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July  2013, 18(5): 1189-1215. doi: 10.3934/dcdsb.2013.18.1189

Optimally swimming stokesian robots

1. 

CMAP UMR 7641, CNRS, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, France

2. 

SISSA, International School of Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy, Italy

Received  November 2011 Revised  October 2012 Published  March 2013

We study self-propelled stokesian robots composed of assemblies of balls, in dimensions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability, and its proof relies on applying Chow's theorem in an analytic framework, similar to what has been done in [4] for an axisymmetric system swimming along the axis of symmetry. We generalize the analyticity result given in [4] to the situation where the swimmers can move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail.
Citation: François Alouges, Antonio DeSimone, Luca Heltai, Aline Lefebvre-Lepot, Benoît Merlet. Optimally swimming stokesian robots. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1189-1215. doi: 10.3934/dcdsb.2013.18.1189
References:
[1]

A. A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry,, Not Yet Published, ().   Google Scholar

[2]

A. A. Agrachev and Y. L. Sachkov, "Control Theory From the Geometric Viewpoint,", Springer Heidelberg, (2004).   Google Scholar

[3]

F. Alouges, A. DeSimone and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers,, Math. Models Methods Appl. Sci., 2 (2011), 361.  doi: 10.1142/S0218202511005088.  Google Scholar

[4]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low Reynolds number swimmers: An example,, J. Nonlinear Sci., 18 (2008), 277.  doi: 10.1007/s00332-007-9013-7.  Google Scholar

[5]

F. Alouges, A. DeSimone and A. Lefebvre, Biological fluid dynamics, nonlinear partial differential equations,, Encyclopedia of Complexity and Systems Science, (2009).   Google Scholar

[6]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymmetric microswimmers,, The European Physical Journal E, 28 (2009), 279.   Google Scholar

[7]

M. Arroyo, L. Heltai, D. Millàn and A. DeSimone, Reverse engineering the euglenoid movement,, Proceedings of the National Academy of Sciences of the United States of America, 109 (2012), 17874.   Google Scholar

[8]

J. E. Avron, O. Gat and O. Kenneth, Optimal swimming at low reynolds numbers,, Phys. Rev. Lett., 93 (2004).   Google Scholar

[9]

W. Bangerth, R. Hartmann and G. Kanschat, deal.II: Differential Equations Analysis Library,, Technical Reference., ().   Google Scholar

[10]

W. Bangerth, R. Hartmann and G. Kanschat, deal.II-a general-purpose object-oriented finite element library,, ACM Trans. Math. Softw., 33 (2007).  doi: 10.1145/1268776.1268779.  Google Scholar

[11]

R. A. Bartlett, Mathematical and high level overview of moocho,, Technical Report SAND2009-3969, (2009), 2009.   Google Scholar

[12]

A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids,, Discrete Contin. Dyn. Syst., 20 (2008), 1.  doi: 10.3934/dcds.2008.20.1.  Google Scholar

[13]

T. Chambrion and A. Munnier, Generalized scallop theorem for Linear swimmers,, preprint, ().   Google Scholar

[14]

G. Dal 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.  doi: 10.1137/10080083X.  Google Scholar

[15]

R. Dreyfus, J. Baudry and H. A. Stone, Purcell's "rotator'': mechanical rotation at low Reynolds number,, The European Physical Journal B-Condensed Matter and Complex Systems, 47 (2005), 161.   Google Scholar

[16]

R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone and J. Bibette, Microscopic artificial swimmer,, Nature, 437 (2005), 862.   Google Scholar

[17]

J. H. Hannay and J. F. Nye, Fibonacci numerical integration on a sphere,, J. Phys. A, 37 (2004), 11591.  doi: 10.1088/0305-4470/37/48/005.  Google Scholar

[18]

J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media,", Prentice-Hall Inc., (1965).   Google Scholar

[19]

R. M. Harshey, Bacterial motility on a surface: many ways to a common goal,, Annual Reviews in Microbiology, 57 (2003), 249.   Google Scholar

