Page 477 - Handbook of Biomechatronics

Bioinspired and Biomimetic Micro-Robotics for Therapeutic Applications 471

Another useful phenomenon to characterize is the ratio of convective
displacement to diffusive displacement which is given by Peclet number,
Pe (Kasyap et al., 2014). This number indicates the effectiveness of a
micro-swimmer towing a particle in the viscous medium, and the larger
it is the more beneficial to use a micro-swimmer to deliver a cargo to a
predefined destination. It is important to note that, indeed, there is a theo-
retical limit to Pe number because Re number cannot exceed unity if Scallop
theorem and Stokesian-flow assumption are expected to be valid. The Pe
number is given as
UL
Pe ¼ (10)
D
2
where D (m /s) stands for the diffusivity coefficient of the cargo in question.
Diffusivity is a term related to the mass transfer of species and very small
particles, and it is dependent on temperature and the dynamic viscosity
of the liquid medium (Probstein, 2003) where the micro-swimmer is
submerged in.
The last dimensionless number we will review is the Deborah number,
De (Teran et al., 2010; Liu et al., 2011). De provides qualitative informa-
tion on the physical condition of the flow field: De number being smaller
than unity indicates that the relaxation time of the liquid is short enough so
the applied stress is conducted without spatial or temporal attenuation in
the liquid bulk, thus the shear rate is not time dependent. On the other
hand, De number larger than unity signifies rheological behavior where
the stress is not propagating in the liquid with isotropic fashion (Guyon
et al., 2015) leading to spatially and temporally inhomogenous stress ten-
2
sor, σ (N/m ), and time-dependent viscous shear stress. Also, there exists
an optimum De for planar wave propagation: undulating flagellum is sup-
posed to be more efficient in non-Newtonian fluids up to De 3after
which propulsive thrust gradually decreases for a micro-swimmer in the
non-Newtonian environment (Teran et al., 2010). The flagellum can be
elastic or rigid, and still, propagate helical waves under which condition
again an optimum expected to present itself (Liuetal.,2011). The De
number is given as

De ¼ δ t f (11)
where δ t (s) is the relaxation time of the liquid which can be a polymer solu-
tion or a Boger fluid which has a shear rate-dependent dynamic viscosity
(Teran et al., 2010; Liu et al., 2011). This also means that, if the other