The Stokesian hydrodynamics of flexing, stretching filaments

A central element of many fundamental problems in physics and biology lies in the interaction of a viscous fluid with slender, elastic filaments. Examples arise in the dynamics of biological fibers, the motility of microscopic organisms, and in phase transitions of liquid crystals. When considering the dynamics on the scale of a single filament, the surrounding fluid can often be assumed to be inertialess and hence governed by the Stokes' equations. A typical simplification then is to assume a local relation, along the filament, between the force per unit length exerted by the filament upon the fluid and the velocity of the filament. While this assumption can be justified through slender-body theory as the leading-order effect, this approximation is only logarithmically separated (in aspect ratio) from the next-order contribution capturing the first effects of non-local interactions mediated by the surrounding fluid; non-local interactions become increasingly important as a filament comes within proximity to itself, or another filament. Motivated by a pattern forming system in isotropic to smectic-A phase transitions, we consider the non-local Stokesian dynamics of a growing elastica immersed in a fluid. The non-local interactions of the filament with itself are included using a modification of the slender-body theory of Keller and Rubinow. This modification is asymptotically equivalent, and removes an instability of their formulation at small, unphysical length-scales. Within this system, the filament lives on a marginal stability boundary, driven by a continual process of growth and buckling. Repeated bucklings result in filament flex, which, coupled to the non-local interactions and mediated by elastic response, leads to the development of space-filling patterns. We develop numerical methods to solve this system accurately and efficiently, even in the presence of temporal stiffness and the close self-approach of the filament. While we have ignored many of the thermodynamic aspects of this system, our simulations show good qualitative agreement with experimental observations. Our results also suggest that non-locality, induced by the surrounding fluid, will be important to understanding the dynamics of related filament systems.