  We analyze experimentally the shape of a long elastic filament rotating in a
viscous liquid. We identify a continuous but sharp transition from a straight
to an helical shape, resulting from the competition between viscous stresses
and elastic forces. This induced helicity generates a propulsive force along
the axis of rotation. In addition, we show that the shape transition is
associated with an unstable branch in the force-torque relation. A linearized
model of the fluid-structure interaction is proposed to account for all the
features of the non-linear filament dynamics.
