Optimal strokes for the 4-sphere swimmer at low Reynolds number in the regime of small deformations
MathematicS In Action, Volume 11 (2022) no. 1, pp. 167-192.

The paper deals with the optimal control problem that arises when one studies the 4 sphere artificial swimmer at low Reynolds number. Composed of four spheres at the end of extensible arms, the swimmer is known to be able to swim in all directions and orientations in the 3D space. In this paper, optimal strokes, in terms of the energy expended by the swimmer to reach a prescribed net displacement, are fully described in the regime of small strokes. In particular, we introduce a bivector formalism to model the displacements that turns out to be elegant and practical. Numerical simulations are also provided that confirm the theoretical predictions.

Published online:
DOI: 10.5802/msia.23
Classification: 15-04,  34H05,  49K15,  76Z10,  93C15
Keywords: Low Reynolds number swimming, optimal strokes, periodic control
François Alouges 1; Aline Lefebvre-Lepot 1; Philipp Weder 2

1 CMAP, Ecole polytechnique et CNRS, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau Cedex, France
2 EPFL, Rue Louis-Favre 4, CH-1024 Ecublens, Switzerland
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Fran\c{c}ois Alouges and Aline Lefebvre-Lepot and Philipp Weder},
     title = {Optimal strokes for the 4-sphere swimmer at low {Reynolds} number in the regime of small deformations},
     journal = {MathematicS In Action},
     pages = {167--192},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {11},
     number = {1},
     year = {2022},
     doi = {10.5802/msia.23},
     language = {en},
     url = {https://msia.centre-mersenne.org/articles/10.5802/msia.23/}
AU  - François Alouges
AU  - Aline Lefebvre-Lepot
AU  - Philipp Weder
TI  - Optimal strokes for the 4-sphere swimmer at low Reynolds number in the regime of small deformations
JO  - MathematicS In Action
PY  - 2022
DA  - 2022///
SP  - 167
EP  - 192
VL  - 11
IS  - 1
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://msia.centre-mersenne.org/articles/10.5802/msia.23/
UR  - https://doi.org/10.5802/msia.23
DO  - 10.5802/msia.23
LA  - en
ID  - MSIA_2022__11_1_167_0
ER  - 
%0 Journal Article
%A François Alouges
%A Aline Lefebvre-Lepot
%A Philipp Weder
%T Optimal strokes for the 4-sphere swimmer at low Reynolds number in the regime of small deformations
%J MathematicS In Action
%D 2022
%P 167-192
%V 11
%N 1
%I Société de Mathématiques Appliquées et Industrielles
%U https://doi.org/10.5802/msia.23
%R 10.5802/msia.23
%G en
%F MSIA_2022__11_1_167_0
François Alouges; Aline Lefebvre-Lepot; Philipp Weder. Optimal strokes for the 4-sphere swimmer at low Reynolds number in the regime of small deformations. MathematicS In Action, Volume 11 (2022) no. 1, pp. 167-192. doi : 10.5802/msia.23. https://msia.centre-mersenne.org/articles/10.5802/msia.23/

[1] Andrei A. Agrachev; Yuri L. Sachkov Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer, 2004 | DOI | Zbl

[2] François Alouges; Antonio DeSimone; Luca Heltai; Aline Lefebvre-Lepot; Benoît Merlet Optimally swimming stokesian robots, Discrete Contin. Dyn. Syst., Ser. B, Volume 18 (2013) no. 5, pp. 1189-1215 | DOI | MR | Zbl

[3] François Alouges; Antonio DeSimone; Aline Lefebvre Optimal Strokes for Low Reynolds Number Swimmers: An Example, J. Nonlinear Sci., Volume 18 (2008) no. 3, pp. 277-302 | DOI | MR | Zbl

[4] François Alouges; Giovanni Di Fratta Parking 3-sphere swimmer. I. Energy minimizing strokes, Discrete Contin. Dyn. Syst., Volume 23 (2018) no. 4 | MR | Zbl

