Locomotion over a Washboard

The purpose of this thesis is to study the problem when a microorganism swims very close to a shaped boundary. In this problem, we model the swimmer to be a two-dimensional, infinite periodic waving sheet. For simplicity, we only consider the case where the fluid between the swimmer and the washboard is Newtonian and incompressible. We assume that the swimmer propagates waves along its body and propels itself in the opposite direction. We consider two cases in our swimming sheet problem and the lubrication approximation is applied for both cases. In the first case, the swimmer has a known fixed shape. Various values of wavenumber, amplitude of the restoring force and amplitude of the topography were considered. We found the instantaneous swimming speed behaved quite differently as the wavenumber was varied. The direction of the swimmer was also found to depend on the amplitude of the restoring force. We also found some impact of the topographic amplitude on the relationship between average swimming speed and the wavenumber. We extended the cosine wave shaped washboard to be a more general shape and observed how it affected the swimming behaviour. In the second case, the swimmer is assumed to be elastic. We were interested to see how different values of wavenumber, stiffness and amplitude of the restoring force could change the swimming behaviour. With normalized stiffness and wavenumber, we found the swimmer remained in a periodic state with small forcing amplitude. While the swimmer reached a steady state with unit swimming speed for high forcing amplitude. However, for other values of stiffness and wavenumber, we found the swimmer’s swimming behaviour was very different.