  Many microorganisms propel through complex media by deformations of their
flagella. The beat is thought to emerge from interactions between forces of the
surrounding fluid, passive elastic response from deformations of the flagellum,
and active forces from internal molecular motors. The beat varies in response
to changes in the fluid rheology, including elasticity, but there is limited
data on how systematic changes in elasticity alters the beat. This work
analyzes a related problem with fixed-strength driving force: the emergence of
beating of an elastic planar filament driven by a follower force at the tip in
a viscoelastic fluid. This analysis examines how the onset of oscillations
depends on the strength of the force and viscoelastic parameters. Compared to a
Newtonian fluid, it takes more force to induce the instability in viscoelastic
fluids, and the frequency of the oscillation is higher. The linear analysis
predicts that the frequency increases with the fluid relaxation time. Using
numerical simulations, the model predictions are compared with experimental
data on frequency changes in bi-flagellated alga Chlamydomonas reinhardtii. The
model shows the same trends in response to changes in both fluid viscosity and
Deborah number, and thus provides a mechanistic understanding of the
experimental observations.