[5] François Alouges; Giovanni Di Fratta Parking 3-sphere swimmer: II. The long-arm asymptotic regime, Eur. Phys. J. E, Volume 43 (2020) no. 2 | DOI

[6] Alaex Arsenovic; Hugo Hadfield; Eric Wieser; Robert Kern; The Pygae Team pygae/clifford: v1.3.1, 2020 | DOI

[7] Joseph E. Avron; Omri Gat; Oded Kenneth Optimal Swimming at Low Reynolds Numbers, Phys. Rev. Lett., Volume 93 (2004) no. 18 | DOI

[8] Bernard Bonnard; Monique Chyba; Jérémy Rouot Geometric and Numerical Optimal Control: Application to Swimming at Low Reynolds Number and Magnetic Resonance Imaging, SpringerBriefs in Mathematics, Springer, 2018 | DOI

[9] Giancarlo Cicconofri; Antonio DeSimone Modelling biological and bio-inspired swimming at microscopic scales: Recent results and perspectives, Comput. Fluids, Volume 179 (2019), pp. 799-805 | DOI | MR | Zbl

[10] Rémi Dreyfus; Jean Baudry; Howard A. Stone Purcell’s “rotator”: mechanical rotation at low Reynolds number, Eur. Phys. J. B, Condens. Matter Complex Syst., Volume 47 (2005) no. 1, pp. 161-164 | DOI

[11] Gerhard Gompper; Roland Winkler; Thomas Speck; Alexandre Solon; Cesare Nardini; Fernando Peruani; Hartmut Löwen; Ramin Golestanian; Benjamin Kaupp; Luis Alvarez; Thomas Kiørboe; Eric Lauga; Wilson Poon; Antonio DeSimone; Santiago Muiños-Landin; Alexander Fischer; Nicola Söker; Frank Cichos; Raymond Kapral; Pierre Gaspard; Marisol Ripoll; Francesc Sagues; Amin Doostmohammadi; Julia Yeomans; Igor Aranson; Clemens Bechinger; Holger Stark; Charlotte Hemelrijk; François Nedelec; Trinish Sarkar; Thibault Aryaksama; Mathilde Lacroix; Guillaume Duclos; Victor Yashunsky; Pascal Silberzan; Marino Arroyo; Sohan Kale The 2020 Motile Active Matter Roadmap, J. Phys.: Condens. Matter, Volume 32 (2020), p. 193001

[12] Brian C. Hall Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics, 222, Springer, 2015 | DOI

[13] Hansjörg Kielhöfer Calculus of variations. An introduction to the one-dimensional theory with examples and exercises, Texts in Applied Mathematics, 67, Springer, 2018 | DOI | MR

[14] Eric Lauga; Thomas R. Powers The hydrodynamics of swimming microorganisms, Rep. Prog. Phys., Volume 72 (2009) no. 9 | MR

[15] Michael J. Lighthill On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers, Commun. Pure Appl. Math., Volume 5 (1952) no. 2, pp. 109-118 | DOI | MR | Zbl

[16] Jérôme Lohéac; Alexandre Munnier Controllability of 3D low Reynolds swimmers, ESAIM, Control Optim. Calc. Var., Volume 20 (2014) no. 1, pp. 236-268 | DOI | Numdam | MR | Zbl

[17] Pertti Lounesto Clifford Algebras and Spinors, Cambridge University Press, 2006, 352 pages

[18] Ali Najafi; Ramin Golestanian Simple swimmer at low Reynolds number: Three linked spheres, Phys. Rev. E, Volume 69 (2004) no. 6 | DOI

[19] Edward M. Purcell Life at low Reynolds number, Am. J. Phys., Volume 45 (1977) no. 1, pp. 3-11 | DOI

[20] Massimilliano Rossi; Giancarlo Cicconofri; Alfred Beran; Giovanni Noselli; Antonio DeSimone Kinematics of flagellar swimming in Euglena gracilis: Helical trajectories and flagellar shapes, Proc. Natl. Acad. Sci. USA, Volume 114 (2017) no. 50, pp. 13085-13090 | DOI

Cited by Sources: