Biological locomotion for both major classes of organisms, prokaryotes and eukaryotes, is three-dimensional and is dominated by the presence of slender filaments, flagella, which are rotating and waving in viscous fluids. In this fifth chapter, we return to the biological movements seen in Chapter 1 and consider the relationship between the cellular shapes and their motion. Using elementary concepts from Stokes flows, we propose an intuitive physical and mathematical interpretation for the generation of propulsive forces by moving flagella and for the natural occurrence of helices and waves in cellular propulsion.

In this tenth chapter we address the impact of external flows on cell locomotion. We start by considering the dynamics of spherical swimmers in arbitrary external flows. In this case, the impact on cell translation and rotation can be obtained exactly (Faxén's laws), which we then use to address cell trajectories in simple canonical flows. We next examine the case of elongated swimmers, which may be analysed when the flow is linear, a limit relevant to many situations where the typical length scale over which the flow varies is much larger than the size of the organism. For slender swimmer shapes, we derive a simplified version of Jeffery's exact equation for ellipsoids in linear flows. Jeffery's equation is then used, in agreement with experiments, to characterise the angular dynamics of elongated bodies in shear flows and address the trajectories of elongated swimmers in elementary flows. We finally consider the case where swimmers have a preferential swimming direction, modelled by an additional external torque, which may then lead to cell trapping in high-shear regions and to hydrodynamic focusing.

After having examined the deterministic motility of swimming cells, we now turn to their interactions with a fluctuating environment. We consider in this thirteenth chapter the motion of small microorganisms subject to thermal noise, a situation relevant to the locomotion of small bacteria. This allows us to introduce two modelling approaches, namely a discrete framework (along with ensemble averaging) and a continuum probabilistic framework, both of which we adapt for the modelling of collective dynamics in the next chapter. We first review Brownian motion in translation and rotation for a passive particle, introduce all the relevant timescales for its dynamics, show how the statistical properties of its trajectory can be captured with both discrete and continuum frameworks, and apply these concepts to the diffusion of cells. By adding a swimming velocity to the particle, we next show how thermal noise affects the motion of swimming microorganisms and in turn how the noisy run-and-tumble motion of bacteria can be described as an effective diffusive process.

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.