Page 467 - Handbook of Biomechatronics

Bioinspired and Biomimetic Micro-Robotics for Therapeutic Applications 461

viscous drag that reigns over the momentum transfer from arms to the liquid
at each cycle as articulated with the “Scallop theorem”(Purcell, 1977).
It has been observed that, in order to overcome this energy-loss obstacle,
a majority of natural micro-swimmers employ microstructures and chemical
motors to generate wave propagation, either planar or helical, along with
slender structures hanging from their cell membrane. Thus, they break spa-
tial and temporal symmetry to harness forward thrust from viscous friction.
That friction is present as long as the wave is propagating, meaning that once
the bacterium or sperm cell ceases propagation of waves the swimming
motion is supposed to stop instantaneously (Purcell, 1977; Lighthill, 1976).
The very first comprehensive theoretical work exploring the physics to
understand this phenomenon was presented by Sir Taylor in 1951 (Taylor,
2
1951). He introduced the particular stream function, Ψ (m /s); a mathemat-
ical function representing the flow field around a moving object in viscous
domains, as depicted in Fig. 1, for simple two-dimensional (2D) wave prop-
agation and combined it with force- and torque-free swimming concept to
demonstrate the asymptotic relationship between wave propagation and
swimming efficiency. In a sense, the stream function represents the diffusion
of flow field within the liquid by representing it via a cross-sectional view of
the bulk. From that point onward, in over 60 years, the efforts mounted
toward the very first computer-controlled artificial cybernetic micro-
swimmer, towing a red blood cell, presented by Dreyfus et al. (2005).
The body of work presented in this subsection can be grouped into the
following categories: observations of bacteria and spermatozoa, mathemat-
ical models explaining the swimming behavior based on observations,

Fig. 1 The flow field, that is, black lines, around a stream-lined object immersed in vis-
2
cous flow. The black lines can be represented by a mathematical function, Ψ (m /s), from
which the local velocity could be deduced, for example, the x-component is obtained by
u x ¼dΨ/dy and the y-component is given by u y ¼ dΨ/dx, in Cartesian coordinates. The
arrows show the flow direction of the far field.