The deformations of flagella are important in the motility of single- and multi-flagellated bacteria. Existing numerical methods have treated flagella as extensible filaments with a large extensional modulus, resulting in a stiff numerical problem and long simulation times. However, flagella are nearly inextensible, so to avoid large extensional stiffness, we introduce inextensible elastic rod models with hydrodynamics treated by a surface distribution of regularized Stokeslets. We benchmark this new model against previously described models of extensible elastic rods with hydrodynamics treated by a centerline distribution of regularized Stokeslets and rotlets, as well as a surface distribution of regularized Stokeslets. We compare the accuracy of the inextensible model with the extensible models and illustrate for which ratios of stretching/bending stiffness (which depend on the diameter of filament) the inextensible model is accurate and more efficient. We show that our inextensible approach can be markedly more efficient than the extensible models for many biological filaments. We also compare the accuracy of the centerline distribution of the Stokeslets and rotlets to the more accurate surface distribution of Stokeslets for modeling fluid-structure interactions of filaments.
We present a new diffuse interface model for the dynamics of inextensible vesicles in a viscous fluid. A new feature of this work is the implementation of the local inextensibility condition in the diffuse interface context. Local inextensibility is enforced by using a local Lagrange multiplier, which provides the necessary tension force at the interface. To solve for the local Lagrange multiplier, we introduce a new equation whose solution essentially provides a harmonic extension of the local Lagrange multiplier off the interface while maintaining the local inextensibility constraint near th…
We consider a classical elastohydrodynamic model of an inextensible filament undergoing planar motion in $\mathbb{R}^3$. The hydrodynamics are described by resistive force theory, and the fiber elasticity is governed by Euler-Bernoulli beam theory. Our aim is twofold: (1) Serve as a starting point for developing the mathematical analysis of filament elastohydrodynamics, particularly the analytical treatment of an inextensibility constraint, and (2) As an application, prove conditions on internal fiber forcing that allow a free-ended filament to swim. Our analysis of fiber swimming speed is sup…
Flexible filaments moving in viscous fluids are ubiquitous in the natural microscopic world. For example, the swimming of bacteria and spermatozoa as well as important physiological functions at organ-level, such as the cilia-induced motion of mucus in the lungs, or individual cell-level, such as actin filaments or microtubules, all employ flexible filaments moving in viscous fluids. As a result of fluid-structure interactions, a variety of nonlinear phenomena may arise in the dynamics of such moving flexible filaments. In this paper we derive the mathematical tools required to study filament-…
Fluid-structure simulations of slender inextensible filaments in a viscous fluid are often plagued by numerical stiffness. Recent coarse-graining studies have reduced the computational requirements of simulating such systems, though have thus far been limited to the motion of planar filaments. In this work we extend such frameworks to filament motion in three dimensions, identifying and circumventing coordinate-system singularities introduced by filament parameterisation via repeated changes of basis. The resulting methodology enables efficient and rapid study of the motion of flexible filamen…
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 s…
We study the collapse of an axisymmetric liquid filament both analytically and by means of a numerical model. The liquid filament, also known as ligament, may either collapse stably into a single droplet or break up into multiple droplets. The dynamics of the filament are governed by the viscosity and the aspect ratio, and the initial perturbations of its surface. We find that the instability of long viscous filaments can be completely explained by the Rayleigh-Plateau instability, whereas a low viscous filament can also break up due to end pinching. We analytically derive the transition betwe…
The morphological dynamics, instabilities and transitions of elastic filaments in viscous flows underlie a wealth of biophysical processes from flagellar propulsion to intracellular streaming, and are also key to deciphering the rheological behavior of many complex fluids and soft materials. Here, we combine experiments and computational modeling to elucidate the dynamical regimes and morphological transitions of elastic Brownian filaments in a simple shear flow. Actin filaments are employed as an experimental model system and their conformations are investigated through fluorescence microscop…
Peritrichous bacteria swim in viscous fluids by rotating multiple helical flagellar filaments. As the bacterium swims forward, all its flagella rotate in synchrony behind the cell in a helical bundle. When the bacterium changes its direction, the flagellar filaments unbundle and reorient the cell for a short period of time before returning to their bundled state and resuming swimming. This rapid bundling and unbundling is, at its heart, a mechanical process whereby hydrodynamic interactions balance with elasticity to determine the time-varying deformation of the filaments. Inspired by this bio…
We analyze experimentally the shape of a long elastic filament rotating in a viscous liquid. We identify a continuous but sharp transition from a straight to an helical shape, resulting from the competition between viscous stresses and elastic forces. This induced helicity generates a propulsive force along the axis of rotation. In addition, we show that the shape transition is associated with an unstable branch in the force-torque relation. A linearized model of the fluid-structure interaction is proposed to account for all the features of the non-linear filament dynamics.