[20]

M. A. Heroux et al, An overview of the Trilinos project,, ACM Trans. Math. Softw., 31 (2005), 397.  doi: 10.1145/1089014.1089021.  Google Scholar

[21]

V. Jurdjevic, "Geometric Control Theory,", Cambridge Univ. Press, (1997).   Google Scholar

[22]

A. Y. Khapalov, Local controllability for a "Swimming'' model,, SIAM J. Control Optim., 46 (2007), 655.  doi: 10.1137/050638424.  Google Scholar

[23]

J. Koiller, K. Ehlers and R. Montgomery, Problems and progress in microswimming,, J. Nonlinear Sci., 6 (1996), 507.  doi: 10.1007/s003329900021.  Google Scholar

[24]

E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms,, Rep. Prog. Phys., 72 (2009).  doi: 10.1088/0034-4885/72/9/096601.  Google Scholar

[25]

A. Lefebvre-Lepot and B. Merlet, A stokesian submarine,, ESAIM: Proc., 28 (2009), 150.   Google Scholar

[26]

M. J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers,, Comm. Pure Appl. Math., 5 (1952), 109.   Google Scholar

[27]

J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low reynolds numbers swimmers,, Acta Appl. Math., 123 (2013), 175.  doi: 10.1007/s10440-012-9760-9.  Google Scholar

[28]

A. Najafi and R. Golestanian, Simple swimmer at low Reynolds number: Three linked spheres,, Phys. Rev. E, 69 (2004).   Google Scholar

[29]

C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow,", Cambridge Texts in Applied Mathematics. Cambridge Univ. Press, (1992).  doi: 10.1017/CBO9780511624124.  Google Scholar

[30]

C. Pozrikidis, "A practical Guide to Boundary Element Methods with the Software Library BEMLIB,", Chapman & Hall/CRC, (2002).  doi: 10.1201/9781420035254.  Google Scholar

[31]

E. M. Purcell, Life at low Reynolds numbers,, Am. J. Phys, 45 (1977), 3.   Google Scholar

[32]

J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms,, Quart. Appl. Math, 65 (2007), 405.   Google Scholar

[33]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number,, J. Fluid Mech., 198 (1989), 557.  doi: 10.1017/S002211208900025X.  Google Scholar

[34]

G. Taylor, Analysis of the swimming of microscopic organisms,, Proc. Roy. Soc. London. Ser. A, 209 (1951), 447.   Google Scholar

show all references

References:
[1]

A. A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry,, Not Yet Published, ().   Google Scholar

[2]

A. A. Agrachev and Y. L. Sachkov, "Control Theory From the Geometric Viewpoint,", Springer Heidelberg, (2004).   Google Scholar

[3]

F. Alouges, A. DeSimone and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers,, Math. Models Methods Appl. Sci., 2 (2011), 361.  doi: 10.1142/S0218202511005088.  Google Scholar

[4]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low Reynolds number swimmers: An example,, J. Nonlinear Sci., 18 (2008), 277.  doi: 10.1007/s00332-007-9013-7.  Google Scholar

[5]

F. Alouges, A. DeSimone and A. Lefebvre, Biological fluid dynamics, nonlinear partial differential equations,, Encyclopedia of Complexity and Systems Science, (2009).   Google Scholar

[6]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymmetric microswimmers,, The European Physical Journal E, 28 (2009), 279.   Google Scholar

[7]

M. Arroyo, L. Heltai, D. Millàn and A. DeSimone, Reverse engineering the euglenoid movement,, Proceedings of the National Academy of Sciences of the United States of America, 109 (2012), 17874.   Google Scholar

[8]

J. E. Avron, O. Gat and O. Kenneth, Optimal swimming at low reynolds numbers,, Phys. Rev. Lett., 93 (2004).   Google Scholar

[9]

W. Bangerth, R. Hartmann and G. Kanschat, deal.II: Differential Equations Analysis Library,, Technical Reference., ().   Google Scholar

[10]

