Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-20T01:27:16.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Controllability of 3D low Reynolds number swimmers

Published online by Cambridge University Press:  27 January 2014

Jérôme Lohéac  and
Alexandre Munnier
Jérôme Lohéac
Affiliation:
Both authors are with Institut Élie Cartan UMR 7502, Université de Lorraine, CNRS, INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France. jerome.loheac@univ-lorraine.fr
Alexandre Munnier
Affiliation:
INRIA Nancy Grand Est, Projet CORIDA, France; Authors both supported by ANR CISIFS Second author supported by ANR GAOS. alexandre.munnier@univ-lorraine.fr
Get access

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.

Keywords

Locomotionbiomechanicsstokes fluidgeometric control theory
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 236 - 268
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint, vol. 87 of Encyclopaedia Math. Sci. Springer-Verlag, Berlin (2004).
Alouges, F., DeSimone, A. and Lefebvre, A., Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18 (2008) 277–302. Google Scholar
Brenner, H.. The stokes resistance of a slightly deformed sphere. Chem. Engrg. Sci. 19 (1964) 519–539. Google Scholar
H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973).
Chambrion, T. and Munnier, A., Locomotion and control of a self-propelled shape-changing body in a fluid. J. Nonlinear Sci. 21 (2011) 325–385. Google Scholar
Chambrion, T. and Munnier, A., Generic controllability of 3d swimmers in a perfect fluid. SIAM J. Control Optim. 50 (2012) 2814–2835. Google Scholar
S. Childress, Mechanics of swimming and flying, vol. 2 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge (1981).
Girault, V. and Sequeira, A., A well-posed problem for the exterior Stokes equations in two and three dimensions. Arch. Rational Mech. Anal. 114 (1991) 313–333. Google Scholar
J. Happel and H. Brenner, Low Reynolds number hydrodynamics with special applications to particulate media. Prentice-Hall Inc., Englewood Cliffs, N.J. (1965).
H. Lamb, Hydrodynamics. Cambridge Mathematical Library. 6th edition. Cambridge University Press, Cambridge, (1993).
J. Lighthill, Mathematical biofluiddynamics. Society for Industrial and Applied Mathematics. Philadelphia, Pa. (1975).
Lighthill, M.J., On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Math. 5 (1952) 109–118. Google Scholar
Maso, G., DeSimone, A. and Morandotti, M., An existence and uniqueness result for the motion of self-propelled microswimmers. SIAM J. Math. Anal. 43 (2011) 1345–1368. Google Scholar
Purcell, E.M., Life at low reynolds number. Amer. J. Phys. 45 (1977) 3–11. Google Scholar
T. Roubíček, Nonlinear partial differential equations with applications, vol. 153 of Internat. Ser. Numer. Math. Birkhäuser Verlag, Basel (2005).
Shapere, A. and Wilczek, F., Geometry of self-propulsion at low Reynolds number. J. Fluid Mech. 198 (1989) 557–585. Google Scholar
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.
Taylor, G., Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond., Ser. A 209 (1951) 447–461. Google Scholar
Whittlesey, E.F., Analytic functions in Banach spaces. Proc. Amer. Math. Soc. 16 (1965) 1077–1083. Google Scholar