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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 and 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, (). 

[2]

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

[3]

F. Alouges, A. DeSimone and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers, Math. Models Methods Appl. Sci., 2 (2011), 361-387. doi: 10.1142/S0218202511005088.

[4]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low Reynolds number swimmers: An example, J. Nonlinear Sci., 18 (2008), 277-302. doi: 10.1007/s00332-007-9013-7.

[5]

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

[6]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymmetric microswimmers, The European Physical Journal E, 28 (2009), 279-284.

[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), 17874-17879.

[8]

J. E. Avron, O. Gat and O. Kenneth, Optimal swimming at low reynolds numbers, Phys. Rev. Lett., 93 (2004), 186001.

[9]

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

[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.

[11]

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

[12]

A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discrete Contin. Dyn. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1.

[13]

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

[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), 1345-1368. doi: 10.1137/10080083X.

[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), 161-164.

[16]

R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone and J. Bibette, Microscopic artificial swimmer, Nature, 437 (2005), 862-865.

[17]

J. H. Hannay and J. F. Nye, Fibonacci numerical integration on a sphere, J. Phys. A, 37 (2004), 11591-11601. doi: 10.1088/0305-4470/37/48/005.

[18]

J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media," Prentice-Hall Inc., Englewood Cliffs, N.J., 1965.

[19]

R. M. Harshey, Bacterial motility on a surface: many ways to a common goal, Annual Reviews in Microbiology, 57 (2003), 249-273.

[20]

M. A. Heroux et al, An overview of the Trilinos project, ACM Trans. Math. Softw., 31 (2005), 397-423. doi: 10.1145/1089014.1089021.

[21]

V. Jurdjevic, "Geometric Control Theory," Cambridge Univ. Press, 1997.

[22]

A. Y. Khapalov, Local controllability for a "Swimming'' model, SIAM J. Control Optim., 46 (2007), 655-682. doi: 10.1137/050638424.

[23]

J. Koiller, K. Ehlers and R. Montgomery, Problems and progress in microswimming, J. Nonlinear Sci., 6 (1996), 507-541. doi: 10.1007/s003329900021.

[24]

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

[25]

A. Lefebvre-Lepot and B. Merlet, A stokesian submarine, ESAIM: Proc., 28 (2009), 150-161.

[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), 109-118.

[27]

J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low reynolds numbers swimmers, Acta Appl. Math., 123 (2013), 175-200. doi: 10.1007/s10440-012-9760-9.

[28]

A. Najafi and R. Golestanian, Simple swimmer at low Reynolds number: Three linked spheres, Phys. Rev. E, 69 (2004), 062901.

[29]

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

[30]

C. Pozrikidis, "A practical Guide to Boundary Element Methods with the Software Library BEMLIB," Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035254.

[31]

E. M. Purcell, Life at low Reynolds numbers, Am. J. Phys, 45 (1977), 3-11.

[32]

J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math, 65 (2007), 405-424.

[33]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, J. Fluid Mech., 198 (1989), 557-585. doi: 10.1017/S002211208900025X.

[34]

G. Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London. Ser. A, 209 (1951), 447-461.

show all references

References:
[1]

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

[2]

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

[3]

F. Alouges, A. DeSimone and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers, Math. Models Methods Appl. Sci., 2 (2011), 361-387. doi: 10.1142/S0218202511005088.

[4]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low Reynolds number swimmers: An example, J. Nonlinear Sci., 18 (2008), 277-302. doi: 10.1007/s00332-007-9013-7.

[5]

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

[6]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymmetric microswimmers, The European Physical Journal E, 28 (2009), 279-284.

[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), 17874-17879.

[8]

J. E. Avron, O. Gat and O. Kenneth, Optimal swimming at low reynolds numbers, Phys. Rev. Lett., 93 (2004), 186001.

[9]

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

[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.

[11]

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

[12]

A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discrete Contin. Dyn. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1.

[13]

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

[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), 1345-1368. doi: 10.1137/10080083X.

[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), 161-164.

[16]

R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone and J. Bibette, Microscopic artificial swimmer, Nature, 437 (2005), 862-865.

[17]

J. H. Hannay and J. F. Nye, Fibonacci numerical integration on a sphere, J. Phys. A, 37 (2004), 11591-11601. doi: 10.1088/0305-4470/37/48/005.

[18]

J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media," Prentice-Hall Inc., Englewood Cliffs, N.J., 1965.

[19]

R. M. Harshey, Bacterial motility on a surface: many ways to a common goal, Annual Reviews in Microbiology, 57 (2003), 249-273.

[20]

M. A. Heroux et al, An overview of the Trilinos project, ACM Trans. Math. Softw., 31 (2005), 397-423. doi: 10.1145/1089014.1089021.

[21]

V. Jurdjevic, "Geometric Control Theory," Cambridge Univ. Press, 1997.

[22]

A. Y. Khapalov, Local controllability for a "Swimming'' model, SIAM J. Control Optim., 46 (2007), 655-682. doi: 10.1137/050638424.

[23]

J. Koiller, K. Ehlers and R. Montgomery, Problems and progress in microswimming, J. Nonlinear Sci., 6 (1996), 507-541. doi: 10.1007/s003329900021.

[24]

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

[25]

A. Lefebvre-Lepot and B. Merlet, A stokesian submarine, ESAIM: Proc., 28 (2009), 150-161.

[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), 109-118.

[27]

J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low reynolds numbers swimmers, Acta Appl. Math., 123 (2013), 175-200. doi: 10.1007/s10440-012-9760-9.

[28]

A. Najafi and R. Golestanian, Simple swimmer at low Reynolds number: Three linked spheres, Phys. Rev. E, 69 (2004), 062901.

[29]

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

[30]

C. Pozrikidis, "A practical Guide to Boundary Element Methods with the Software Library BEMLIB," Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035254.

[31]

E. M. Purcell, Life at low Reynolds numbers, Am. J. Phys, 45 (1977), 3-11.

[32]

J. San Martín, T. Takahashi and M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math, 65 (2007), 405-424.

[33]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, J. Fluid Mech., 198 (1989), 557-585. doi: 10.1017/S002211208900025X.

[34]

G. Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. London. Ser. A, 209 (1951), 447-461.

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