Page 471 - Handbook of Biomechatronics

Bioinspired and Biomimetic Micro-Robotics for Therapeutic Applications 465

momentumequationgoverningtheviscousdomain,alsoknownastheStokes
equations for incompressible creeping flow conditions (Lighthill, 1976):
2
0 ¼ rp + μr U + f (3)
The approximated local flow field is usually represented by stream functions,
Ψ, such as the one Sir Taylor made use of (Taylor, 1951). The stream func-
tion is a geometric representation of the path of fluid packets around a sta-
tionary or moving object. The reason the term “fluid packet” is preferred is to
look at the flow field at a safe distance, with a zoomed enough view, while
staying clear of the quantum limit where it is possible to observe the motion
of individual molecules. In such a case, usual conservation equations do not
apply as the continuity assumption is violated. Hence, when a singularity or
stream function is modeled, the volume of interest is much larger than a sin-
gle molecule and all intermolecular interactions are lumped as bulk proper-
ties of flowing matter at an arbitrary location. For instance, Sir Lighthill
(Lighthill, 1976) articulated the “Stokeslet velocity distribution” associated with
the dipole force, f, as follows:

1
f 2 f
U ¼rΨ ¼ + r rr (4)
f
6πμr f 4 6πμr f
where r f (m) denotes the radial position from the singularity.
Similarly, it is possible to come up with a “rotlet function” that represents
the flow field based on the local rotation of a specific point. The magnitude
of the Stokeslet function is inversely proportional to the distance to the
selected point of analysis whereas the magnitude of the rotlet function is
inversely proportional to the cube of the same distance. The overall hydro-
dynamic force acting on the flagellum is found by the integration of forces,
f(s,t), represented by the resultant singularity distribution throughout the
flagellum, s,atany given instance, t,(Higdon, 1979; Johnson and Brokaw,
1979). The intricacies of modeling the dipoles and obtaining the solution
of the Stokes equation are left to the reader.
Once the local force distribution is obtained, it is possible to reevaluate
the local force-velocity relation in terms of local resistance force coefficients,
which brings us to the next approach known as the RFT. RFT is a
“suboptimal solution” to the SBT and allows us to model the local force in
2
the form of f(s,t)¼Cu, where C (Ns/m ) denotes the diagonal matrix of
local resistive force coefficients written in Frenet-Serret coordinates, that
is, c t , c n , and c b , and u (m/s) represent the local velocity vector of the flagellum