Slender-body theory for slow viscous flow

Keller, Joseph B.; Rubinow, S. I.

doi:10.1017/s0022112076000475

Cited by 336 publications

(393 citation statements)

References 8 publications

Supporting

Mentioning

390

Contrasting

Order By: Relevance

“…First, the hydrodynamics is simplified to the level of resistive-force theory [3][4][5]34], a version of the equations of slender-body hydrodynamics [36][37][38][39][40][41] where only the leading term in an expansion of hydrodynamic forces and moments in powers of 1/ log(L/r) is conserved. In that case, the filament hydrodynamics is completely described by two drag coefficients, ξ ⊥ and ξ , relating linearly the drag forces per unit length of the filament to the local velocity relative to the fluid, for motion perpendicular and parallel to the filament respectively.…”

Section: A Assumptionsmentioning

confidence: 99%

Floppy swimming: Viscous locomotion of actuated elastica

2007

View full text Add to dashboard Cite

Actuating periodically an elastic filament in a viscous liquid generally breaks the constraints of Purcell's scallop theorem, resulting in the generation of a net propulsive force. This observation suggests a method to design simple swimming devices -which we call "elastic swimmers" -where the actuation mechanism is embedded in a solid body and the resulting swimmer is free to move. In this paper, we study theoretically the kinematics of elastic swimming. After discussing the basic physical picture of the phenomenon and the expected scaling relationships, we derive analytically the elastic swimming velocities in the limit of small actuation amplitude. The emphasis is on the coupling between the two unknowns of the problems -namely the shape of the elastic filament and the swimming kinematics -which have to be solved simultaneously. We then compute the performance of the resulting swimming device, and its dependance on geometry. The optimal actuation frequency and body shapes are derived and a discussion of filament shapes and internal torques is presented. Swimming using multiple elastic filaments is discussed, and simple strategies are presented which result in straight swimming trajectories. Finally, we compare the performance of elastic swimming with that of swimming microorganisms. I. INTRODUCTIONThe fluid mechanics of microorganism locomotion, pioneered more than fifty years ago by G.I. Taylor, has become one of the most successful branches of biomechanics, with success in both the basic physical understanding of flow behavior and the quantitative prediction of kinematics and energetics of locomotion [1][2][3][4][5][6][7][8].Recent technical advances have led to ever more precise fabrication at small scales (microns or less), prompting both theorists [9-13] and experimentalists [14] to design and analyze a series of simple low-Reynolds number swimmers. The experiment of Dreyfus et al. [14], in particular, reported locomotion in a sperm-like micro-swimmer, composed of a cargo (red blood cell) and a slender flexible filament made of a series of paramagnetic beads. In that case, actuation by oscillating transverse magnetic fields led to the generation of bending waves propagating along the filament and resulted in the motion of the micro-swimmer. In this system, the right-left symmetry was broken by the presence of a cargo and led to a preferential tip-to-base propagation of the bending waves, resulting in locomotion in the direction base-to-tip.An alternative way to break the symmetry in a similar system would be to build-in the asymmetry in the actuation. In particular, if an elastic filament is periodically actuated at one of its extremities in a viscous liquid, the resulting motion will lead to the propagation of bending waves and, in general, propulsive forces. This idea was originally proposed by Edward Purcell [8]. Physically, actuating an elastic filament allows one to break the constraints of the "scallop theorem" -which states that a body performing a reciprocal motion at low Reynolds number cannot p...

show abstract

Section: A Assumptionsmentioning

confidence: 99%

Floppy swimming: Viscous locomotion of actuated elastica

2007

View full text Add to dashboard Cite

show abstract

“…with c a constant of order unity [19]. Likewise, the axial elastic torque per unit length m = CΩ s [14,15] ζ r ≃ 4πηa 2 [19].…”

mentioning

confidence: 99%

“…Thus arise two dimensionless ratios: Γ ≡ C/A, and the aspect ratio a/L. At zero Reynolds number, elastic forces per length f ≡ −δE/δr [18] balance the viscous drag from slender-body hydrodynamics [19]: …”

