Coarse graining the dynamics of immersed and driven fiber assemblies

Abstract

An important class of fluid-structure problems involve the dynamics of ordered arrays of immersed, flexible fibers. While specialized numerical methods have been developed to study fluid-fiber systems, they become infeasible when there are many, rather than a few, fibers present, nor do these methods lend themselves to analytical calculation. Here, we introduce a coarse-grained continuum model, based on local-slender body theory, for elastic fibers immersed in a viscous Newtonian fluid. It takes the form of an anisotropic Brinkman equation whose skeletal drag is coupled to elastic forces. This model has two significant benefits: (1) the density effects of the fibers in a suspension become analytically manifest, and (2) it allows for the rapid simulation of dense suspensions of fibers in regimes inaccessible to standard methods. As a first validation, without fitting parameters, we achieve very reasonable agreement with 3D Immersed Boundary simulations of a bed of anchored fibers bent by a shear flow. Secondly, we characterize the effect of density on the relaxation time of fiber beds under oscillatory shear, and find close agreement to results from full numerical simulations. We then study buckling instabilities in beds of fibers, using our model both numerically and analytically to understand the role of fiber density and the structure of buckling transitions. We next apply our model to study the flow-induced bending of inclined fibers in a channel, as has been recently studied as a flow rectifier, examining the nature of the internal flows within the bed, and the emergence of inhomogeneous permeability. Finally, we extend the method to study a simple model of metachronal waves on beds of actuated fibers, as a model for ciliary beds. Our simulations reproduce qualitatively the pumping action of coordinated waves of compression through the bed. I. INTRODUCTIONMany fundamental hydrodynamic phenomena, particularly in biology, involve the interaction of structured arrays of immersed fibers, often anchored to a substrate [21]. For example, in eukaryotic cells, arrays of aligned microtubules in the spindle orchestrate the segregation of chromosomes, while those around centrosomes help position the spindle prior to cell division [14]. During midoogenesis of Drosophila, kinesin motors interacting with the microtubule cytoskeleton drive largescale coherent flows known as cytoplasmic streaming [10]. Another example is the beds of driven cilia that pump fluid, move mucus, or propel microorganisms [4]. In microfluidic engineering, fabricated arrays of flexible fibers are the elements of proposed soft flow rectifiers [1]. A central aspect of all of these examples is that the relevant dynamics is collective and not well-described by the dynamics of a single fiber.Given the importance of fluid-fiber systems, specialized computational methods have been developed to treat them, most especially in the zero Reynolds limit where flows are governed by the Stokes equation. These approaches include the use of nonlocal slender body the...

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