Abstract
Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers which is the regime of interest for micro-organisms and micro-robots. We focus on self-propelled stokesian robots composed of assemblies of balls and we prove that the presence of a wall has an effect on their motility. To rest on what has been done in Alouges et al. (Discrete Contin. Dyn. Syst., Ser. B 18(5):1189–1215, 2013) for such systems swimming on R 3, we demonstrate that a controllable swimmer remains controllable in a half space whereas the reachable set of a non fully controllable one is increased by the presence of a wall.
Similar content being viewed by others
Notes
Here and in the sequel, we use the definition for the H 1/2(∂B) norm:
$$\|\mathbf {v}\|_{H^{1/2}(\partial B)}=\min_{\mathbf{w}\in H^1(B,\mathbb {R}^3),\mathbf{w}=\mathbf {v}\mbox{ on }\partial B}{\|\mathbf{w} \|_{H^1(B)}}. $$
References
Alouges, F., DeSimone, A., Lefebvre, A.: Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18, 277–302 (2008)
Alouges, F., DeSimone, A., Lefebvre, A.: Optimal strokes for axisymmetric microswimmers. Eur. Phys. J. E 28, 279–284 (2009)
Alouges, F., DeSimone, A., Lefebvre, A.: Biological fluid dynamics, nonlinear partial differential equations. In: Encyclopedia of Complexity and Systems Science (2009)
Alouges, F., DeSimone, A., Heltai, L., Lefebvre, A., Merlet, B.: Optimally swimming stokesian robots. Discrete Continuous Dyn. Syst., Ser. B 18(5), 1189–1215 (2013)
Alouges, F., DeSimone, A., Heltai, L.: Numerical strategies for stroke optimization of axisymmetric microswimmers. Math. Models Methods Appl. Sci. 2, 361–387 (2011)
Berke, A.P., Allison, P.: Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102 (2008)
Berke, A.P., Turner, L., Berg, H.C., Lauga, E.: Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101(3), 038102 (2008)
Blake, J.R.: A note on the image system for a Stokeslet in a no-slip boundary. Proc. Gamb. Phil. Soc. 70, 303 (1971)
Chambrion, T., Munnier, A.: Generic controllability of 3D swimmers in a perfect fluid. Preprint (2011). arXiv:1103.5163
Coron, J.M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. Am. Math. Soc., Providence (2007)
Dal Maso, G., DeSimone, A., Morandotti, M.: An existence and uniqueness result for the selfpropelled motion of micro-swimmers. SIAM J. Math. Anal. 43, 1345–1368 (2011)
Gaffney, E.A., Gadêlha, H., Smith, D.J., Blake, J.R., Kirkman-Brown, J.C.: Mammalian sperm motility: observation and theory. Ann. Rev. Fluid Mech. 43, 501–528 (2011)
Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Prentice-Hall, Englewood Cliffs (1965)
Golestanian, R., Ajdari, A.: Analytic results for the three-sphere swimmer at low Reynolds. Phys. Rev. E 77, 036308 (2008)
Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, Cambridge (1997)
Leshansky, A.M., Kenneth, O.: Surface tank treading: propulsion of Purcell’s toroidal swimmer. Phys. Fluids 20, 063104 (2008)
Lohéac, J., Munnier, A.: Controllability of 3D low Reynolds swimmers. Preprint (2012). arXiv:1202.5923
Lohéac, J., Scheid, J.F., Tucsnak, M.: Controllability and time optimal control for low Reynolds numbers swimmers. Preprint Hal 00635981 (2011)
Munnier, A.: Locomotion of deformable bodies in an ideal fluid: Newtonian versus Lagrangian formalisms. J. Nonlinear Sci. 19(6), 665–715 (2009)
Najafi, A., Golestanian, R.: Simple swimmer at low Reynolds number: three linked spheres. Phys. Rev. E 69(6), 062901 (2004)
Najafi, A., Zargar, R.: Two-sphere low Reynolds-number propeller. Phys. Rev. E 81(6), 067301 (2010)
Or, Y., Murray, M.: Dynamics and stability of a class of low Reynolds number swimmers near a wall. Phys. Rev. E 79, 045302(R) (2009)
Purcell, E.M.: Life at low Reynolds number. Am. J. Phys. 45, 3–11 (1977)
Rothschild, L.: Non-random distribution of bull spermatozoa in a drop of sperm suspension. Nature 198, 1221–1222 (1963)
Sauvage, J.P.: Molecular Machines and Motors. Springer, Berlin (2001)
Shum, H., Gaffney, E.A., Smith, D.J.: Modeling bacterial behavior close to a no-slip plane boundary: the influence of bacterial geometry. Proc. Royal Soc. A 466, 1725–1748 (2010)
Smith, D.J., Blake, J.R.: Surface accumulation of spermatozoa: a fluid dynamic phenomenon. Preprint (2010). arXiv:1007.2153v1
Smith, D.J., Gaffney, E.A., Blake, J.R., Kirkman-Brown, J.C.: Human sperm accumulation near surfaces: a simulation study. J. Fluid Mech. 621, 289–320 (2009)
Taylor, G.: Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461 (1951)
Watson, B., Friend, J., Yeo, L.: Piezoelectric ultrasonic resonant motor with stator diameter less than 250 μm: the Proteus motor. J. Micromech. Microeng. 19, 022001 (2009)
Winet, H., Bernstein, G.S., Head, J.: Observation on the response of human spermatozoa to gravity, boundaries and fluid shear. Reproduction 70, 511–523 (1984)
Zargar, R., Najafi, A., Miri, M.: Three-sphere low Reynolds number swimmer near a wall. Phys. Rev. E 80, 026308 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been partially supported by the Direction Générale de l’Armement (DGA).
Rights and permissions
About this article
Cite this article
Alouges, F., Giraldi, L. Enhanced Controllability of Low Reynolds Number Swimmers in the Presence of a Wall. Acta Appl Math 128, 153–179 (2013). https://doi.org/10.1007/s10440-013-9824-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-013-9824-5