3 research outputs found

    Efficient Implementation of Elastohydrodynamics via Integral Operators

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    The dynamics of geometrically non-linear flexible filaments play an important role in a host of biological processes, from flagella-driven cell transport to the polymeric structure of complex fluids. Such problems have historically been computationally expensive due to numerical stiffness associated with the inextensibility constraint, as well as the often non-trivial boundary conditions on the governing high-order PDEs. Formulating the problem for the evolving shape of a filament via an integral equation in the tangent angle has recently been found to greatly alleviate this numerical stiffness. The contribution of the present manuscript is to enable the simulation of non-local interactions of multiple filaments in a computationally efficient manner using the method of regularized stokeslets within this framework. The proposed method is benchmarked against a non-local bead and link model, and recent code utilizing a local drag velocity law. Systems of multiple filaments (1) in a background fluid flow, (2) under a constant body force, and (3) undergoing active self-motility are modeled efficiently. Buckling instabilities are analyzed by examining the evolving filament curvature, as well as by coarse-graining the body frame tangent angles using a Chebyshev approximation for various choices of the relevant non-dimensional parameters. From these experiments, insight is gained into how filament-filament interactions can promote buckling, and further reveal the complex fluid dynamics resulting from arrays of these interacting fibers. By examining active moment-driven filaments, we investigate the speed of worm- and sperm-like swimmers for different governing parameters. The MATLAB(R) implementation is made available as an open-source library, enabling flexible extension for alternate discretizations and different surrounding flows.Comment: 37 pages, 17 figure

    Doing more with less: the flagellar end piece enhances the propulsive effectiveness of human spermatozoa

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    Spermatozoa self-propel by propagating bending waves along a predominantly active elastic flagellum. The organized structure of the "9 + 2" axoneme is lost in the most-distal few microns of the flagellum, and therefore this region is unlikely to have the ability to generate active bending; as such it has been largely neglected in biophysical studies. Through elastohydrodynamic modeling of human-like sperm we show that an inactive distal region confers significant advantages, both in propulsive thrust and swimming efficiency, when compared with a fully active flagellum of the same total length. The beneficial effect of the inactive end piece on these statistics can be as small as a few percent but can be above 430%. The optimal inactive length, between 2-18% of the total length, depends on both wavenumber and viscous-elastic ratio, and therefore is likely to vary in different species. Potential implications in evolutionary biology and clinical assessment are discussed.Comment: To Appear, Physical Review Fluids. 25 pages, 14 figure

    Elastohydrodynamics of actuated slender bodies in Stokes flows: methods, tools, and simulations of microscale motility

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    Countless microorganisms use slender elastic filaments to affect and traverse their surroundings. From the metachronal synchronisation of tracheal cilia to transport mu- cus, to the propulsive undulatory motion of sperm flagella, cells of all shapes and sizes use filament-like structures in a variety of fluid environments. The key physics in such environments arise from the interactions between slender-body filament-like elastic structures comprising the organism and surrounding viscous fluid. Understanding the coupled elastohydrodynamics of microscale propulsive mechanisms can provide in- sights into cell ultrastructure and rheology, paving the way for experimental studies and data interpretation, and suggesting novel diagnostics useful to researchers and clinicians alike. In this thesis, we present two mathematical models for describing the dynamics of elastohydrodynamic filaments, in particular for simulating the motion of human sper- matozoa. Both methods are accompanied by bespoke implementations in open source MATLAB® code. The methods, dubbed the EIF and SPX methods, differ principally in dependent variables, which is the angle made between the filament centreline and a fixed axis in the EIF, and nonplanar centreline position, tension, and twist curvature in the SPX model. Key considerations in both approaches are (a) accuracy, improving upon many de-facto standard approaches by considering nonlocal hydrodynamic in- teractions and nonlinear geometries, (b) generalisability, so that the proposed methods can be applied to a variety of problems in a straightforward manner, and (c) efficiency, to reduce the formidable computational requirements that have historically stood as barriers to entry for rapid and reliable simulation studies. The methods developed de- rive from exploiting the slenderness property of the propulsive structures (i.e. cilia and flagella); auxiliary structures such as cell bodies or heads are not required to be slender provided the overall cell length is much longer than the width. The EIF method is formulated and applied to simulate groups of planar active and passive filaments in quiescent fluid, shear flows, and sedimenting due to gravity. By using novel discretisation techniques and exploitation of optimised MATLAB® built-in algorithms, the resulting numerical implementation enables rapid simulations on even modest readily-available computer hardware. Expanding upon the EIF, the SPX method provides a model for simulating filaments and monoflagellate cells moving in three dimensions. Maintaining accuracy and non- local interactions in the fluid dynamics through the method of regularised stokeslets, the elasticity model is generalised to account for arbitrary bend and twist deforma- tions. The presented tools and methodology are applied to simulate human spermato- zoa locomoting through a rhythmic twist and bend motion. Results indicate that the rate of axial rolling exhibited by nonplanar swimmers is reduced in the presence of a plane wall, with the rate of reduction dependent on the angle of approach towards the boundary and a dimensionless parameter characterising the relative ratio of twist drag to viscous drag in the system
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