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.