The three-dimensional motion of slender filaments

Abstract

This paper models the dynamics of a three-dimensional inextensible filament, as a function of a given velocity distribution along its centreline. This geometric problem has various applications in physics, engineering and biology, e.g. the dynamics of vortex lines, the motion of bacterial flagella, and the motion of eukaryotic cilia and flagella. A previously proposed solution to the problem used time evolution equations for the centreline of the filament, which were expressed in the Frenet co-ordinate system. Unfortunately, this approach is limited because singularities occur when the filament straightens, even partially. To overcome the singularity problem we express the dynamics of the filament in a different intrinsic co-ordinate system, named ‘body co-ordinates’. This also enables us to take into account the twisting motion of the filament. We apply our method to simulate three-dimensional motion of cilia, while taking into account the hydrodynamic effect of the flat surface from which they emerge and the hydrodynamic interactions between the cilia. This is the first model that enables robust and consistent simulations of three-dimensional ciliary motion with non-local hydrodynamics, at a computational cost which is comparable with that of the planar model. We provide here a complete and detailed description and derivation of our model equations together with all the necessary details for numerical implementation, hoping that this modelling template would become a useful tool for further studies of the internal mechanism of cilia and flagella. Copyright © 2001 John Wiley & Sons, Ltd.