Shape Transition and Propulsive Force of an Elastic Rod Rotating in a Viscous Fluid

Abstract

The deformation of thin rods in a viscous liquid is central to the mechanics of motility in cells ranging from Escherichia coli to sperm. Here we use experiments and theory to study the shape transition of a flexible rod rotating in a viscous fluid driven either by constant torque or at constant speed. The rod is tilted relative to the rotation axis. At low applied torque, the rod bends gently and generates small propulsive force. At a critical torque, the rotation speed increases abruptly and the rod forms a helical shape with much greater propulsive force. We find good agreement between theory and experiment.PACS numbers: 87.16. Qp, 46.70.Hg Understanding how flagella and cilia work is a central aim of the field of cell motility. The problem may be split into two parts, both of which involve physics: the means of actuation, and the fluid-structure interaction. In this letter we consider the fluid-structure interaction for thin filaments in a viscous fluid. At micron scales, viscous effects dominate inertia, and the fluid-structure interaction problem simplifies because the Stokes equations governing the fluid motion are linear. Gray and Hancock used this linearity to develop a simple theory that successfully predicted the swimming speed of a sperm cell with a loadindependent pattern of bending waves propagating along the flagellum [1]. Soon after, Machin considered the fluid-structure interaction [2]. He argued that the motors must be distributed all along the length of the flagellum, since, for small amplitudes, a passive flexible rod waved at one end has an exponentially decaying envelop of deflection, whereas the amplitude of deflection in real flagellar bending waves increases slightly with distance from the head [2]. The shapes and propulsive forces of a passive rod actuated at one end have recently been examined theoretically [3,4] and experimentally [4]. Although sperm flagella are not passive, the results of [2,3,4] are important for modeling real flagella since the modes that Machin found also enter models in which the flagellum is actuated along its entire length [5].Rotating flagella are also common. For example, nodal cilia [6] have an internal structure similar to that of sperm flagella. However, instead of beating in a plane like most sperm flagella, nodal cilia rotate along the surface of an imaginary cone. The flow set up by these flagella has been implicated in the formation of left-right asymmetry in developing embryos (see [6] and references therein). Bacterial flagella provide another example. These flagella are helical, much thinner than eukaryotic flagella, and driven by a rotary motor embedded in the cell wall. Fluid-structure interactions are important for polymorphic transformations in swimming bacteria [7] and the bundling of multiple flagella [8].Complementary to the problem of understanding how biological flagella work is the problem of building an arti-

High-speed propulsion of flexible nanowire motors: Theory and experiments

“…6 The error bars in the Figure 2 are attributed to the uncertainty of the position of the ABF based on the resolution of the recorded images. The fluctuations in the curves are attributed to unmodeled boundary conditions such as wall effects [23][24][25] and intermolecular interactions of the ABF with the substrate. The maximum velocity that we have achieved with a 2.0 mT field is 18 µm/s, which is comparable to bacteria, such as E. coli, that swim by rotating their flagella with a frequency of about 100 Hz at room temperature.…”

“…This simple physical argument shows that a combination of rotational actuation and nanowire flexibility is critical for this mode of propulsion. Recently, the dynamics of tethered elastic filaments actuated by precessing magnetic fields has been studied [19][20][21][22][23][24][25][26] and chiral deformation along the filament has been found to produce propulsive force and fluid pumping. The swimming behaviours of an untethered flexible magnetic filament displaying chiral deformation was also addressed computationally [27].…”

“…We parameterized the doublet in terms of the coordinates of the centers of the two spheres x a and x b , and the the bounding plane, b=að<1Þ accounts for the size difference in the two particles of the doublet, and à AE að1 þ Þ=½hð1 þ AE3 Þ [24]. The precession velocity of the doublet director is imposed by the applied magnetic field.…”

Propulsive force measurements and flow behavior of undulatory swimmers at low Reynolds number

“…10,11 In recent years, analytical studies on the motility of micro-organisms at low Re have been complimented by a growing number of experimental investigations. For example, scaled-up models of elastic tails 12,13 and bacterial flagella 14 are commonly used to measure filament shapes, velocity fields, and propulsive forces. 15 At smaller scales, the kinematics of single bacterium have been investigated 16 and the shapes of an oscillating passive actin filament have been probed.…”

Numerical modelling of chirality-induced bi-directional swimming of artificial flagella

“…The quest for a simple and viable design of biomimetic bacterial micro-swimmers has attracted considerable attention in the literature [9,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Various experimental and analytical studies have been performed to analyse the evolution of the chiral shape for an elastic filament or rod-like structure subjected to external actuation [18,19,[21][22][23][25][26][27][28]. Also, in an experimental study performed by Garstecki et al [15] a flexible planar structure is deformed into a chiral shape on-the-fly owing to the opposing torques imposed by the externally applied magnetic field and the resisting viscous forces of the surrounding fluid.…”