Energetics of synchronized states in three-dimensional beating flagella

During collective locomotion, beating flagella of spermatozoa interact hydrodynamically and are observed experimentally to synchronize. G. I. Taylor used a small-amplitude two-dimensional sheet model to show that the rate at which swimmers do work against the fluid is minimal for in-phase beating. We use a semianalytical approach based on hydrodynamic reflections to extend these results to the small-amplitude three-dimensional beating of infinite flagellar filaments. We first consider a configuration of two parallel filaments. In the case where the beating of both flagella occurs in the same plane as that defined by their axis, in-phase beating is found to lead to an overall minimum of energy dissipation, while opposite-phase leads to a maximum. If we allow the orientation of the beating planes to vary, we find that the minimum of energy dissipation is obtained for either the in-phase or opposite-phase conformation, in a manner that depends on the flagella orientation and their relative distance. We further characterize numerically the set of optimal relative orientations. Quantitatively analogous results are obtained using a simple model based on the beating of two spheres interacting hydrodynamically in the far field. Exploiting the linearity of Stokes equation, we then extend our results to the case of three beating flagella in an aligned and triangular conformation. Consistent with Taylor's two-dimensional work, our results suggest that, from a hydrodynamic standpoint, it is more energetically favorable for spermatozoa with three-dimensional flagella to swim close to each other and with synchronized, parallel, in-phase beating.