  Flexible filaments moving in viscous fluids are ubiquitous in the natural
microscopic world. For example, the swimming of bacteria and spermatozoa as
well as important physiological functions at organ-level, such as the
cilia-induced motion of mucus in the lungs, or individual cell-level, such as
actin filaments or microtubules, all employ flexible filaments moving in
viscous fluids. As a result of fluid-structure interactions, a variety of
nonlinear phenomena may arise in the dynamics of such moving flexible
filaments. In this paper we derive the mathematical tools required to study
filament-driven propulsion in the asymptotic limit of stiff filaments. Motion
in the rigid limit leads to hydrodynamic loads which deform the filament and
impact the filament propulsion. We first derive the general mathematical
formulation and then apply it to the case of a helical filament, a situation
relevant for the swimming of flagellated bacteria and for the transport of
artificial, magnetically actuated motors. We find that, as a result of
flexibility, the helical filament is either stretched or compressed (conforming
previous studies) and its axis also bends, a new result which we interpret
physically. We then explore and interpret the dependence of the perturbed
propulsion speed due to the deformation on the relevant dimensionless dynamic
and geometric parameters.
