A control theoretic approach to the swimming of microscopic organisms
Authors:
Jorge San Martín, Takéo Takahashi and Marius Tucsnak
Journal:
Quart. Appl. Math. 65 (2007), 405-424
MSC (2000):
Primary 76D05
DOI:
https://doi.org/10.1090/S0033-569X-07-01045-9
Published electronically:
May 21, 2007
MathSciNet review:
2354880
Full-text PDF Free Access
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Abstract: In this paper, we give a control theoretic approach to the slow self-propelled motion of a rigid body in a viscous fluid. The control of the system is the relative velocity of the fluid with respect to the solid on the boundary of the rigid body (the thrust). Our main results show that there exists a large class of finite-dimensional input spaces for which the system is exactly controllable, i.e., one can find controls steering the rigid body into any final position with a prescribed velocity field. The equations we use are motivated by models of swimming of micro-organisms like cilia. We give a control theoretic interpretation of the swimming mechanism of these organisms, which takes place at very low Reynolds numbers.
References
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References
- J. R. Blake and S. R. Otto, Ciliary propulsion, chaotic filtration and a “blinking” Stokeslet, J. Engrg. Math. 30 (1996), no. 1-2, 151–168, The centenary of a paper on slow viscous flow by the physicist H. A. Lorentz. MR 1396365 (97e:76098)
- J. R. Blake, A finite model for ciliated micro-organisms, J. Biomech. 6 (1973), 133–140.
- Christopher Brennen, An oscillating-boundary-layer theory for ciliary propulsion, J. Fluid Mech. 65 (1974), 799–824.
- Christopher Brennen and H. Winet, Fluid mechanics of propulsion by cilia and flagella, Ann. Rev. Fluid Mech. 9 (1977), 339–398.
- Stephen Childress, Mechanics of swimming and flying, Cambridge Studies in Mathematical Biology, vol. 2, Cambridge University Press, Cambridge, 1981. MR 665099 (84a:76050)
- Caroline Fabre and Gilles Lebeau, Prolongement unique des solutions de l’equation de Stokes, Comm. Partial Differential Equations 21 (1996), no. 3-4, 573–596. MR 1387461 (97c:35156)
- Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994, Linearized steady problems. MR 1284205 (95i:35216a)
- ---, On the steady self-propelled motion of a body in a viscous incompressible fluid, Arch. Rational Mech. Anal. 148 (1999), no. 1, 53–88. MR 1715453 (2000h:76186)
- ---, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of mathematical fluid dynamics. Vol. I, North-Holland, Amsterdam, 2002, pp. 653–791. MR 1942470 (2003j:76024)
- John Happel and Howard Brenner, Low Reynolds number hydrodynamics with special applications to particulate media, Prentice-Hall Inc., Englewood Cliffs, N.J., 1965. MR 0195360 (33:3562)
- S. R. Keller and T. Y. Wu, A porous prolate-spheroidal model for ciliated micro-organisms, J. Fluid Mech. 80 (1977), 259–278.
- Gen Komatsu, Analyticity up to the boundary of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 32 (1979), no. 5, 669–720. MR 533297 (82j:35084)
- James Lighthill, Mathematical biofluiddynamics, Society for Industrial and Applied Mathematics, Philadelphia, 1975. MR 0469320 (57:9113)
- Charles B. Morrey, Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511 (34:2380)
- Jorge Alonso San Martín, Victor Starovoitov, and Marius Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Rational Mech. Anal. 161 (2002), no. 2, 113–147. MR 1870954 (2002j:35259)
- Denis Serre, Chute libre d’un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math. 4 (1987), no. 1, 99–110. MR 899206 (89m:76032)
- Ana Leonor Silvestre, On the slow motion of a self-propelled rigid body in a viscous incompressible fluid, J. Math. Anal. Appl. 274 (2002), no. 1, 203–227. MR 1936694 (2003j:76030)
- Eduardo D. Sontag, Mathematical control theory, second ed., Texts in Applied Mathematics, vol. 6, Springer-Verlag, New York, 1998, Deterministic finite-dimensional systems. MR 1640001 (99k:93001)
- Takéo Takahashi and Marius Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech. 6 (2004), no. 1, 53–77. MR 2027754 (2004k:76035)
- Geoffrey Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London Ser. A. 209 (1951), 447–461. MR 0045521 (13:596a)
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Additional Information
Jorge San Martín
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile
Email:
jorge@dim.uchile.cl
Takéo Takahashi
Affiliation:
Institut Élie Cartan UMR7502, Université Henri Poincaré Nancy 1, BP239, 54506 Vandœuvre-lès-Nancy Cedex, France
Email:
takahash@iecn.u-nancy.fr
Marius Tucsnak
Affiliation:
Institut Élie Cartan UMR7502, Université Henri Poincaré Nancy 1, BP239, 54506 Vandœuvre-lès-Nancy Cedex, France
MR Author ID:
175105
Email:
tucsnak@loria.fr
Received by editor(s):
July 25, 2005
Published electronically:
May 21, 2007
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.