Elastohydrodynamical instabilities of active filaments, arrays, and carpets analyzed using slender-body theory

The rhythmic motions and wavelike planar oscillations in filamentous soft structures are ubiquitous in biology. Inspired by these, recent work has focused on the creation of synthetic colloid-based active mimics that can be used to move, transport cargo, and generate fluid flows. Underlying the functionality of these mimics is the coupling between elasticity, geometry, dissipation due to the fluid, and active force or moment generated by the system. Here, we use slender-body theory to analyze the linear stability of a subset of these-active elastic filaments, filament arrays and filament carpets-animated by follower forces. Follower forces can be external or internal forces that always act along the filament contour. The application of slender-body theory enables the accurate inclusion of hydrodynamic effects, screening due to boundaries, and interactions between filaments. We first study the stability of fixed and freely suspended sphere-filament assemblies, calculate neutral stability curves separating stable oscillatory states from stable straight states, and quantify the frequency of emergent oscillations. The results from the slender-body theory differ from that obtained using an approximate theory used often in the literature to study dynamics of filaments, referred to as the resistance force theory, in which the tangential and normal components of the fluid traction at a point on the filament are proportional to the tangential and normal components of the velocity of the filament. Next, we examine the onset of instabilities in a small cluster of filaments attached to a wall and examine how the critical force for onset of instability and the frequency of sustained oscillations depend on the number of filaments and the spacing between the filaments. Our results emphasize the role of hydrodynamic interactions in driving the system toward perfectly in-phase or perfectly out-of-phase responses depending on the nature of the instability. Specifically, the first bifurcation corresponds to filaments oscillating in-phase with each other. We then extend our analysis to filamentous (line) array and (square) carpets of filaments and investigate the variation of the critical parameters for the onset of oscillations and the frequency of oscillations on the interfilament spacing. The square carpet also produces a uniform flow at infinity and we determine the ratio of the mean-squared flow at infinity to the energy input by active forces. We conclude by analyzing t