  Inspired by microscopic paramecies which use trichocyst extrusion to propel
themselves away from thermal aggressions, we propose a macroscopic experiment
to study the stability of a slender beam extruded in a highly viscous fluid.
Piano wires were extruded axially at constant speed in a tank filled with corn
syrup. The force necessary to extrude the wire was measured to increase
linearly at first until the compressive viscous force causes the wire to
buckle. A numerical model, coupling a lengthening elastica formulation with
resistive force theory, predicts a similar behaviour. The model is used to
study the dynamics at large time when the beam is highly deformed. It is found
that at large time, a large deformation regime exists in which the force
necessary to extrude the beam at constant speed becomes constant and
length-independent. With a proper dimensional analysis, the beam can be shown
to buckle at a critical length based on the extrusion speed, the bending
rigidity and the dynamic viscosity of the fluid. Hypothesising that the
trichocysts of paramercies must be sized to maximise their thrust per unit
volume as well as avoid buckling instabilities, we predict that their bending
rigidity must be about $3\times 10^{-9}~\mathrm{N\cdot \mu m^2}$. The
verification of this prediction is left for future work.

Fluid Dynamics