  Synchronisation is often observed in the swimming of flagellated cells,
either for multiple appendages on the same organism or between the flagella of
nearby cells. Beating cilia are also seen to synchronise their dynamics. In
1951, Taylor showed that the observed in-phase beating of the flagella of
co-swimming spermatozoa was consistent with minimisation of the energy
dissipated in the surrounding fluid. Here we revisit Taylor's hypothesis for
three models of flagella and cilia: (1) Taylor's waving sheets with both
longitudinal and transverse modes, as relevant for flexible flagella; (2)
spheres orbiting above a no-slip surface to model interacting flexible cilia;
and (3) whirling rods above a no-slip surface to address the interaction of
nodal cilia. By calculating the flow fields explicitly, we show that the rate
of working of the model flagella or cilia is minimised in our three models for
(1) a phase difference depending on the separation of the sheets and precise
waving kinematics; (2) in-phase or opposite-phase motion depending on the
relative position and orientation of the spheres; and (3) in-phase whirling of
the rods. These results will be useful in future models probing the dynamics of
synchronisation in these setups.
