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A three-dimensional model of flagellar swimming in a Brinkman fluid

Published online by Cambridge University Press:  14 February 2019

NguyenHo Ho ,
Karin Leiderman  and
Sarah Olson Open the ORCID record for Sarah Olson [Opens in a new window]
NguyenHo Ho
Affiliation:
Department of Mathematics, Bridgewater State University, 131 Summer Street, Bridgewater, MA 02325, USA
Karin Leiderman
Affiliation:
Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois Street, Golden, CO 80401, USA
Sarah Olson*
Affiliation:
Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA
*
†Email address for correspondence: sdolson@wpi.edu
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Abstract

We investigate three-dimensional flagellar swimming in a fluid with a sparse network of stationary obstacles or fibres. The Brinkman equation is used to model the average fluid flow where a flow-dependent term, including a resistance parameter that is inversely proportional to the permeability, captures the effects of the fibres on the fluid. To solve for the local linear and angular velocities that are coupled to the flagellar motion, we extend the method of regularized Brinkmanlets to incorporate a Kirchhoff rod, discretized as point forces and torques along a centreline. Representing a flagellum as a Kirchhoff rod, we investigate emergent waveforms for different preferred strain and twist functions. Since the Kirchhoff rod formulation allows for out-of-plane motion, in addition to studying a preferred planar sine wave configuration, we also study a preferred helical configuration. Our numerical method is validated by comparing results to asymptotic swimming speeds derived for an infinite-length cylinder propagating planar or helical waves. Similar to the asymptotic analysis for both planar and helical bending, we observe that with small amplitude bending, swimming speed is always enhanced relative to the case with no fibres in the fluid (Stokes) as the resistance parameter is increased. For regimes not accounted for with asymptotic analysis, i.e. large amplitude planar and helical bending, our model results show a non-monotonic change in swimming speed with respect to the resistance parameter; a maximum swimming speed is observed when the resistance parameter is near one. The non-monotonic behaviour is due to the emergent waveforms; as the resistance parameter increases, the swimmer becomes incapable of achieving the amplitude of its preferred configuration. We also show how simulation results of slower swimming speeds for larger resistance parameters are actually consistent with the asymptotic swimming speeds if work in the system is fixed.

JFM classification

Biological Fluid Dynamics: Micro-organism dynamics Low-Reynolds-number flows: Porous media Biological Fluid Dynamics: Propulsion
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 864 , 10 April 2019 , pp. 1088 - 1124
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Copyright
© 2019 Cambridge University Press 

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