W. Bangerth, R. Hartmann and G. Kanschat, deal.II-a general-purpose object-oriented finite element library,, ACM Trans. Math. Softw., 33 (2007).  doi: 10.1145/1268776.1268779.  Google Scholar

[11]

R. A. Bartlett, Mathematical and high level overview of moocho,, Technical Report SAND2009-3969, (2009), 2009.   Google Scholar

[12]

A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids,, Discrete Contin. Dyn. Syst., 20 (2008), 1.  doi: 10.3934/dcds.2008.20.1.  Google Scholar

[13]

T. Chambrion and A. Munnier, Generalized scallop theorem for Linear swimmers,, preprint, ().   Google Scholar

[14]

G. Dal 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.  doi: 10.1137/10080083X.  Google Scholar

[15]

R. Dreyfus, J. Baudry and H. A. Stone, Purcell's "rotator'': mechanical rotation at low Reynolds number,, The European Physical Journal B-Condensed Matter and Complex Systems, 47 (2005), 161.   Google Scholar

[16]

R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone and J. Bibette, Microscopic artificial swimmer,, Nature, 437 (2005), 862.   Google Scholar

[17]

J. H. Hannay and J. F. Nye, Fibonacci numerical integration on a sphere,, J. Phys. A, 37 (2004), 11591.  doi: 10.1088/0305-4470/37/48/005.  Google Scholar

[18]

J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media,", Prentice-Hall Inc., (1965).   Google Scholar

[19]

R. M. Harshey, Bacterial motility on a surface: many ways to a common goal,, Annual Reviews in Microbiology, 57 (2003), 249.   Google Scholar

[20]

M. A. Heroux et al, An overview of the Trilinos project,, ACM Trans. Math. Softw., 31 (2005), 397.  doi: 10.1145/1089014.1089021.  Google Scholar

[21]

V. Jurdjevic, "Geometric Control Theory,", Cambridge Univ. Press, (1997).   Google Scholar

[22]

A. Y. Khapalov, Local controllability for a "Swimming'' model,, SIAM J. Control Optim., 46 (2007), 655.  doi: 10.1137/050638424.  Google Scholar

[23]

J. Koiller, K. Ehlers and R. Montgomery, Problems and progress in microswimming,, J. Nonlinear Sci., 6 (1996), 507.  doi: 10.1007/s003329900021.  Google Scholar

[24]

E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms,, Rep. Prog. Phys., 72 (2009).  doi: 10.1088/0034-4885/72/9/096601.  Google Scholar

[25]

A. Lefebvre-Lepot and B. Merlet, A stokesian submarine,, ESAIM: Proc., 28 (2009), 150.   Google Scholar

[26]

M. J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers,, Comm. Pure Appl. Math., 5 (1952), 109.   Google Scholar

[27]

J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low reynolds numbers swimmers,, Acta Appl. Math., 123 (2013), 175.  doi: 10.1007/s10440-012-9760-9.  Google Scholar

[28]

A. Najafi and R. Golestanian, Simple swimmer at low Reynolds number: Three linked spheres,, Phys. Rev. E, 69 (2004).   Google Scholar

[29]

C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow,", Cambridge Texts in Applied Mathematics. Cambridge Univ. Press, (1992).  doi: 10.1017/CBO9780511624124.  Google Scholar

[30]

C. Pozrikidis, "A practical Guide to Boundary Element Methods with the Software Library BEMLIB,", Chapman & Hall/CRC, (2002).  doi: 10.1201/9781420035254.  Google Scholar

[31]

E. M. Purcell, Life at low Reynolds numbers,, Am. J. Phys, 45 (1977), 3.   Google Scholar

[32]

J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms,, Quart. Appl. Math, 65 (2007), 405.   Google Scholar

[33]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number,, J. Fluid Mech., 198 (1989), 557.  doi: 10.1017/S002211208900025X.  Google Scholar

[34]

G. Taylor, Analysis of the swimming of microscopic organisms,, Proc. Roy. Soc. London. Ser. A, 209 (1951), 447.   Google Scholar

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