Page 470 - Handbook of Biomechatronics

464 Ahmet Fatih Tabak

equation happens to be due to the (U r)U term which also dictates that the
acceleration is not necessarily a time-dependent phenomenon in the
explicit form.
However, taking into account the fact that computers and the numerical
methods being used in abundance today were not simply available at the
time when the problem was first studied, mathematicians had to avoid tack-
ling this highly nonlinear equation as a whole, and came up with ingenious
simplified models to represent the problem without higher-order effects
which are expected to be insignificant due to dimensions and the effective vis-
cosity in micro-realm. In other words, one aspect that engineers try to elim-
inate from macroscale systems whenever convenient, that is, friction, gave
rise to a completely new and much simpler modeling approach for micro-
swimmers. Effective viscosity can be explained best by the analogy of “real
feel” of the temperature due to humidity levels, and it is directly related to the
famous Reynolds number (Purcell, 1977), which will be discussed at length
later. Although this simplification will be of great help, it will be evident later
that including all possible physical stimuli acting on the micro-swimmer
renders the modeling laborious and delicate, nevertheless.

2.2.1 Mathematical Models for Hydrodynamics of Flagellar Motion
From the mathematical and, arguably, the chronological points of view, the
first approach to explore is the slender-body theory (SBT). In 1953, Hancock
talked about singularities representing the flow fields around slender objects
and relatively simple geometries in order to explain the fluid dynamics of the
swimmers (Hancock, 1953). Although it was in 1955 that the resistive force
theory (RFT) approach was fully articulated with the use of local force coef-
ficientsintroducedbyGrayandHancock,asamathematicalconclusionof the
previous analysis followed by a series of simplifications and assumptions (Gray
andHancock,1955), thecore oftheSBTmethodlingeredin theliterature for
22 yearsbefore SirLighthill brilliantly took it to the next step (Lighthill, 1976)
while referring to the paper published by Hancock (1953) on the distribution
of Stokeslets along a beating flagellum. SBT employs “Stokeslet functions” that
represent an approximation for the flow field invoked by a singularity, that is,
dipoleforce,f,whicharticulatesthephysicalstimulidrivingtheresultantlocal
flow field, U. Hence, the flow field can be modeled in terms of that particular
singularity associated with the local structural deformation and the dynamic

viscosity, μ, of the surrounding liquid. The solution could be obtained after f
is introduced as an external force term in the simplified conservation of