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Shape of optimal active flagella

Published online by Cambridge University Press:  01 August 2013

Eric Lauga  and
Christophe Eloy
Eric Lauga
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
Christophe Eloy*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
*
†Email address for correspondence: Christophe.Eloy@irphe.univ-mrs.fr
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Abstract

Many eukaryotic cells use the active waving motion of flexible flagella to self-propel in viscous fluids. However, the criterion governing the selection of particular flagellar waveforms among all possible shapes has proved elusive so far. To address this question, we derive computationally the optimal shape of an internally forced periodic planar flagellum deforming as a travelling wave. The optimum is here defined as the shape leading to a given swimming speed with minimum energetic cost. To calculate the energetic cost, we consider the irreversible internal power expended by the molecular motors forcing the flagellum, only a portion of which is dissipated in the fluid. This optimization approach allows us to derive a family of shapes depending on a single dimensionless number quantifying the relative importance of elastic to viscous effects: the Sperm number. The computed optimal shapes are found to agree with the waveforms observed on spermatozoon of marine organisms, thus suggesting that these eukaryotic flagella might have evolved to be mechanically optimal.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Biological Fluid Dynamics: Micro-organism dynamics Biological Fluid Dynamics: Swimming/flying

Information

Type
Rapids
Information
Journal of Fluid Mechanics , Volume 730 , 10 September 2013 , R1
Copyright
©2013 Cambridge University Press 

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Footnotes

‡

Current address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK. Email address for correspondence: e.lauga@damtp.cam.ac.uk

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