  Colonies of the green alga $Volvox$ are spheres that swim through the beating
of pairs of flagella on their surface somatic cells. The somatic cells
themselves are mounted rigidly in a polymeric extracellular matrix, fixing the
orientation of the flagella so that they beat approximately in a meridional
plane, with axis of symmetry in the swimming direction, but with a roughly 15
degree azimuthal offset which results in the eponymous rotation of the colonies
about a body-fixed axis. Experiments on colonies held stationary on a
micropipette show that the beating pattern takes the form of a symplectic
metachronal wave (Brumley et al. (2012)). Here we extend the Lighthill/Blake
axisymmetric, Stokes-flow model of a free-swimming spherical squirmer
(Lighthill (1952), Blake (1971b)) to include azimuthal swirl. The measured
kinematics of the metachronal wave for 60 different colonies are used to
calculate the coefficients in the eigenfunction expansions and hence predict
the mean swimming speeds and rotation rates, proportional to the square of the
beating amplitude, as functions of colony radius. As a test of the squirmer
model, the results are compared with measurements (Drescher et al. (2009)) of
the mean swimming speeds and angular velocities of a different set of 220
colonies, also given as functions of colony radius. The predicted variation
with radius is qualitatively correct, but the model underestimates both the
mean swimming speed and the mean angular velocity unless the amplitude of the
flagellar beat is taken to be larger than previously thought. The reasons for
this discrepancy are discussed.
