# Tricks and tips for faster small-scale swimming: Complex fluids and elasticity

Thesis

Many cells exploit the bending or rotation of flagellar filaments in order to self-propel in viscous fluids. Often swimming occurs in complex, nonlinear fluids, e.g. mucus. Futhermore even in simple Newtonian fluids, if swimming appendages are deformable then locomotion is subject to fluid-structure interactions. The fundamental question addressed in this thesis is how exactly locomotion is impacted, in particular if it is faster or slower, with or without these effects. First we study locomotion in shear-thinning and viscoelastic fluids with rigid swimming appendages. Following the introductory Chapter, in Chapter 2 we propose empirical extensions of the classical Newtonian resistive-force theory to model the waving of slender filaments in non-Newtonian fluids, based on experimental measurements for the motion of rigid rods in non-Newtonian fluids and on the Carreau fluid model. We then use our models to address waving locomotion in shear-thinning fluids, and show that the resulting swimming speeds are systematically lowered a result which we are able to capture asymptotically and to interpret physically. In Chapter 3 we consider swimming using small-amplitude periodic waves in a viscoelastic fluid described by the Oldroyd-B constitutive relationship. Using Taylor’s swimming sheet model, we show that if all travelling waves move in the same direction, the locomotion speed of the organism is systematically decreased. However, if we allow waves to travel in two opposite directions, we show that this can lead to enhancement of the swimming speed, which is physically interpreted as due to asymmetric viscoelastic damping of waves with different frequencies. A change of the swimming direction is also possible. Secondly we consider the affect of fluid-structure interactions. In Chapter 4, we use Taylor’s swimming sheet model to describe an active swimmer immersed in an Oldroyd-B fluid. We solve for the shape of an active swimmer as a balance between the external fluid stresses, the internal driving moments, and the passive elastic resistance. We show that this dynamic balance leads to a generic transition from hindered rigid swimming to enhanced flexible locomotion. The results are physically interpreted as due to a viscoelastic suction increasing the swimming amplitude in a non-Newtonian fluid and overcoming viscoelastic damping. In Chapter 5 we consider peritrichously flagellated bacteria, such as Escherichia coli. The rotation of each motor is transmitted to a flexible rod called the hook which in turns transmits it to a helical filament, leading to swimming. The motors are randomly distributed over the body of the organism, and thus one expects the propulsive forces from the filament to almost cancel out leading to negligible swimming. We show that the transition to swimming is an elasto-hydrodynamic instability arising when the flexibility of the hook is below a critical threshold.