In this fluid dynamics video we explore an array of different possible dynamics for a flexible filament sedimenting in a viscous fluid. The time-dependent shapes and trajectories of the filament are determined analytically and numerically by balancing viscous, elastic and gravitational forces in a slender-body theory for zero-Reynolds number flows. The dynamics are characterized by a single dimensionless elasto-gravitation number. The video shows the process of filament relaxation and reorientation, the formation of particle clouds, and finally a buckling instability.
Filaments are ubiquitous within the microscopic world. They occur frequently in both biological and industrial environments and display varied and rich dynamics. Their wide range of applications has spurred the development of a special branch of asymptotics focused on the behaviour of filaments, called slender-body theory. Slender-body theories are typically computationally efficient and focus on the mechanics of an isolated fibre that is not too curved. However, slender-body theories that work beyond these standard limits are needed to explore more complex systems. Recently, we developed tubu…
We present a numerical study on the rheology of semi-dilute and concentrated filament suspensions of different bending stiffness and Reynolds number, with the immersed boundary method used to couple the fluid and solid. The filaments are considered as one-dimensional inextensible slender bodies with fixed aspect ratio, obeying the Euler-Bernoulli beam equation. To understand the global suspension behavior we relate it to the filament microstructure, deformation and elastic energy and examine the stress budget to quantify the effect of the elastic contribution. At fixed volume fraction, the vis…
Kelvin wave and soliton propagation in classical viscous vortex filaments
Université Côte d’Azur（蔚蓝海岸大学）；Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange（蔚蓝海岸天文台，法国国家科学研究中心，拉格朗日实验室）；Université Côte d’Azur, CNRS, Institut de Physique de Nice (INPHYNI)（蔚蓝海岸大学，法国国家科学研究中心，尼斯物理研究所）
A filament of liquid is usually unstable and breaks up into small droplets, while a filament of polymer solution is known to be quite stable against such instability, and they form a stable configuration of filament connecting two spherical droplets. If the droplets are fixed in space, the liquid flows from the filament region to the droplet region to reduce the surface energy and the filament gets thinner. If the whole liquid is placed in another viscous fluid, the droplets approach each other, and the filament can get thicker. Here we study the dynamics of such system. We derive time evoluti…
Reversibility, Chaos, and Attractors in Periodically Sheared Elastic Filaments
University of Perugia（佩鲁贾大学）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究所国际理论科学中心）；PMMH-ESPCI, Université PSL, CNRS, Sorbonne Université, Université Paris Cité（巴黎高等物理化工学院国立科学研究中央机构索邦大学巴黎西岱大学）；University of California San Diego（加州大学圣地亚哥分校）
The inertialess fluid-structure interactions of active and passive inextensible filaments and slender- rods are ubiquitous in nature, from the dynamics of semi-flexible polymers and cytoskeletal filaments to cellular mechanics and flagella. The coupling between the geometry of deformation and the phys- ical interaction governing the dynamics of bio-filaments is complex. Governing equations negotiate elastohydrodynamical interactions with non-holonomic constraints arising from the filament inex- tensibility. Such elastohydrodynamic systems are structurally convoluted, prone to numerical erros, …
The equilibrium state of a flexible fiber settling in a viscous fluid is examined using a combination of macroscopic experiments, numerical simulations and scaling arguments. We identify three regimes having different signatures on this equilibrium configuration of the elastic filament: weak and large deformation regimes wherein the drag is proportional to the settling velocity as expected in Stokes flow and an intermediate elastic reconfiguration regime where the filament deforms to adopt a shape with a smaller drag which is no longer linearly proportional to the velocity.
The dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared to the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic, and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. Filaments of both non-uniform and uniform cross-sectiona…
We present a generalisation of efficient numerical frameworks for modelling fluid-filament interactions via the discretisation of a recently-developed, non-local integral equation formulation to incorporate regularised Stokeslets with half-space boundary conditions, as motivated by the importance of confining geometries in many applications. We proceed to utilise this framework to examine the drag on slender inextensible filaments moving near a boundary, firstly with a relatively-simple example, evaluating the accuracy of resistive force theories near boundaries using regularised Stokeslet seg…
Many species of bacteria swim through viscous environments by rotating multiple helical flagella. The filaments gather behind the cell body and form a close helical bundle, which propels the cell forward during a "run". The filaments inside the bundle cannot be continuously actuated, nor can they easily unbundle, if they are tangled around one another. The fact that bacteria can passively form coherent bundles, i.e. bundles which do not contain tangled pairs of filaments, may appear surprising given that flagella are actuated by uncoordinated motors. In this article, we establish the theoretic…
We consider sedimentation of a rigid helical filament in a viscous fluid under gravity. In the Stokes limit, the drag forces and torques on the filament are approximated within the resistive-force theory. We develop an analytic approximation to the exact equations of motion that works well in the limit of a sufficiently large number of turns in the helix (larger than two, typically). For a wide range of initial conditions, our approximation predicts that the centre of the helix itself follows a helical path with the symmetry axis of the trajectory being parallel to the direction of gravity. Th…
We study the behaviour of circular flexible loops sedimenting in a viscous fluid by numerical simulations and linear stability analysis. The numerical model involves a local slender-body theory approximation for the flow coupled to the Euler-Bernoulli elastic forces for an inextensible fibre. Starting from an inclined circle, we simulate the dynamics using truncated Fourier modes to observe three distinct regimes of motion: absolute stability, two-, and three-dimensional dynamics, depending on therelative importance of elastic and gravitational forces. We identify the governing parameter and d…
Theory and numerical simulations of the thinning of liquid threads at high superficial concentration of surfactants suggest the existence of an asymptotic regime where surface tension balances surface viscous stresses, leading to an exponential thinning with an $e$-fold time $F(Θ)\,(3μ_s + κ_s)/σ$, where $μ_s$ and $κ_s$ are the surface shear and dilatational viscosity coefficients, $σ$ is the interfacial tension, $Θ=κ_s/μ_s$, and $F(Θ)$ is a universal function. The potential use of this phenomenon to measure the surface viscosity coefficients is discussed.
The effects of elasticity on filament thinning and breakup are investigated in microchannel cross flow. When a viscous solution is stretched by an external immiscible fluid, a low 100 ppm polymer concentration strongly affects the breakup process, compared to the Newtonian case. Qualitatively, polymeric filaments show much slower evolution, and their morphology features multiple connected drops. Measurements of filament thickness show two main temporal regimes: flow- and capillary-driven. At early times both polymeric and Newtonian fluids are flow-driven, and filament thinning is exponential. …
Confinement and wall effects are known to affect the kinematics and propulsive characteristics of swimming microorganisms. When a solid body is dragged through a viscous fluid at constant velocity, the presence of a wall increases fluid drag, and thus the net force required to maintain speed has to increase. In contrast, recent optical trapping experiments have revealed that the propulsive force generated by human spermatozoa is decreased by the presence of boundaries. Here, we use a series of simple models to analytically elucidate the propulsive effects of a solid boundary on passively actua…
A vortex filament immersed in a non-ideal fluid, thanks to viscous diffusion, experiences a slipping motion with respect to the fluid. In recognition of this, in this paper, the effect of this slipping motion on the Hasimoto soliton propagating on a vortex filament is investigated, and an exact solution is given to describe this process. A strong slipping motion is shown to prevent the existence of the Hasimoto soliton. The critical slipping speed (above which the Hasimoto soliton fails to exist) is shown to be equal to the torsion.
