Dynamics of chiral particles in viscous fluids

Colloidal suspensions --- micron sized particles in a molecular solvent, typically water --- are found everywhere in nature, e.g. milk, and in artificial materials, e.g. paint. The dynamics of colloidal particles are therefore of interest to both academia and industry. The sedimentation of particles in suspensions is relevant in multiple fields, such as its application to the transport and separation of biological particles (viruses, bacteria, etc.). Many classical fluid mechanics studies have looked at the sedimentation, and/or tumbling due to shear, of a single particle, either using analytic methods or, more recently, numerical techniques. By now it is well understood that the particle shape uniquely determines its dynamics, but the precise trajectory and orientational dynamics are only known for a limited set of shapes. This is because, barring the simplest of shapes, the analytic calculations are challenging and often involve reducing approximations. The way chiral particles sediment and behave under shear are still unknown, and in the process of being studied and understood. This is true, in particular, for conditions under which the trajectories of such particles are chiral. In this thesis, therefore, the focus will be on the behaviour of chiral particles suspended in fluids at low Reynolds number --- where the viscosity dominates over the inertia. The dynamics of chiral objects is studied using Resistive Force Theory, which assumes that the body can be partitioned into segments but ignores the hydrodynamic couplings between the parts of the particle. These studies are then compared to a numerical calculation by Palanisamy and Den Otter that accounts for the hydrodynamic interactions using the Rotne-Prager-Yamakawa approximation. For a sedimenting helix, great agreement is found between the analytic and numerical results. Helices, for most initial conditions, sediment performing a superhelical trajectory --- a helical path with the symmetry axis parallel to the direction of gravity --- for which the handedness is opposite to that of the helix. It is also observed that a helix, in an almost horizontal configuration, is either attracted to the horizontal orientation, in which it sediments in a straight line in the direction of gravity, or to trajectories that form an unstable helical-like path. Alternatively, when a helix is in a simple shear flow it travels performing Jeffery-like orbits with a lateral drift perpendicular to the plane of shear. To better understand the result for the helix sedimentation, the settling of L- and C-shapes is also considered. Here it shown that an object does not need to be chiral for its sedimentation trajectory to possess chirality, in agreement with the findings by Krapf et al. Counter-intuitively, it was observed that the result, by Taylor, of a sedimenting rod --- the rod does not reorient --- is not obtained by taking the limit of an L-shape with a vanishing short leg. This is because any minute perturbation away from the rod limit leads to the emergence of a fixed point to the dynamics at infinite time due to orientational couplings in the Grand Mobility Matrix that persist for all perturbations. Thanks to this understanding of simple (chiral) objects, insight was gained into the sedimentation behaviour of a complicated shape: a Möbius strip. This object has a rich state diagram for its settling behaviour, which is strongly dependent on its initial orientation. This diagram, for a single Möbius strip, is portrayed and insight into the identified trends is given. Overall, it was shown that the sedimentation of anisotropic or chiral particles is chiral and future work could include finding analytic expressions to describe these trajectories. Whereas, for a helix in a shear flow, the properties and dependencies of the lateral drift observed can be further studied.