Microscopic-scale swimming has been a very active area of research in the last couple of decades. The interest in this topic has been driven partly by experimental advances, which have allowed scientists to observe the motion of microorganisms with unprecedented detail, partly by the great potential for applications (e.g., in the biomedical sciences and biomimetic engineering), and partly by a desire to understand the fundamentals of life starting from unicellular organisms.

Flagellar propulsion is the most common form of locomotion amongst unicellular organisms. On the one hand, eukaryotic organisms have flexible flagella which they undulate in a remarkable variety of patterns - from the whip-like motion of a spermatozoon tail, to the breast-stroke pattern of biflagellate algae such as Chlamydomonas, down to the metachronal waves formed by thousands of cilia on the surface of a Paramecium. On the other hand, bacteria are equipped with corkscrew-shaped flagella, which they can rotate rigidly in viscous fluids to generate forward motion.

In this thesis, we consider the role played by the number of flagella in the swimming of multi-flagellated bacteria, for which the model organism is Escherichia coli. The work presented here is theoretical and consists mainly of asymptotic calculations (usually exploiting a small parameter that represents the aspect ratio of flagellar filaments), assisted or verified by numerical simulations. We use two well-established theories for the hydrodynamics of slender filaments, namely resistive-force theory (RFT) for the analytical calculations and slender-body theory (SBT) for the numerical simulations.

The first part of the thesis, comprising three chapters, deals with the geometric aspects of having multiple helical filaments in close proximity to one another. We derive geometric constraints on the rate of synchronisation between rotating helical filaments as they come together to form a bundle, as well as the geometric constraints on the entanglement of a pair of helical filaments. Our results suggest that bacterial flagella are typically too few, and hence anchored too far apart on the cell body, to be able to form tangled bundles based on their intrinsic, undeformed geometry alone.

The next three chapters of the thesis focus on the hydrodynamic interactions (HIs) between flagellar filaments, and the effect that HIs have on the synchronisation and the propulsive capacity of a bundle of parallel filaments. We first derive an asymptotic theory for the HIs between rigid filaments separated by a distance larger than their contour length. Next, using this theory, we propose a novel analytical model for the synchronisation of elastically tethered rotating helices. Remarkably, we find that there is an optimum strength of the elastic compliance which minimises the time scale for synchronisation, and that the flagellar filament, although more rigid than the hook, may play a more important role in the synchronisation of bacterial flagella and stability of flagellar bundles rotating in-phase.

In the final chapter on hydrodynamics we investigate how the hydrodynamic drag and thrust associated with a bundle of parallel helical filaments depend on the number of filaments. Remarkably, our findings reveal that the torque-speed relationship of the bacterial flagellar motor plays an important role in the swimming of multiflagellated bacteria. Because HIs within a circular bundle of filaments reduce the hydrodynamic resistance of each filament to rotation about its own axis, the bacterial flagellar motors actuating the bundle transition from the high-load to the low-load regime at a critical number of filaments within the biologically relevant range, which leads to a peak in the swimming speed followed by a monotonic decay with increasing number of filaments.