Many microorganisms swim in a highly heterogeneous environment with obstacles such as fibers or polymers. To better understand how this environment affects microorganism swimming, we study propulsion of a cylinder or filament in a fluid with a sparse, stationary network of obstructions modeled by the Brinkman equation. The mathematical analysis of swimming speeds is investigated by studying an infinite-length cylinder propagating lateral or spiral displacement waves. For fixed bending kinematics, we find that swimming speeds are enhanced due to the added resistance from the fibers. In addition…
Many cells exploit the bending or rotation of flagellar filaments in order to self-propel in viscous fluids. While appropriate theoretical modelling is available to capture flagella locomotion in simple, Newtonian fluids, formidable computations are required to address theoretically their locomotion in complex, nonlinear fluids, e.g.~mucus. Based on experimental measurements for the motion of rigid rods in non-Newtonian fluids and on the classical Carreau fluid model, we propose empirical extensions of the classical Newtonian resistive-force theory to model the waving of slender filaments in n…
We perform experiments to investigate the relaxation of a highly deformed elastic filament at a liquid-air interface. The dynamics for filaments of differing length, diameter and elastic modulus collapse to a single curve when the time-dependence is scaled by a time scale $τ= 8 πμL_o^4/B$. The relaxation, however, is completed in a very small fraction of the time $τ$. Even though the time scale $τ$ can be obtained by balancing the linear bending and viscous forces, it appears to control the highly nonlinear regime of our experiments. Nonlinear numerical simulations show that the force due to t…
We study experimentally the dynamics of one and two ball-chains settling under gravity in a very viscous fluid at a Reynolds number much smaller than unity. We demonstrate that single ball-chains in most cases do not tend to be planar and often rotate, not keeping the ends at the same horizontal level. Shorter ball-chains usually form shapes resembling distorted U, and longer ones in the early stage of the evolution form a shape resembling distorted W, and later deform non-symmetrically and significantly out of plane. This behaviour is reproduced in our numerical simulations for a single very …
Parametrically excited standing waves are observed on a cylindrical fluid filament. These are the cylindrical analog of Faraday instability in a flat surface or spherical droplet. Using the Floquet technique, linear stability analysis has been investigated on a viscous cylindrical fluid surface, which is subjected to a time-periodic radial acceleration. Viscosity has a significant impact on the critical forcing amplitude as well as the dispersion relation of the non-axis symmetric patterns. The effect of viscosity on onset parameters of the pattern with azimuthal wavenumber, $m=1$, has shown d…
A highly organized and densely packed lattice of molecular machinery within the sarcomeres of muscle cells powers contraction. Although many of the proteins that drive contraction have been studied extensively, the mechanical impact of fluid shearing within the lattice of molecular machinery has received minimal attention. It was recently proposed that fluid flow augments substrate transport in the sarcomere, however, this analysis used analytical models of fluid flow in the molecular machinery that could not capture its full complexity. By building a finite element model of the sarcomere, we …
The breakup of a jet of a viscous fluid with viscosity $μ_{1}$ immersed into another viscous fluid with viscosity $μ_{2}$ is considered in the limit when the viscosity ratio $λ=μ_{1}/μ_{2}$ is close to zero. We show that, in this limit, a transition from ordinary continuous selfsimilarity to discrete selfsimilarity takes place as $λ$decreases. The result being that instead of a single point breakup, the rupture of the inner jet occurs through the appearance of an infinite sequence of filaments of decreasing size that will eventually produce infinite sequences of bubbles of the inner fluid insi…
The range and speed of direct ink writing, the workhorse of 3d and 4d printing, is limited by the practice of liquid extrusion from a nozzle just above the surface to prevent instabilities to cause deviations from the required print path. But what if could harness the ``fluid rope trick", whence a thin stream of viscous fluid falling from a height spontaneously folds or coils, to write specified patterns on a substrate? Using Deep Reinforcement Learning we control the motion of the extruding nozzle and thence the fluid patterns that are deposited on the surface. The learner (nozzle) repeatedly…
Peritrichous bacteria such as Escherichia coli swim in viscous fluids by forming a helical bundle of flagellar filaments. The filaments are spatially distributed around the cell body to which they are connected via a flexible hook. To understand how the swimming direction of the cell is determined, we theoretically investigate the elastohydrodynamic motility problem of a multi-flagellated bacterium. Specifically, we consider a spherical cell body with a number N of flagella which are initially symmetrically arranged in a plane in order to provide an equilibrium state. We analytically solve the…
