where к (J/m) is the bending rigidity of the flagellum and ω (rad/s) is the
angular frequency of the actuation. Next comes the magnetoelastic number,
Mn e (Dreyfus et al., 2005; Roper et al., 2006). This number is also known as
the “dimensionless magnetic strength” or square of the ratio of characteristic
length to the “magnetoelastic persistence length,” and it becomes important if
the elastic flagellum has significant magnetic properties and reacts to the
external magnetic field: as Mn e increases, the forward velocity is also
expected to increase; and vice versa. However, distinct regions with structural
stability are observed, and it has been reported that buckling is expected to
2
occur with Mn e ¼π /4 (Roper et al., 2006). The Mn e number is given as
2
2π rBLÞ χ χ + χ χ =4
ð
t
t n
n
Mn e ¼ (8)
3μ κ ð 1 χ =6Þ 1+ χ =12ð n Þ
0
t
where χ denotes the direction-dependent susceptibility given by respective
subscripts, r (m) is the radius of the tail given that it has a somewhat homo-
geneous magnetic quality, B (T) is the magnitude of the applied magnetic
field-density, and μ 0 (H/m) is the permeability.
In order to quantify the magneto-hydrodynamic behavior of the micro-
swimmer, we use the Mason number, Mn h , which is the ratio of the viscous
torque to magnetic torque, and akin to magnetoelastic number, structural
stability is only observed under certain values of Mn h after exceeding which
the artificial flagellum is expected to rupture (Ido et al., 2016). The Mn h
number is given as
32μω
Mn h ¼ (9)
2 2
jj
0
μ χ H
where H (A/m) is the magnetic field-strength vector acting on the magnetic
flagellum or at the joint between the magnetic head and the flagellum.