Page 509 - Handbook of Biomechatronics

Bioinspired and Biomimetic Micro-Robotics for Therapeutic Applications 503

F m tðÞ ð M rÞB tðÞ
¼ (23)
T m tðÞ M B tðÞ
if we assume that center of mass and volumetric center of the magnetic mate-
rial coincide. The sixth term on both rows, F e (t) (N) and T e (t) (Nm), denote
the electrostatic effects, that is, if there is a charge potential, between the
micro-swimmer and boundaries nearby. This effect can be simply represen-
ted as

2 2 3
q

F e tðÞ n wall tðÞ
2
¼ 4πεh tðÞ 5 (24)
4
T e tðÞ c
p tðÞ F e tðÞ
where q (C) denotes the charge difference and h c (t) (m) is the proximity of
the geometric center of the charged surfaces to the nearest boundary (Sadelli
et al., 2017). Next effect to consider arises due to gravitational attraction and
buoyancy which some bacteria species known to exploit to move simply up
and down (Khalil et al., 2017a; Sadelli et al., 2017; Ning and Cannon, 1998):

F g tðÞ m eff g tðÞ
¼ (25)
T g tðÞ t ðÞ F g tðÞ
cov
p
where m eff (kg) is the effective mass of the fully submerged micro-swimmer
after buoyancy effect is included and p cov (t) (m) stands for the position vec-
tor of volumetric center with respect to the center of mass of the micro-
swimmer. In other words, it is assumed that the micro-swimmer is not a
homogeneously distributed construct. It should be acknowledged that grav-
ity, although the volumetric force acting on the mass of the micro-swimmer,
cannot be lightly eliminated from the equations because gravitational accel-
eration is always present unless the experiment is conducted in an environ-
ment under microgravity conditions. Fig. 14 shows the effect of gravitational
pull inside confined spaces for an artificial micro-swimmer of dimensions
that of the Escherichia coli minicell (Chattopadhyay and Wu, 2009). Further-
more, the reader should keep in mind that the gravitational attraction is writ-
ten in the frame of reference of the micro-swimmer that is experiencing
complex rotations in the lab frame. Thus, the time-dependent nature of
2
g(t) (m/s ) is justified.
The final term in the equation of motion gives us the inertial effects
within the flow field, which are present as long as the wave propagation
exists, owing to the fact that the head most often follows a helical trajectory,
that is, an oscillating motion, due to geometric aberrations or hydrodynamic