  The dynamics of a flexible filament sedimenting in a viscous fluid are
explored analytically and numerically. Compared to the well-studied case of
sedimenting rigid rods, the introduction of filament compliance is shown to
cause a significant alteration in the long-time sedimentation orientation and
filament geometry. A model is developed by balancing viscous, elastic, and
gravitational forces in a slender-body theory for zero-Reynolds-number flows,
and the filament dynamics are characterized by a dimensionless
elasto-gravitation number. Filaments of both non-uniform and uniform
cross-sectional thickness are considered. In the weakly flexible regime, a
multiple-scale asymptotic expansion is used to obtain expressions for filament
translations, rotations, and shapes. These are shown to match excellently with
full numerical simulations. Furthermore, we show that trajectories of
sedimenting flexible filaments, unlike their rigid counterparts, are restricted
to a cloud whose envelope is determined by the elasto-gravitation number. In
the highly flexible regime we show that a filament sedimenting along its long
axis is susceptible to a buckling instability. A linear stability analysis
provides a dispersion relation, illustrating clearly the competing effects of
the compressive stress and the restoring elastic force in the buckling process.
The instability travels as a wave along the filament opposite the direction of
gravity as it grows and the predicted growth rates are shown to compare
favorably with numerical simulations. The linear eigenmodes of the governing
equation are also studied, which agree well with the finite-amplitude buckled
shapes arising in simulations.
