Controllability of 3D low Reynolds number swimmers

Jérôme Lohéac; Alexandre Munnier

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 1, page 236-268
  • ISSN: 1292-8119

Abstract

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In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies on short time intervals, for a wide class of cost functionals.

How to cite

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Lohéac, Jérôme, and Munnier, Alexandre. "Controllability of 3D low Reynolds number swimmers." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 236-268. <http://eudml.org/doc/272911>.

@article{Lohéac2014,
abstract = {In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies on short time intervals, for a wide class of cost functionals.},
author = {Lohéac, Jérôme, Munnier, Alexandre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {locomotion; biomechanics; stokes fluid; geometric control theory; Stokes fluid},
language = {eng},
number = {1},
pages = {236-268},
publisher = {EDP-Sciences},
title = {Controllability of 3D low Reynolds number swimmers},
url = {http://eudml.org/doc/272911},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Lohéac, Jérôme
AU - Munnier, Alexandre
TI - Controllability of 3D low Reynolds number swimmers
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 1
SP - 236
EP - 268
AB - In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies on short time intervals, for a wide class of cost functionals.
LA - eng
KW - locomotion; biomechanics; stokes fluid; geometric control theory; Stokes fluid
UR - http://eudml.org/doc/272911
ER -

References

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  1. [1] A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint, vol. 87 of Encyclopaedia Math. Sci. Springer-Verlag, Berlin (2004). Zbl1062.93001MR2062547
  2. [2] F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci.18 (2008) 277–302. Zbl1146.76062MR2411380
  3. [3] H. Brenner. The stokes resistance of a slightly deformed sphere. Chem. Engrg. Sci.19 (1964) 519–539. 
  4. [4] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973). Zbl0252.47055MR348562
  5. [5] T. Chambrion and A. Munnier, Locomotion and control of a self-propelled shape-changing body in a fluid. J. Nonlinear Sci.21 (2011) 325–385. Zbl1222.74019MR2823860
  6. [6] T. Chambrion and A. Munnier, Generic controllability of 3d swimmers in a perfect fluid. SIAM J. Control Optim.50 (2012) 2814–2835. Zbl1320.76131MR3022088
  7. [7] S. Childress, Mechanics of swimming and flying, vol. 2 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge (1981). Zbl0499.76118MR665099
  8. [8] V. Girault and A. Sequeira, A well-posed problem for the exterior Stokes equations in two and three dimensions. Arch. Rational Mech. Anal.114 (1991) 313–333. Zbl0731.35078MR1100798
  9. [9] J. Happel and H. Brenner, Low Reynolds number hydrodynamics with special applications to particulate media. Prentice-Hall Inc., Englewood Cliffs, N.J. (1965). Zbl0612.76032MR195360
  10. [10] H. Lamb, Hydrodynamics. Cambridge Mathematical Library. 6th edition. Cambridge University Press, Cambridge, (1993). Zbl0828.01012MR1317348JFM36.0817.07
  11. [11] J. Lighthill, Mathematical biofluiddynamics. Society for Industrial and Applied Mathematics. Philadelphia, Pa. (1975). Zbl0312.76076MR469320
  12. [12] M.J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Math.5 (1952) 109–118. Zbl0046.41908MR51076
  13. [13] G. Maso, A. DeSimone and M. Morandotti, An existence and uniqueness result for the motion of self-propelled microswimmers. SIAM J. Math. Anal.43 (2011) 1345–1368. Zbl05969681MR2821587
  14. [14] E.M. Purcell, Life at low reynolds number. Amer. J. Phys.45 (1977) 3–11. 
  15. [15] T. Roubíček, Nonlinear partial differential equations with applications, vol. 153 of Internat. Ser. Numer. Math. Birkhäuser Verlag, Basel (2005). Zbl1087.35002
  16. [16] A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number. J. Fluid Mech.198 (1989) 557–585. Zbl0674.76114MR982199
  17. [17] J. Simon, Domain variation for drag in stokes flow, in vol. 159 of Control Theory of Distributed Parameter Systems and Applications, Lecture Notes in Control and Information Sciences, edited by X. Li and J. Yong. Springer Berlin/Heidelberg (1991) 28–42. Zbl0801.76075MR1129956
  18. [18] G. Taylor, Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond., Ser. A 209 (1951) 447–461. Zbl0043.40302MR45521
  19. [19] E.F. Whittlesey, Analytic functions in Banach spaces. Proc. Amer. Math. Soc.16 (1965) 1077–1083. Zbl0143.15302MR184092

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