Most motile bacteria swim in viscous fluids by rotating multiple helical flagellar filaments. These semi-rigid filaments repeatedly join ('bundle') and separate ('unbundle'), resulting in a two-gait random walk-like motion of the cell. In this process, hydrodynamic interactions between the filaments are known to play an important role and can be categorised into two distinct types: direct interactions mediated through flows that are generated through the actuation of the filaments themselves, and indirect interactions mediated through the motion of the cell body (i.e. flows induced in the swim…
Fluid-suspended microorganisms have evolved different swimming and feeding strategies in order to cope with an environment dominated by viscous effects. For instance ciliated organisms rely on the collective motion of flexible appendices to move and feed. By performing a non-reciprocal motion, flexible filaments can produce a net propulsive force, or pump fluid, in the absence of inertia. Inspired by such fundamental concept, we propose a strategy to produce macroscopic pumping and mixing in creeping flow. We measure experimentally the net motion of a Newtonian viscous fluid induced by the rec…
Mechanical properties of cellular structures, including the cell cytoskeleton, are increasingly used as biomarkers for disease diagnosis and fundamental studies in cell biology. Recent experiments suggest that the cell cytoskeleton and its permeating cytosol, can be described as a poroelastic (PE) material. Biot theory is the standard model used to describe PE materials. Yet, this theory does not account for the fluid viscous stress, which can lead to inaccurate predictions of the mechanics in the dilute filamentous network of the cytoskeleton. Here, we adopt a two-phase model that extends Bio…
A thin thread of viscous fluid falling onto a moving belt generates a surprising variety of patterns depending on the belt speed, fall height, flow rate, and fluid properties. Here we simulate this experiment numerically using the Discrete Viscous Threads method that can predict the non-steady dynamics of thin viscous filaments, capturing the combined effects of inertia, stretching, bending and twisting. Our simulations successfully reproduce nine out of ten different patterns previously seen in the laboratory, and agree closely with the experimental phase diagram of Morris et al.\ (2008). We …
Motile eukaryotic cells propel themselves in viscous fluids by passing waves of bending deformation down their flagella. An infinitely long flagellum achieves a hydrodynamically optimal low-Reynolds number locomotion when the angle between its local tangent and the swimming direction remains constant along its length. Optimal flagella therefore adopt the shape of a helix in three dimensions (smooth) and that of a sawtooth in two dimensions (non-smooth). Physically, biological organisms (or engineered micro-swimmers) must expend internal energy in order to produce the waves of deformation respo…
The pinch-off dynamics of a liquid thread has been studied through numerical simulations and theoretical analysis. Occurring at small length scales, the pinch-off dynamics admits similarity solutions that can be classified into the Stokes regime and the diffusion-dominated regime, with the latter being recently experimentally observed in aqueous two-phase systems [Phys. Rev. Lett. 123, 134501 (2019)]. Derived by applying Onsager's variational principle, the Cahn-Hilliard-Navier-Stokes model is employed as a minimal model capable of describing the interfacial motion driven by not only advection…
Cilia and flagella are highly conserved slender organelles that exhibit a variety of rhythmic beating patterns from non-planar cone-like motions to planar wave-like deformations. Although their internal structure, composed of a microtubule-based axoneme driven by dynein motors, is known, the mechanism responsible for these beating patterns remains elusive. Existing theories suggest that the dynein activity is dynamically regulated, via a geometric feedback from the cilium's mechanical deformation to the dynein force. An alternative, open-loop mechanism based on a 'flutter' instability was rece…
Painting is a fluid mechanical process. The action of covering a solid surface with a layer of a viscous fluid is one of the most common human activities; virtually all man-made surfaces are painted to provide protection against the environment or simply for decoration. This process, in an industrial context, has been vastly studied and it is well understood. In case of artistic painting the purpose is different. Painters learn how to manipulate the non-uniform deposition of paint onto a surface, through lengthy empirical testing of the action and modifying the physical properties of the fluid…
We present results from a systematic direct-numerical simulation study of pressure fluctuations in an unforced, incompressible, homogeneous, and isotropic, three-dimensional turbulent fluid. At cascade completion, isosurfaces of low pressure are found to be organised as slender filaments, whereas the predominant isostructures appear sheet-like. We exhibit several new results, including plots of probability distributions of the spatial pressure-difference, the pressure-gradient norm, and the eigenvalues of the pressure-hessian tensor. Plots of the temporal evolution of the mean pressure-gradien…
