Coiling of elastic rods on rigid substrates

Jawed, Mohammad; Da, Fang; Joo, Jungseock; Grinspun, Eitan; Reis, Pedro

doi:10.1073/pnas.1409118111

Cited by 95 publications

(75 citation statements)

References 28 publications

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“…Similar coiling patterns can be found in a number of industrial or everyday situations, such as the production of nonwoven textiles [9], the laying down of "squiggles" of icing on cakes, Jackson Pollock's action painting, in which paint from a moving brush dribbles onto a stationary horizontal canvas [10], or when transoceanic fiber-optic cables are deposited from a vessel onto the ocean bed [11]. The latter are elastic rather than viscous [12][13][14], showing that the patterns are robust with respect to a change in the thread rheology.…”

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confidence: 70%

Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities

et al. 2015

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Thin, viscous fluid threads falling onto a moving belt behave in a way reminiscent of a sewing machine, generating a rich variety of periodic stitchlike patterns including meanders, W patterns, alternating loops, and translated coiling. These patterns form to accommodate the difference between the belt speed and the terminal velocity at which the falling thread strikes the belt. Using direct numerical simulations, we show that inertia is not required to produce the aforementioned patterns. We introduce a quasistatic geometrical model which captures the patterns, consisting of three coupled ordinary differential equations for the radial deflection, the orientation, and the curvature of the path of the thread's contact point with the belt. The geometrical model reproduces well the observed patterns and the order in which they appear as a function of the belt speed.

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Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities

et al. 2015

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“…A slender elastic rod deployed onto a moving substrate can form a variety of geometrically nonlinear periodic patterns [1][2][3] that are reminiscent of the stitches of a sewing machine. As such, this problem has come to be known as the elastic sewing machine (ESM) [1].…”

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confidence: 99%

A Geometric Model for the Coiling of an Elastic Rod Deployed Onto a Moving Substrate

Jawed

Brun

Reis

2015

Journal of Applied Mechanics

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We report results from a systematic numerical investigation of the nonlinear patterns that emerge when a slender elastic rod is deployed onto a moving substrate; a system also known as the elastic sewing machine (ESM). The discrete elastic rods (DER) method is employed to quantitatively characterize the coiling patterns, and a comprehensive classification scheme is introduced based on their Fourier spectrum. Our analysis yields physical insight on both the length scales excited by the ESM, as well as the morphology of the patterns. The coiling process is then rationalized using a reduced geometric model (GM) for the evolution of the position and orientation of the contact point between the rod and the belt, as well as the curvature of the rod near contact. This geometric description reproduces almost all of the coiling patterns of the ESM and allows us to establish a unifying bridge between our elastic problem and the analogous patterns obtained when depositing a viscous thread onto a moving surface; a well-known system known as the fluid-mechanical sewing machine (FMSM).

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“…Both DER and LSBT were independently validated against precision experiments in Ref. [24] and in Ref. [12], respectively.…”

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

Jawed¹,

Khouri²,

Da³

et al. 2015

Phys. Rev. Lett.

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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...

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Coiling of elastic rods on rigid substrates

Cited by 95 publications

References 28 publications

Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities

Liquid Ropes: A Geometrical Model for Thin Viscous Jet Instabilities

A Geometric Model for the Coiling of an Elastic Rod Deployed Onto a Moving Substrate

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

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