  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. 102, 168101 (2009)] 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
$\theta$-dependent tangential velocity being imposed on the spherical surface
(radius $a$; $\theta$ is the polar angle), a time-independent and uniform
tangential shear stress is applied to the fluid on a sphere of radius
$(1+\epsilon)a, \epsilon \ll 1$, where $\epsilon a$ represents the length of
the flagella. The fluid must satisfy the no-slip condition on the sphere at
radius $a$. In addition to the shear stress, the motions depend on two
dimensionless parameters that describe the effect of gravity on a colony:
$F_g$, proportional to the ratio of the sedimentation speed of a non-swimming
colony to its swimming speed, and $G_{bh}$, that represents the fact that
colonies are bottom-heavy...
