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Geometric phase methods with Stokes theorem for a general viscous swimmer

Koens, Lyndon; Lauga, Eric

Authors

Eric Lauga



Abstract

The geometric phase techniques for swimming in viscous flows express the net displacement of a swimmer as a path integral of a field in configuration space. This representation can be transformed into an area integral for simple swimmers using the Stokes theorem. Since this transformation applies for any loop, the integrand of this area integral can be used to help design these swimmers. However, the extension of this Stokes theorem technique to more complicated swimmers is hampered by problems with variables that do not commute and by how to visualise and understand the higher-dimensional spaces. In this paper, we develop a treatment for each of these problems, thereby allowing the displacement of general swimmers in any environment to be designed and understood similarly to simple swimmers. The net displacement arising from non-commuting variables is tackled by embedding the integral into a higher-dimensional space, which can then be visualised through a suitability constructed surface. These methods are developed for general swimmers and demonstrated on three benchmark examples: Purcell's two-hinged swimmer, an axisymmetric squirmer in free space and an axisymmetric squirmer approaching a free interface. We show in particular that, for swimmers with more than two modes of deformation, there exists an infinite set of strokes that generate each net displacement. Hence, in the absence of additional restrictions, general microscopic swimmers do not have a single stroke that maximises their displacement.

Citation

Koens, L., & Lauga, E. (2021). Geometric phase methods with Stokes theorem for a general viscous swimmer. Journal of Fluid Mechanics, 916, Article A17. https://doi.org/10.1017/jfm.2021.181

Journal Article Type Article
Acceptance Date Feb 24, 2021
Online Publication Date Apr 6, 2021
Publication Date 2021
Deposit Date Jan 25, 2022
Publicly Available Date Mar 29, 2024
Journal Journal of Fluid Mechanics
Print ISSN 0022-1120
Electronic ISSN 1469-7645
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 916
Article Number A17
DOI https://doi.org/10.1017/jfm.2021.181
Public URL https://hull-repository.worktribe.com/output/3916637

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