Propulsion and Instability of a Flexible Helical Rod Rotating in a Viscous Fluid

Abstract

We combine experiments with simulations to investigate the fluid-structure interaction of a flexible helical rod rotating in a viscous fluid, under low Reynolds number conditions. Our analysis takes into account the coupling between the geometrically nonlinear behavior of the elastic rod with a nonlocal hydrodynamic model for the fluid loading. We quantify the resulting propulsive force, as well as the buckling instability of the originally helical filament that occurs above a critical rotation velocity. A scaling analysis is performed to rationalize the onset of this instability. A universal phase diagram is constructed to map out the region of successful propulsion and the corresponding boundary of stability is established. Comparing our results with data for flagellated bacteria suggests that this instability may be exploited in nature for physiological purposes. DOI: 10.1103/PhysRevLett.115.168101 PACS numbers: 47.63.Gd, 46.32.+x, 46.70.Hg, 47.63.mf Bacteria often rely on the deformation of filamentary helical structures, called flagella, for locomotion [1]. The propulsion arises from a complex fluid-structure interaction between the structural flexibility of the flagellum and the viscous forces generated by the flow. This fluid-structure interaction may lead to geometrically nonlinear deformations [2,3], which in turn can be exploited for turning [4], tumbling [5], bundle formation [6], and polymorphic transformations [7,8].Resistive force theories (RFT) [9,10] are often used to model the role of viscous forces on flexible filaments [11], at low Reynolds number. These simplify the viscous loading by introducing local geometry-dependent drag coefficients. More sophisticated descriptions consider nonlocal hydrodynamic effects, albeit typically assuming that the filament is rigid such that elastic forces are ignored [12,13]. The few studies that have coupled long-range hydrodynamics with elasticity either assume small deflections [13] or approximate the filament as a network of springs [14,15], thereby oversimplifying the mechanics of the problem. One exception is the study of buckling of a straight elastic filament loaded by viscous stresses [16]. More recently, a systematic computational study has been performed on a discretized model based on Kirchhoff's theory for elastic rods (in the form of a chain of connected spheres), coupled with RFT [3]. This study was significant in that it was the first, to the best of our knowledge, to report a series of buckling instabilities of the flagellum that arise during locomotion and suggested its relevance to the biological system. Moreover, it addressed the important rotation-translation coupling. However, recent experiments [12,17] Here, motivated by the locomotion of uniflagellated bacteria, we perform a combined experimental and numerical investigation of the dynamics of a helical elastic filament rotated in a viscous fluid. Our goal is to predictively understand the underlying mechanical instabilities. In our precision model experiments, we reproduce and syste...

Buckling Instabilities and Complex Trajectories in a Simple Model of Uniflagellar Bacteria

“…Because the resultant drag from the cell body exactly balances the propulsive force, the hook is under tension for pullers and under compression for pushers. Experiments with both elastic and rigid flagella have shown that the hook will buckle under large enough compression, one consequence being misalignment between the cell body and flagellum (3,6). Buckling has been observed in Vibrio alginolyticus when the hook unwinds and significantly weakens during a pull-to-push transition.…”

“…Because assigning configurations to swimmers necessarily precludes the exploration of any hook dynamics or mechanics, other studies instead use various models incorporating elasticity to examine the stability of the hook and of the overall flagellar filament (12)(13)(14)(15). Indeed, models of a standalone elastic flagellum show instability in the equilibrium helical shape above a critical applied torque load/angular velocity (6,14). However, for flagella attached to cell bodies, experiments show that the hook can buckle even without significant deformation to the flagellar filament, highlighting the key role the local hook flexibility plays in determining the stability of swimming (3).…”

A Perspective on the Revival of Structural (In)Stability With Novel Opportunities for Function: From Buckliphobia to Buckliphilia

“…In this case, the instability is localized at the highly compliant hook that connects the slender flagellum to the rotary motor inside the cell body. Subsequently, by combining computer simulations and precision model experiments, we have demonstrated [25] that it is possible for the actual flagella to buckle, itself a helical filament that is highly flexible ought to its slenderness. Here, the instability arises due to the compressive stresses that derive from the viscous loading during the rotation of the helical filament.…”

Bending transition in the penetration of a flexible intruder in a two-dimensional dense granular medium

“…In all these cases, the flexible body experiences shape reconfiguration with self-streamlining and reduction of the surface area exposed to the flow, thus resulting in a drastic reduction of the drag force exerted on the object [10]. An immediate field of application of these problems can be found in bio-physical domains, when a passive or an active elastic part of a body interacts with the flow [6,[11][12][13]. In particular, in animal locomotion, the flexibility of wings or fins intervene in the flying of birds and insects or the swimming of fishes or eels.…”