mentioning

confidence: 99%

Twirling and Whirling: Viscous Dynamics of Rotating Elastic Filaments

2000

View full text Add to dashboard Cite

Motivated by diverse phenomena in cellular biophysics, including bacterial flagellar motion and DNA transcription and replication, we study the overdamped nonlinear dynamics of a rotationally forced filament with twist and bend elasticity. Competition between twist injection, twist diffusion, and writhing instabilities is described by a novel pair of coupled PDEs for twist and bend evolution. Analytical and numerical methods elucidate the twist/bend coupling and reveal two dynamical regimes separated by a Hopf bifurcation: (i) diffusion-dominated axial rotation, or twirling, and (ii) steady-state crankshafting motion, or whirling. The consequences of these phenomena for selfpropulsion are investigated, and experimental tests proposed.PACS numbers: 46.70.Hg, 47.15.Gf, Dynamics and stability of rotationally forced elastic filaments arise in several important biological settings involving bend and twist elasticity at low Reynolds number. In the context of DNA replication, when two daughter strands are produced from a duplex, it was noted [1] long ago that energy dissipation for rotations about the filament axis is so much smaller than that for transverse motions that axial "speedometer-cable" motions are favored, and are energetically and topologically feasible. During DNA transcription, in which a polymerase protein moves down the double-stranded filament, progressive unwinding of the helix can lead to an accumulation of local twist that may induce "writhing" instabilities of the filament [2]. Energetic and dynamical aspects of these processes are of great current interest [3,4].At the cellular level, bacteria are propelled through fluids by helical flagella turned by rotary motors in the cell wall [5]. Recent studies [6] have revealed the details of two competing crystal structures assumed by flagellin, the protein building block of flagella, corresponding to helices of opposite chirality. Both local and distributed torques can change the conformation of flagella; during swimming these motors episodically reverse direction [7], and the resultant torques can induce transformations between these states [8], while uniform flow past a pinned flagellum may induce such chirality inversions [9].To elucidate fundamental processes common to these systems, we consider here the model problem shown in Fig. 1: a slender elastic filament in a fluid of viscosity η, rotated at one end at frequency ω 0 with the other free. We study competition between three processes: twist injection at the rotated end, twist diffusion, and writhing. Analytical and numerical methods reveal two dynamical regimes of motion: twirling, in which the straight but twisted rod rotates about its centerline, and whirling, in which the centerline of the rod writhes and crankshafts around the rotation axis in a steady state.This work is a natural outgrowth of recent studies of forced elastica in the plane [10,11], and dynamic twistbend coupling [12][13][14]. The balance considered between elastic and viscous stresses complements that between elasticity and ...

show abstract

“…The recent literature has been populated by new and more refined results, both theoretical and experimental, in the two limit regimes. Concerning the viscous one, on which we concentrate in this paper, we recall that approximated theories, such as slender body approximation [4,14], resistive force theory [11], and also others [16,20], have been developed, and a number of biological experiments has been run to understand swimming strategies.…”

Section: Introductionmentioning

confidence: 99%

Self-propelled micro-swimmers in a Brinkman fluid

Morandotti

2012

Journal of Biological Dynamics

View full text Add to dashboard Cite

We prove an existence, uniqueness, and regularity result for the motion of a self-propelled micro-swimmer in a particulate viscous medium, modelled as a Brinkman fluid. A suitable functional setting is introduced to solve the Brinkman system for the velocity field and the pressure of the fluid by variational techniques. The equations of motion are written by imposing a self-propulsion constraint, thus allowing the viscous forces and torques to be the only ones acting on the swimmer. From an infinite-dimensional control on the shape of the swimmer, a system of six ordinary differential equations for the spatial position and the orientation of the swimmer is obtained. This is dealt with standard techniques for ordinary differential equations, once the coefficients are proved to be measurable and bounded. The main result turns out to extend an analogous result previously obtained for the Stokes system.

show abstract

Slender-body theory for slow viscous flow

Cited by 336 publications

References 8 publications

Floppy swimming: Viscous locomotion of actuated elastica

Floppy swimming: Viscous locomotion of actuated elastica

Twirling and Whirling: Viscous Dynamics of Rotating Elastic Filaments

Self-propelled micro-swimmers in a Brinkman fluid

Contact Info

Product

Resources

About