This fourth chapter introduces the second canonical model of low Reynolds number swimming, namely that of the squirming sphere. Originally proposed by Lighthill (1952), and later extended by Blake (1971), this model is a variation of the waving sheet adapted to a finite-size swimmer. Here, the surface of a spherical body undergoes periodic small-amplitude deformations, leading to instantaneous velocity boundary conditions applied on an effective spherical frame. Since Lighthill's original paper, the squirmer model has been used and extended in a variety of setups, in particular to provide an alternative envelope model of the metachronal waves of cilia for finite-size organisms. We first derive the classical swimming squirmer model for a translating swimmer before discussing its extension to rotational motion. We then show how to link the motion of points on the surface of a deformable sphere described in a Lagrangian fashion to the squirmer model written naturally in an Eulerian framework. We finish by comparing the results of the model with flow measurements around the flagellated green alga Volvox.

In this fourteenth chapter we turn to the collective dynamics of cell populations such as bacterial colonies. Specifically, we use the results from Chapters 9 and 10 on hydrodynamic interactions to adapt the discrete and continuum frameworks introduced in the previous chapter to the case of collective cell locomotion. Swimming cells create flows, which advect and rotate neighbouring organisms, and since the flow induced by each cell depends on its location and orientation, this coupling leads to complex nonlinear swimming dynamics. In the discrete case, we derive a first-principle model of cells interacting in the dilute limit, demonstrate the different ways in which two swimming microorganisms affect each other hydrodynamically, and show how the model can be used to explain clustering instabilities of swimming algae. We then develop a continuum approach coupling the dynamics of the fluid with the distribution in position and orientation of the cell population. After relating the model to alternative phenomenological descriptions based on symmetry arguments, we use this continuum framework to capture collective cell instabilities.

In many biological situations, self-propelled cells and slender appendages interact with complex, non-Newtonian fluids. In this fifteenth and final chapter, we revisit the Newtonian hydrodynamic principles from Chapter 2 for fluids characterised by a nonlinear, or non-instantaneous, relationship between stress and deformation. We first consider linear viscoelastic fluids. We show how the presence of memory in the fluid affects the hydrodynamic forces and the energy of slender filaments, and compare a modified active-filament model of spermatozoa to experiments in viscoelastic fluids. We next address fluids with nonlinear rheological properties and their impact on the locomotion of Taylor's waving sheet. When the wave kinematics are prescribed, the nonlinear fluid leads to decreases in both speed and energy expenditure, which compares favourably to experiments using small nematodes. If instead the wave results from a balance between activity, elasticity and fluid forces, a transition to enhanced motion is possible for sufficiently flexible swimmers. We finish by addressing heterogeneous fluids and show how the multiscale nature of the fluid systematically enhances flow and transport.

In this first chapter, we give a brief biological introduction to understand the context of the mathematical models developed in the following chapters of the book and to also appreciate the relevance of the biophysical problems addressed. The chapter includes also a short overview of the role played by fluid dynamics in biology.

We have so far described the fluid dynamics relevant to the propulsion mechanisms of individual microorganisms, with a focus on the physical principles dictating cell locomotion. The length scales involved ranged from tens of nanometres to a few microns. In contrast, the dynamics of cell populations is characterised by much larger length scales, typically hundreds of microns and above. The flow disturbances induced by swimming cells on these large length scales play important biophysical roles, from governing the mixing and transport of nutrients to impacting the physical interactions of cells with their environment and the collective dynamics of populations. In this ninth chapter we address the consequences of the force-free and torque-free swimming constraints of swimming cells on the flows they create. We develop the framework to describe these flow signatures mathematically on length scales larger than that of the organisms, compare them with experimental measurements, explain how to find better approximations near the cells and discuss implications for the hydrodynamic stresses induced by suspensions of swimming microorganisms.

The viscous drag acting on a moving filament is the key to predicting correctly the direction of flagellar propulsive forces. In this sixth chapter we provide the mathematical basis for this result by evaluating asymptotically the hydrodynamic forces experienced by slender filaments. After revisiting the classical solution for Stokes flow due to the translation of a rigid sphere, we capture the flow along weakly bent, slender filaments using a centreline distribution of two hydrodynamic singularities. We show that in the slender limit the flow near any cross section of the filament is dominated by a local anisotropic force density, which arises from hydrodynamic singularities in the vicinity of the cross section, whose magnitudes depend logarithmically on the aspect ratio of the filament (resistive-force theory). The introduction of additional singularities far from the local cross section of the filament allows us to derive an improved nonlocal integral relationship relating the velocity of the filament to the distribution of hydrodynamic forces (slender-body theory). We close by comparing theoretical predictions with experiments on rotating helices.

We consider in this eighth chapter bacterial locomotion powered by the rotation of helical filaments. We focus on the canonical case of a cell body rotating a single filament. Bacterial locomotion features two differences from the swimming of spermatozoa. First, bacterial swimming does not involve time-varying shape changes but may be understood physically as due to the relative rotation of rigid bodies. Second, in order to balance hydrodynamic torques, a bacterium needs a cell body of finite size to swim. We derive the Stokes resistance matrix of a rigid helix as predicted by resistive-force theory. We use it to compute the velocity of a bacterium moving along a straight line and compare our results with experimental measurements on E. coli. We employ our theoretical estimates to address the energy expended by the motors powering the rotation of filaments. Extending the theory to the 3D motion of finite-size bacteria, we obtain helical trajectories and compare to measurements for B. subtilis. We also show that there exists an optimal size of the cell body and that we can define an intrinsic efficiency for the helical propeller, allowing us to rationalise the shapes of natural flagellar filaments.

Motivated by the propulsion of spermatozoa, the seventh chapter focuses on the planar waving of eukaryotic flagella. We first show how to apply resistive-force theory to compute the swimming speeds of simple flagellar waves and detail how wave geometry influences locomotion. Next we introduce a measure of swimming efficiency and show how to use it in order to derive the shape of the optimal waving motion. Finally we model the eukaryotic flagellum as an active filament where actuation from molecular motors is accounted for in a continuum manner and the waving motion is obtained as a mechanical balance between molecular forcing, fluid dynamics and passive elasticity. We close with a demonstration of how that model allows us to quantify cellular energy consumption.

The presence of surfaces near a swimmer can impact dramatically its ability to generate propulsive forces for locomotion. In this eleventh chapter, we review theways in which boundaries influence cellular propulsion from a hydrodynamic standpoint. At the cellular level, both the distribution of cells and their swimming kinematics are affected. On smaller length scales, boundaries govern the ability of appendages such as cilia to produce net forces and flow. We first consider length scales much larger than those of the cells. We show how the method of images for hydrodynamic singularities can be used to demonstrate that long-range hydrodynamic interactions lead to the attraction of swimming cells by boundaries and to a change in the swimming kinematics of bacteria from straight to circular. We then examine the dynamics of swimming cells in shear flows and explain how the presence of a surface leads to cell reorientation and upstream swimming. We next revisit the waving sheet model near a boundary to show how increased friction impacts locomotion kinematics. We finish by zooming in to the sub-cellular level and addressing the role played by surfaces on force and flow generation by cilia.

This second chapter is devoted to an overview of the fundamental physical concepts and mathematical equations required to construct models of cell locomotion. We start from the general setup of locomotion in fluids at arbitrary length scales before focusing on the low Reynolds number limit. We show in particular how the Stokes limit of hydrodynamics can be exploited along with symmetry arguments to derive general properties of microscopic locomotion.

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.