Spontaneous dynamics of an active filament in simple shear flow

We study the effects of external simple shear flow on the spontaneous oscillations of an inertialess active filament in a highly viscous medium. The filament, clamped to a wall at one end, is subjected to a compressive “follower force” applied tangentially at its free end. Extending my weakly nonlinear analysis of the shear-free case (J. Fluid. Mech., 1007 A65, 2025), I derive a generalized amplitude equation capturing the near-onset dynamics under weak shear. Besides inducing a steady deflection, the shear effectively damps the filament’s intrinsic oscillations. This damping mechanism, which arises from a nonlinear resonance between the shear and the intrinsic oscillations, scales quadratically with the shear rate and is anisotropic—stronger along the shear than normal to it. Without shear, stable whirling states, where the filament tip traces a circular orbit parallel to the wall, and unstable planar-beating states simultaneously emerge at a critical follower force magnitude, with circular whirling typically observed beyond this threshold. Shear breaks this degeneracy, driving a sequence of dynamical transitions: from circular to elliptical whirling, then to transverse beating (normal to the shear), and ultimately to steady deflection.