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 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.