Dynamics of flexible fibers in viscous flows and fluids

Dynamics of flexible fibers in viscous flows and fluids

Table of Contents


The dynamics and deformations of immersed flexible fibers are at the heart of important industrial and biological processes, induce peculiar mechanical and transport properties in the fluids that contain them, and are the basis for novel methods of flow control. Here we focus on the low Reynolds number regime where advances in studying these fiber-fluid systems have been especially rapid. On the experimental side, this is due to new methods of fiber synthesis, microfluidic flow control, and microscope-based tracking measurement techniques. Likewise, there have been continuous improvements in the specialized mathematical modeling and numerical methods needed to capture the interactions of slender flexible fibers with flows, boundaries, and each other.


flexible fibers, microfluidics, flow control, elasticity, buckling, hydrodynamics, complex fluids, suspensions


An important class of micro-scale fluid-structure interactions involves the interactions and deformations of flexible fibers with fluid flows. This is evident in the many biological transport processes, such as microorganismal swimming ([71]) or nuclear positioning in eukaryotic cells ([124]), that involve flexible fibers either actuated or passive. Fibers are the microstructure of many complex fluids – biological, industrial, and synthetic – studied both for their scientific and industrial importance and for their peculiar mechanical responses ([11]). Such suspensions are particularly challenging to study because the suspended fibers have many degrees of freedom in deformation, and can exhibit macroscopic instabilities. This makes the interaction of fibers with background flows surprisingly complex. Flexible fibers, both anchored and freely suspended, have been studied of late for the purposes of microfluidic flow control, and when actuated can exhibit complex collective and transport dynamics. This review focuses on the low Reynolds number regime, as progress on fiber-fluid interactions in this regime has been especially rapid. On the experimental side, this is due to improvements in fiber synthesis and characterization, microfluidic flow control, and improved microscope-based measurement and particle tracking techniques. By taking advantage of the relative simplicity of the Stokes equations (as opposed to Navier-Stokes), combined with adaptive resolution and fast summation approaches, there has been a likewise rapid improvement in numerical methods for simulating the deformations and interactions of fibers with flows, as well as with each other and other immersed structures. Coarse-grained descriptions of fiber assemblies and suspensions remain in their early stages, but their development is being sped by these new numerical methodologies.

In outline, we first discuss the current state-of-art in experimental synthesis and measurement techniques, and in numerical methods for dynamical simulation fiber-fluid interactions. We then describe the results of both experiments and theory for the dynamics of free and anchored fibers, followed by a review of the current state of research in many-fiber/fluid systems. We close with a discussion of future directions.

Setting the stage

Here we briefly introduce a few important physical parameters and relations. Consider a slender elastic fiber of length L, of circular cross-section with radius a (hence  = a/L  1), and flexural rigidity E = Y I with Y the material Young’s modulus and I the areal moment of inertia (I = πa4/4). This fiber is immersed in a Newtonian fluid of shear viscosity µ with the fluid motion characterized by a strain-rate ˙γ. Neglecting inertial forces, three important forces are at play: Brownian forces ∼ kBT /L (kB is the Boltzmann constant and T the temperature), drag forces ∼ µγL˙ 2, and elasticity forces ∼ E/L2. For most of the work reviewed here, though not all, viscous drag and elasticity forces dominate Brownian forces. That predominance requires that lp/L  1 and P e = 8πµγL˙ 3/kbT  1, where lp = E/kbT is the persistence length of the fiber against thermal fluctuations, and the P´eclet number, P e, is the ratio of viscous to Brownian forces. Taking water as the solvent, a fluid strain-rate of ˙γ = 1 s −1 and fiber of L = 4 µm we find P e ∼ 400. For fibers of a length greater than a few microns P e  1 and center of mass diffusion can thus always be neglected compared to advection by viscous flow. Now for a material modulus of Y = 1 GP a and an aspect ratio  = 10−2, we find lp/L ∼ 105 and Brownian forces can be neglected over elastic forces. This is the case of most of the synthetic fibers treated in this review. Decreasing the aspect ratio to  = 10−3 and Y by an order of magnitude reduces lp/L to lp/L ∼ 1 and shape fluctuations resulting from Brownian forces will become important. This is the case of semi-flexible polymers as for example actin filaments.

Finally, while this dimensionless parameter will appear naturally later in the review, we introduce ˜η = 8πµγL˙ 4/Ec, where c = − ln(e2 ), as the ration between viscous and elastic forces, a control parameter in many fiber-fluid problems.


As should be clear, flexible fibers interacting with flowing liquids present a rich source of problems in fluid/structure interaction. They also present complicated phenomena, which require sophisticated experimental techniques to observe and measure, and complicated theories through which to understand them. There are many areas of open inquiry, among them the interactions of fibers with complex media. Viscoelastic responses are typical of many biological environments, such as the reproductive tract [36] or inside of the cell [143]. There is a developing literature on the swimming of microorganisms in viscoelastic fluids ([33, 134]). Using a 2D slender body actuated elastic model, [135] recently studied the role of body flexibility for undulatory swimming in viscoelastic fluids. There have been few if any studies of flexible fibers interacting with complex flows of complex fluids; Recent simulations of [147] (Fig. 5f) of fiber transport by cellular viscoelastic flow is the first of which we are aware. The fundamental theoretical difficulty is the necessity of evolving bulk elastic stresses via transport nonlinearities. This makes fluid-structure problems, much less those with multiple elastic bodies, very challenging for viscoelastic flows and morally equivalent to those for the Navier-Stokes equations. On the experimental side, it remains challenging to synthesize complex fluids with well-characterized (and simple!) rheological responses.

New kinds of mathematical coarse-grained descriptions need to be developed to describe the collective behavior of flexible fibers in fluids, especially when hydrodynamic interactions are strong. One regime where progress is being made is when the fibers can be considered as well-aligned. In recent work, Stein & Shelley (in preparation) have developed a continuum Brinkman-type model that captures the anisotropic drag from elongated flexible structures to compute the flow feedback to the bending and tensile response of a porous elastic medium ([89, 133]). Figure 9d (top right) shows the model’s result in simulating a soft flow rectified by the bending of a bed of tilted and anchored elastic fibers (Fig. 9d (top middle) from [1]).

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FULL Paper PDF file:


Dynamics of flexible fibers in viscous flows and fluids



O. du Roure,1, A. Lindner,1,2, E.N. Nazockdast3, and M.J. Shelley4,5

1 Laboratoire de Physique et Mecanique des Milieux Heterog‘enes (PMMH), ESPCI Paris, PSL University, CNRS, Sorbonne University, and Paris Diderot University, 75005 Paris, France; email: olivia.duroure@espci.fr

2 Global Station for Soft Matter, Global Institution for Collaborative Research and Education, Hokkaido University, Sapporo 060-0808, Japan

3 Applied Physical Sciences, the University of North Carolina at Chapel Hill, North Carolina 27514, USA

4 The Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA and

5 Center for Computational Biology, Flatiron Institute, New York, NY 10010, USA (Dated: May 23, 2019)




Dynamics of flexible fibers in viscous flows and fluids

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Annual Review of Fluid Mechanics



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Nasim Gazerani was born in 1983 in Arak. She holds a Master's degree in Software Engineering from UM University of Malaysia.

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Professor Siavosh Kaviani was born in 1961 in Tehran. He had a professorship. He holds a Ph.D. in Software Engineering from the QL University of Software Development Methodology and an honorary Ph.D. from the University of Chelsea.

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Somayeh Nosrati was born in 1982 in Tehran. She holds a Master's degree in artificial intelligence from Khatam University of Tehran.