A variety of swimming microorganisms, called ciliates, exploit the bending of
a large number of small and densely-packed organelles, termed cilia, in order
to propel themselves in a viscous fluid. We consider a spherical envelope model
for such ciliary locomotion where the dynamics of the individual cilia are
replaced by that of a continuous overlaying surface allowed to deform
tangentially to itself. Employing a variational approach, we determine
numerically the time-periodic deformation of such surface which leads to
low-Reynolds locomotion with minimum rate of energy dissipation (maximum
efficiency). Employing both Lagrangian and Eulerian points of views, we show
that in the optimal swimming stroke, individual cilia display weak asymmetric
beating, but that a significant symmetry-breaking occurs at the organism level,
with the whole surface deforming in a wave-like fashion reminiscent of
metachronal waves of biological cilia. This wave motion is analyzed using a
formal modal decomposition, is found to occur in the same direction as the
swimming direction, and is interpreted as due to a spatial distribution of
phase-differences in the kinematics of individual cilia. Using additional
constrained optimizations, as well as a constructed analytical ansatz, we
derive a complete optimization diagram where all swimming efficiencies,
swimming speeds, and amplitude of surface deformation can be reached, with the
mathematically optimal swimmer, of efficiency one half, being a singular limit.
Biologically, our work suggests therefore that metachronal waves may allow
cilia to propel cells forward while reducing the energy dissipated in the
surrounding fluid.Comment: 29 pages, 20 figure