Page 480 - Handbook of Biomechatronics

474 Ahmet Fatih Tabak

nondimensional form, that is, dimensions being scaled with a selected char-
acteristic length, so results will be applicable to a wide design space as long as
Re number is smaller than unity.
For instance, Phan-Thien et al. (1987) presented the optimization of a
bacteria-like micro-swimmer, with a single flagellum, based on BEM anal-
ysis. Authors, first, swept the design space for the wave propagation param-
eters, that is, wavelength and wave amplitude, of the tail while attached to a
spherical head. Then, they studied the effect of the shape of the head while
the wave geometry is kept constant. The efficiency definition they used is:
6πμRU 2
η ¼ (13)
Σ τ UÞ
ð
tail
where the denominator stands for the total power required to sustain the
wave propagation along the tail, which is in the definitive form for all pure
hydrodynamic intents and purposes. Moreover, R (m) stands for the diam-
eter of the head. Here, U (m/s) denotes the total velocity vector at each node
along the boundary of the tail as those nodes also coincide with the spatial
2
limits of the fluid domain, and τ (N/m ) signifies the hydrodynamic force
vector associated with each respective node calculated as τ¼σn with the
surface normal of each individual node, n. Thus, the total power is calculated
by summing the hydrodynamic power spent on the nodes at the surface in
contact with the liquid.
A very similar study is carried out by Shum et al. (2010), however, exten-
ding the analysis to the effect of distance of a nearby solid boundary, again
using BEM analysis. The design space analyzed by the authors is the same as
the previous example; however, surface plots showing the effect of simul-
taneous variation of tail and head geometries are presented for the efficiency
instead. Furthermore, the stability of swimming motion, that is, steadiness of
the swimming direction under the influence of hydrodynamic interactions
with the nearby boundary, is investigated by the representation of stable and
unstable geometric combinations. In addition, authors discussed that
efficiency could also be defined in terms of motor torque since the helical
flagellum is being rotated with the bacterial motor present in natural
micro-swimmers. The new definition is given in the form of:

2
8πμR Ω body
η ¼ ð (14)
ð r τÞda
tail