3 Physical background
A B C D
Figure 3.3: Time-sequence of instantaneous velocity elds around a sphere that is moving
with periodic velocity variations. Insets show the sphere velocity (solid
line, black), which varies with time in a sinusoidal pattern around an average
velocity U (dashed line, black). The instantaneous velocities are shown
as red dots. The colour maps show the normalised velocity magnitude v=U.
this instant. At a very low Reynolds number the ow is instantly dissipated such that
there is no trace of the ow pattern from the last instant. History does not matter.
This means that we can construct models of ows around moving swimmers, where the
time-dependence of the ow pattern lies purely in the kinematics of the swimmer. The
kinematics are instant by instant translated into varying boundary conditions that lead
to dierent ow pictures. Figure 3.3 shows an example of a sphere which is moving
forward and backward with a sinusoidal pattern. The instantaneous sphere velocities
lead to dierent instantaneous ows.
Another general feature of the Stokes equations is known as kinematic reversibility,
which also follows from linearity and time-independence. This leads to several surprising
conclusions Stone and Duprat, 2012; Pak and Lauga, 2012. Assume that we
produce a low Reynolds number ow with a certain forcing, e.g. certain boundary
velocities on a swimmer surface. Now if the forcing is reversed (or doubled), kinematic
reversibility means that we will get exactly reversed (or doubled) ow velocities. In
gure 3.3 the forcing is the sphere velocity that changes in a sinusoidal pattern. This
simply leads the ow velocities at each point to change in a sinusoidal pattern. Panel
(C) and (D) have exactly reversed forcings and thus have exactly reversed ow velocities
at every point. Kinematic reversibility led Purcell to formulate the famous scallop
theorem, in which he states that a scallop, which swims (usually at Re > 1) by opening
and closing its rigid shell, would not be able to propel itself at a low Reynolds number
Purcell, 1977. Such a periodic shape change is classied as reciprocal motion. At a
higher Reynolds number, where inertia is important, it matters for the translation how
fast the shell is closed and opened, i.e., how long the opening and closing time is, while
this does not matter in the viscous regime.
We formally dene swimming as continuous translation or rotation due to a periodic
deformation of the body. Only certain deformations lead to non-reciprocal motion and
thus eective swimming. Those deformations are generally dierent than for higher
Reynolds number swimmers. One can not swim in Stokes ow, if in the conguration
space a closed cycle does not enclose a nite area Purcell, 1977; Lauga, 2011.