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Zusammenfassung:
Biflagellate algal cells of the genus Volvox form spherical colonies
that propel themselves, vertically upwards in still fluid, by the
coordinated beating of thousands of flagella, that also cause the
colonies to rotate about their vertical axes. When they are swimming in
a chamber of finite depth, pairs (or more) of Volvox carteri colonies
were observed by Drescher et al. (Phys. Rev. Lett., vol. 102, 2009,
168101) to exhibit hydrodynamic bound states when they are close to a
rigid horizontal boundary. When the boundary is above, the colonies are
attracted to each other and orbit around each other in a 'waltz'; when
the boundary is below they perform more complex 'minuet' motions. These
dances are simulated in the present paper, using a novel 'spherical
squirmer' model of a colony in which, instead of a time-independent but
-dependent tangential velocity being imposed on the spherical surface
(radius ; is the polar angle), a time-independent and uniform tangential
shear stress is applied to the fluid on a sphere of radius , where
represents the length of the flagella. The fluid must satisfy the
no-slip condition on the sphere at radius . In addition to the shear
stress, the motions depend on two dimensionless parameters that describe
the effect of gravity on a colony: , proportional to the ratio of the
sedimentation speed of a non-swimming colony to its swimming speed, and
, that represents the fact that colonies are bottom heavy; is the ratio
of the time scale to swim a distance equal to the radius, to the time
scale for gravitational reorientation of the colony's axis to the
vertical when it is disturbed. In addition to reproducing both of the
dancing modes, the simulations are able to determine values of and for
which they are stable (or not); there is reasonable agreement with the
experiments. A far-field model for the minuet motions is also shown to
have qualitative agreement, but does not describe some features that are
reproduced in the full simulations.
