Shape Optimization of Swimming Sheets Page: 3 of 26
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Shape Optimization of Swimming Sheets 3
n
u = clt
t Pi
elastic membrane
- 4u +-vp =0
.i"nt mV.u=0 h(x)
substrate
---' u = c2t t
x=0 n x=W
(a) (b)
FIGURE 1. (a) Schematic of gastropod locomotion showing a thin viscous fluid sandwiched
between a periodically deforming flexible foot and a rigid substrate. (b) Idealized model depicting
steady state swimming of a sheet via wave propagation. This system is conveniently studied in
the reference frame in which the wave profile F1 remains stationary. In this frame, material
points on the sheet move tangent to F1 with constant speed ci and the wall moves with contant
speed c2.
Fi {(x, h(x))} (i.e. the shape of the foot) remains stationary in time. In this frame,
the wall moves with constant velocity c2 in the x-direction while the sheet moves tangent
to F1 with constant speed c1; thus, the sheet remains inextensible as it swims. Our goal
is to study the effect of the curve shape on the swimming speed and the power required
to swim, and to find optimal shapes that maximize speed or efficiency subject to given
constraints (e.g. holding fixed the membrane length L f0W 1+ h'(x)2 dx and fluid
volume A fj h(x) dx, where h(x) is the local film thickness and subscripts denote
partial derivatives). Since the optimal shapes turn out not to exhibit infinite slopes or
singularities, it does not appear necessary to pose the problem more generally to allow
the wave profile to overturn.
2.1. Swimming Speed and Power Dissipation
In the following sections, we will restrict our analysis to the long wavelength limit. How-
ever, before we consider this limit, it is useful to derive certain quantities and constraints
that are generally true for Stokes flow, independent of geometry. If ci, c2 E IR and h(x) is
a sufficiently smooth periodic function, there is a unique solution to the Stokes' equations
Vp = pV2u V u =0 (2.1)
subject to the boundary conditions shown in Figure 1. Here p denotes pressure, p is the
dynamic viscosity and u is the velocity field within the thin film. Given this solution, we
can calculate the relevant forces and powers required for steady state swimming in terms
of the stress tensor a = (Vu + VuT), namely
P(ci, c2) J t " an ds, F(ci, c2) -j ei " an ds = t " an ds (2.2)
where el is the unit vector in the direction of motion and t and n denote respectively
unit tangent and normal vectors to the membrane/fluid interface. Physically, c1P is the
power required to maintain the steady motion of the sheet, F is the x-component of the
net force exerted by the fluid on the sheet (which is the same as that exerted by the
bottom wall on the fluid), and c2F is the power required to maintain the steady motion
of the bottom wall (in the wave frame). Assuming the forces exerted by the sheet on the
fluid are internally generated (e.g. by the muscles in the snail or by an internal motor in
a mechanical crawler), we require F 0 in steady state. Since the Stokes equations are
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Wilkening, J. & Hosoi, A.E. Shape Optimization of Swimming Sheets, article, March 1, 2005; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc890103/m1/3/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.