Page 475 - Handbook of Biomechatronics

Bioinspired and Biomimetic Micro-Robotics for Therapeutic Applications 469

large-scale mock-ups and with CFD simulations as building and tracking a
self-sustaining and autonomous swimmer in actual micro dimensions are not
always trivial.
The most important dimensionless number is the Reynolds number (Re)
(Purcell, 1977). It provides information on the ratio of inertial forces to shear
forces. If Re number is smaller than unity, Navier-Stokes equations could be
simplified to Stokes equations. This is known as the Stokesian-flow assump-
tion where RFT and SBT methods are also subject to this condition for
validity. It is important to note that the smaller the Re number is, the more
accurate the Stokesian-flow approach will be. The Re number is given as

2
ρUL ρL f
Re ¼ ¼ (5)
μ μ
where L (m) is the characteristic length of the micro-swimmer, such as the
diameter of the body, and U (m/s) is the swimming velocity of the center of
mass of the micro-swimmer. The letter f signifies the frequency, in Hz, of
the wave propagation should the characteristic length is selected to be the
wavelength, λ (m). So, the characteristic length and the characteristic veloc-
ity remains subjective depending on the analysis. Consequently, different Re
numbers can be defined and used. Re number is virtually the most preferred
dimensionless quantity to prove the validity of the experiments and analysis,
numerical or theoretical, in the literature to this date. Moreover, the recip-
rocal of Re number, that is, 1/Re, gives the effective viscosity of the flow
field; as Re number attains smaller values, the effective viscosity of the flow
field increases. Another number that is dependent on velocities is the
Strouhal number (St) (Lovalenti and Brady, 1993; Ido et al., 2016):
Although the St number is associated with high Re number flows, it can
be used as an indication of efficiency. Because, while investigating within
a design space, relatively smaller St number means loss of useful momentum
in terms of “parasitic circulations” within the bulk of the flow. The St number
is given as
fL
St ¼ (6)
U
Next, we study the Sperm number (Sp) (Dreyfus et al., 2005; Wiggins and
Goldstein, 1998). Sp represents the ratio of viscous forces acting on the fla-
gellum to the balancing structural stresses. It is also the ratio of characteristic
length to the “viscous penetration length,” L v (m), when the flagellum is undu-
lating. It defines the shape of the flagellum that will give the optimum