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.

INTRODUCTION

The heat or diffusion equation

models the heat flow in solids and fluids. It also describes the diffusion ofchemical particles. It is also one of the fundamental equations that haveinfluenced the development of the subject of partial differential equations(PDE) since the middle of the last century. Heat and fluidflow problems are important topics in fluid dynamics. Here theheat flow is combined with a fluid flow problem and the resulting equationis termed as energy equation. We begin with a derivation ofone-dimensional heat equation, arising from the analysis of heat flow in athin rod. Further, equation (8.1) is also a prototype in the class ofparabolic equations and hence the importance of studying this equation.

Derivation of One-Dimensional Heat Equation

Consider a thin rod of length L and place it along thex-axis on the interval [0; L]. Weassume that the rod is insulated so that its lateral surface is impenetrableto heat transfer. We also assume that the temperature is the same at allpoints of any cross-sectional area of the rod. Let denote, respectively, themass density, heat capacity and the coefficient of (internal) thermalconductivity, of the rod. Let us analyze the heat balance in an arbitrarysegment [x1; x2] of the rod, with𝛿x = x2 −x1 very small, over a time interval[t, t + 𝛿t];𝛿t small (see Figure 8.1).

Let u(x, t) denote thetemperature in the cross-section with abscissax, at time t. According toFourier's law of heat conduction, the rate ofheat propagation q is proportional to , with

S denoting the area of the cross-section. Here,

where 𝛿Q is the quantity of heat that has passedthrough a cross-section S during a time𝛿t. Thus,

Applying Fourier's law at x1 andx2, we obtain

Thus, the quantity of heat that has passed through the small segment[x1, x2] of therod, during time 𝛿t is given by

This influx of heat during time 𝛿t was spent inraising or lowering the temperature of the rod by𝛿u, say.