Getting in Shape: Swimming with Stokes and Surfing with Brinkman

Abstract

In this dissertation, we investigate the effect of shape on the motion of microscopic particles that perform a swimming motion or ‘surf’, driven by an external flow, through microscopic channels. These motions take place in a fluid, and fluid motion is in general described by the Navier-Stokes equations. However, at small scales (the particles that we investigate are of micrometer dimensional), the hydrodynamic regime is characterized by a low Reynolds number, allowing to simplify the governing equations to the Stokes equations. From the linearity of the Stokes equations, a linear relation can be derived between the forces on, and the velocity of, a microscopic particle. The proportionality factor is in general a tensor, the resistance tensor, which depends on the geometry of the particle, a result that is central in this thesis. In Chapter 2, we give an extensive background of the hydrodynamic theory that is used in this thesis. With the extensiveness of Chapter 2, we aim to make this thesis as self-contained as possible; readers with a strong background in Stokesian hydrodynamics may skip Chapter 2 and proceed to the next Chapters. In Chapter 3, we develop a numerical bead-shell model to calculate the hydrodynamic friction on colloidal particles, which we generalize to systems of (hydrodynamically interacting) particles. This extension is applied in Chapter 4, where we investigate the intricate dependence of the swimming efficiency on the shape of a microswimmer, both for simplified models (three-body swimmers) and biologically inspired swimmers (modelled after E. coli bacteria). In Chapter 5, we collaborate with an experimental group from the Radboud University (Nijmegen) to investigate the motion of chemically self-propelled bead chains, where the motion of these chains is influenced by both the chain shape as well as the internal chain structure. In the second part of the thesis, we focus on the shape dependence of confined particle motion in Hele-Shaw channels. The quasi-two-dimensional fluid flow that drives this motion is described by the Brinkman equation, which is derived from the Stokes equation in the microchannel geometry. In Chapter 6, we set up the theoretical and numerical framework to calculate these trajectories, and compare directly with experimental results obtained by experimental collaborators from the TU Delft. In Chapter 7, we solve the particle equations of motion analytically, and classify the trajectories as far as possible on the basis of only a few geometry-dependent time scales. Finally, in Chapter 8, we generalize this framework to collections of particles in Hele-Shaw channels, and investigate the pairwise and many body-interaction in the channel, as well as the influence of particle shape on these interactions.