  Slender bodies capable of spontaneous motion in the absence of external
actuation in an otherwise quiescent fluid are common in biological, physical
and technological contexts. The interplay between the spontaneous fluid flow,
Brownian motion, and the elasticity of the body presents a challenging
fluid-structure interaction problem. Here, we model this problem by
approximating the slender body as an elastic filament that can impose
non-equilibrium velocities or stresses at the fluid-structure interface. We
derive equations of motion for such an active filament by enforcing momentum
conservation in the fluid-structure interaction and assuming slow viscous flow
in the fluid. The fluid-structure interaction is obtained, to any desired
degree of accuracy, through the solution of an integral equation. A simplified
form of the equations of motion, that allows for efficient numerical solutions,
is obtained by applying the Kirkwood-Riseman superposition approximation to the
integral equation. We use this form of the equation of motion to study
dynamical steady states in free and hinged minimally active filaments. Our
model provides the foundation to study collective phenomena in
momentum-conserving, Brownian, active filament suspensions.
