Onset of spontaneous beating and whirling in the follower force model of an active filament

Abstract: We study the onset of spontaneous dynamics in the follower force model of an active filament, wherein a slender elastic filament in a viscous liquid is clamped normal to a wall at one end and subjected to a tangential compressive force at the other. Clarke, Hwang and Keaveny (Phys. Rev. Fluids, to appear) have recently conducted a thorough investigation of this model using methods of computational dynamical systems; inter alia, they show that the filament first loses stability via a supercritical double-Hopf bifurcation, with periodic 'planar-beating' states (unstable) and 'whirling' states (stable) simultaneously emerging at the critical follower-force value. We complement their numerical study by carrying out a weakly nonlinear analysis close to this unconventional bifurcation, using the method of multiple scales. The main outcome is an 'amplitude equation' governing the slow modulation of small-magnitude oscillations of the filament in that regime. Analysis of this reduced-order model provides insights into the onset of spontaneous dynamics, including the creation of the nonlinear whirling states from particular superpositions of linear planar-beating modes as well as the selection of whirling over planar beating in three-dimensional scenarios.