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Many microorganisms propel through complex media by deformations of theirflagella. The beat is thought to emerge from interactions between forces of thesurrounding fluid, passive elastic response from deformations of the flagellum,and active forces from internal molecular motors. The beat varies in responseto changes in the fluid rheology, including elasticity, but there is limiteddata on how systematic changes in elasticity alters the beat. This workanalyzes a related problem with fixed-strength driving force: the emergence ofbeating of an elastic planar filament driven by a follower force at the tip ina viscoelastic fluid. This analysis examines how the onset of oscillationsdepends on the strength of the force and viscoelastic parameters. Compared to aNewtonian fluid, it takes more force to induce the instability in viscoelasticfluids, and the frequency of the oscillation is higher. The linear analysispredicts that the frequency increases with the fluid relaxation time. Usingnumerical simulations, the model predictions are compared with experimentaldata on frequency changes in bi-flagellated alga Chlamydomonas reinhardtii. Themodel shows the same trends in response to changes in both fluid viscosity andDeborah number, and thus provides a mechanistic understanding of theexperimental observations.

Effect of fluid elasticity on the emergence of oscillations in an active elastic filament