Page 469 - Handbook of Biomechatronics

Bioinspired and Biomimetic Micro-Robotics for Therapeutic Applications 463

The wave patterns are also found to be of different external and inter-
nal stimuli. The wave parameters are not necessarily homogeneous
throughout the entire swimming action. This is caused by either viscosity
or sporadic changes in direction of swimming (Lighthill, 1976; Brennen
and Winet, 1977; Brokaw, 1965; Gibbons and Gibbons, 1980). The
reader is encouraged to study the references for the details of changes
in wave patterns. However, one particular stimulus important for all
intents and purposes is the wall effects. Flow fields with the presence
of a solid boundary nearby the micro-swimmer will induce attenuated
or amplified shear stress on the head or the tail. This will inevitably affect
the wave pattern and the swimming behavior (Brennen and Winet, 1977;
Fauci and McDonald, 1995; Lauga et al., 2006). Unfortunately, it is not
straightforward to determine the overall effect; the local viscous stress is
either inversely proportional to the local proximity to the solid boundary,
h (m), or linearly proportional to the logarithm of 1/h. Only after taking
the integral over the entire flagellumand summingupwiththeeffecton
the head, one can predict the instantaneous influence (Brennen and
Winet, 1977; Lauga et al., 2006)which leadsustothemathematical
modeling to the physics of swimming in micro-realm via wave
propagation.

2.2 Physics of Swimming in Micro-Realm
Normally the most powerful approach, to solve the hydrodynamics of a
complex geometry going through six degrees of freedom (6 DOF) rigid-
body motion while it is carrying out wave propagation on only a certain por-
tion of its surface area, is to summon Navier-Stokes equations with proper
boundary conditions and appropriate meshing. These set of equations artic-
ulate the conservation of momentum and mass, respectively, but for flowing
matter instead of for objects experiencing rigid-body motion:

ρ ∂U ð 2 (1)
∂t + U rÞU ρg ¼ rp + μr U
ρr U ¼ 0 (2)
2 3
where t (s) denotes time, g (m/s ) is the gravity vector, ρ (kg/m ) is the den-
sity, μ (Pas) denotes the dynamic viscosity, p (Pa) is the static pressure of the
fluid, and U (m/s) signifies the flow field vector. It should be noted that these
equations are valid for incompressible liquids. The left-hand side of the first
equation is responsible for the acceleration and inertial effects of the fluid. It
is important to acknowledge that the much-dreaded nonlinearity of the