When a viscoelastic fluid blob is stretched out into a thin horizontal filament, it
sags and falls gradually under its own weight, forming a catenary-like structure that
evolves dynamically. If the ends are brought together rapidly after stretching, the
falling filament tends to straighten by rising. These two effects are strongly influenced
by the elasticity of the fluid and yield qualitatively different behaviours from the case
of a purely viscous filament analysed previously (Teichman & Mahadevan, J. Fluid
Mech. vol. 478, 2003, p. 71). Starting from the bulk equations for the motion of a
viscoelastic fluid, we derive a simplified equation for the dynamics of a viscoelastic
filament and analyse this equation in some simple settings to explain our observations.