It has long been observed that an ensemble of flagella or cilia can synchronise their periodic beating. A long-standing hypothesis is that hydrodynamic interactions may provide a systematic route towards synchronisation. In this twelfth chapter we focus on the role played by fluid mechanics and highlight how interactions through the viscous fluid may lead to synchronised beating consistent with experiments. We start by the case where flagella, or cilia, are anchored on a surface or on an organism. We use a minimal model of spheres undergoing cyclic motion above a surface and interacting hydrodynamically in the far field. We show that in-phase synchronisation can be achieved if the spheres move along compliant paths or if the forcing responsible for their motion is phase-dependent, capturing experimental observations. We then address the synchronisation of free-swimming cells such as spermatozoa. Using a two-dimensional model we show the additional degree of freedom may lead to passive synchronisation in a manner that depends only on the geometry, but might not minimise energy dissipation. In contrast, active synchronisation always leads to in-phase swimming, as observed in experiments.

In this third chapter we introduce the historically important model of swimming at low Reynolds numbers originally proposed by G. I. Taylor (1951), which is now considered classical. In his paper, Taylor set out to investigate the possibility of swimming in a fluid without inertia at all, a possibility that was at odds with physical intuition at the time. Since waves are the fundamental non-reciprocal kinematics, and since microorganisms were observed to deform their flagella in a wave-like fashion, he focused on the simplest setup possible, namely that of a flexible two-dimensional sheet deforming as a travelling wave of transverse displacements. In this chapter, considering waves with both transverse and longitudinal motion, we show that indeed inertia-less swimming is possible, and that the sheet motion can be used to model both swimming using flagella and pumping using cilia. By computing the rate of working of the wave on the fluid, and its optimisation, we then illustrate how this simple two-dimensional model can be exploited to interpret the two modes of deformation of cilia arrays that are observed experimentally.