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

Abstract

The rhythmic motions and wave-like 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. When shadowing effects due to the physical presence of the spherical boundary are taken into account, the results from the slender body theory differ from that obtained using local resistivity theory. 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 towards perfectly in-phase or perfectly out of phase responses depending on the nature of the instability. Specifically, the first bifurcation corresponds to filaments oscillating inphase 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 inter-filament 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 the bending and buckling instabilities of a straight passive filament attached to a wall and placed in a viscous stagnant flow -a problem related to the growth of biofilms, and also to mechanosensing in passive cilia and microvilli. Taken together, our results provide the foundation for more detailed non-linear analyses of spatiotemporal patterns in active filament systems.

Buckling instabilities and spatio-temporal dynamics of active elastic filaments

“…In this context, we have recently started analysis of active filament-cargo assemblies-moving singly or in a cluster-using slender body theory with full hydrodynamic interactions instead of resistivity theory used here. Analytical results from the minimal model-specifically, the critical eigenmodes given by equations (3.5), (3.6), (3.7) and (3.10)-have been used to construct exact numerical solutions [51]. Perhaps most importantly, this approach could prove useful to design filament-cargo assemblies exhibiting a specific behaviour by breaking down the problem into two pieces: first, identify a region in cargo parameter space that produces the desired behaviour; then, look for a cargo shape and size whose drag parameters are in said region.…”

Synchronized oscillations, metachronal waves, and jammed clusters in sterically interacting active filament arrays

“…To focus on the role of steric interactions, we do not consider hydrodynamic interactions and the wall only serves to keep the base of the filament fixed. Note that in a system where full hydrodynamic effects are included, fluid flow generated by beating filaments will alter the motion of the filament [33][34][35] . In our case, we neglect these induced fluid flows and consider, to leading order, just the viscous Stokes drag on the beads comprising the filament as they move.…”

“…These instabilities have been the subject of recent theoretical and computational inquiries. Continuum as well as discrete agent-based models have been used to investigate the emergence of oscillations in single filaments, and coupling-induced synchrony in systems of two rotating filaments [29][30][31][32][33][34][35][36][37][38][39][40] .…”

Flapping, swirling and flipping: Non-linear dynamics of pre-stressed active filaments

“…This is due to the non-self-adjoint nature of the equations and boundary conditions [23][24][25][45][46][47][48][49][50][51]. Instead time dependent evolution equations (dynamical equations) derived using variants of the Kirchhoff-Love theory are appropriate and have been used to analyze the onset of instabilities and non-linear patterns in these systems [29,[52][53][54][55]. As part of this effort, studies have also focussed on elucidating material constitutive laws [56,57].…”

“…However, a significant fundamental gap in the literature exists. Most current theories relate to filaments/rods that are are not pre-stressed (equivalently not pre-strained) and further are only partially constrained; that is, in most cases the base state is an uncompressed rod with a straight shape [29,[52][53][54][55]. In many important contexts however, rods and filaments that are subject to deforming forces and torques start off from shapes that are neither planar nor stress-free.…”