June  2018, 23(4): 1797-1817. doi: 10.3934/dcdsb.2018085

Parking 3-sphere swimmer I. Energy minimizing strokes

CMAP, Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

* Corresponding author: Giovanni Di Fratta

Received  October 2016 Revised  November 2017 Published  March 2018

Fund Project: This work was partially supported by the Labex LMH through grant ANR-11-LABX-0056-LMH in the Programme des Investissements d'Avenir

The paper is about the parking 3-sphere swimmer (sPr3), a low-Reynolds number model swimmer composed of three balls of equal radii. The three balls can move along three horizontal axes (supported in the same plane) that mutually meet at the center of sPr3 with angles of 120°. The governing dynamical system is introduced and the implications of its geometric symmetries revealed. It is then shown that, in the first order range of small strokes, optimal periodic strokes are ellipses embedded in 3d space, i.e., closed curves of the form $t ∈ [0, 2 π] \mapsto (\cos t) u + (\sin t) v$ for suitable vectors u and v of $\mathbb{R}^3$. A simple analytic expression for the vectors u and v is derived.

Citation: François Alouges, Giovanni Di Fratta. Parking 3-sphere swimmer I. Energy minimizing strokes. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1797-1817. doi: 10.3934/dcdsb.2018085
References:
[1]

F. AlougesA. DeSimone and A. Lefebvre-Lepot, Optimal strokes for low reynolds number swimmers: An example, Journal of Nonlinear Science, 18 (2008), 277-302. doi: 10.1007/s00332-007-9013-7. Google Scholar

[2]

F. AlougesA. DeSimoneL. HeltaiA. Lefebvre-Lepot and B. Merlet, Optimally swimming stokesian robots, Discrete and Continuous Dynamical Systems-Series B (DCDS-B), 18 (2013), 1189-1215. doi: 10.3934/dcdsb.2013.18.1189. Google Scholar

[3]

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

[4]

J. E. Avron, O. Gat and O. Kenneth, Optimal swimming at low reynolds numbers, Physical Review Letters, 93 (2004), 186001. doi: 10.1103/PhysRevLett.93.186001. Google Scholar

[5]

J. E. Avron and O. Raz, A geometric theory of swimming: Purcell's swimmer and its symmetrized cousin, New Journal of Physics, 10 (2008), 63016. doi: 10.1088/1367-2630/10/6/063016. Google Scholar

[6]

L. E. BeckerS. A. Koehler and H. A. Stone, On self-propulsion of micro-machines at low reynolds number: Purcell's three-link swimmer, Journal of Fluid Mechanics, 490 (2003), 15-35. doi: 10.1017/S0022112003005184. Google Scholar

[7]

A. DeSimoneF. Alouges and A. Lefebvre-Lepot, Biological fluid dynamics, non-linear partial differential equations, in Mathematics of Complexity and Dynamical Systems, SpringerVerlag, New York, (2012), 26-31. doi: 10.1007/978-1-4614-1806-1_3. Google Scholar

[8]

R. DreyfusJ. 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. doi: 10.1140/epjb/e2005-00302-5. Google Scholar

[9]

L. Giraldi, P. Martinon and M. Zoppello, Optimal design of purcell's three-link swimmer, Physical Review E, 91 (2015), 23012, 6pp. doi: 10.1103/PhysRevE.91.023012. Google Scholar

[10]

E. Gutman and Y. Or, Symmetries and gaits for Purcell's three-link microswimmer model, IEEE Transactions on Robotics, 32 (2016), 53-69. doi: 10.1109/TRO.2015.2500442. Google Scholar

[11]

E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms, Reports on Progress in Physics, 72 (2009), 96601, 36pp. doi: 10.1088/0034-4885/72/9/096601. Google Scholar

[12]

A. Lefebvre-Lepot and B. Merlet, A stokesian submarine, ESAIM: Proceedings, 28 (2009), 150-161. doi: 10.1051/proc/2009044. Google Scholar

[13]

M. J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers, Communications on Pure and Applied Mathematics, 5 (1952), 109-118. doi: 10.1002/cpa.3160050201. Google Scholar

[14]

A. Najafi and R. Golestanian, Simple swimmer at low reynolds number: Three linked spheres, Physical Review E, 69 (2004), 62901. doi: 10.1103/PhysRevE.69.062901. Google Scholar

[15]

E. M. Purcell, Life at low reynolds number, AIP Conference Proceedings, 28 (1976), p49. doi: 10.1063/1.30370. Google Scholar

[16]

L. Schwartz, Les Tenseurs, Herman, Paris, 1975. Google Scholar

[17]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low reynolds number, Journal of Fluid Mechanics, 198 (1989), 557-585. doi: 10.1017/S002211208900025X. Google Scholar

[18]

A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low reynolds number, Journal of Fluid Mechanics, 198 (1989), 587-599. doi: 10.1017/S0022112089000261. Google Scholar

[19]

D. Tam and A. E. Hosoi, Optimal stroke patterns for Purcell's three-link swimmer, Physical Review Letters, 98 (2007), 68105. doi: 10.1103/PhysRevLett.98.068105. Google Scholar

[20]

G. Taylor, Analysis of the swimming of microscopic organisms, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 209 (1951), 447-461. doi: 10.1098/rspa.1951.0218. Google Scholar

show all references

References:
[1]

F. AlougesA. DeSimone and A. Lefebvre-Lepot, Optimal strokes for low reynolds number swimmers: An example, Journal of Nonlinear Science, 18 (2008), 277-302. doi: 10.1007/s00332-007-9013-7. Google Scholar

[2]

F. AlougesA. DeSimoneL. HeltaiA. Lefebvre-Lepot and B. Merlet, Optimally swimming stokesian robots, Discrete and Continuous Dynamical Systems-Series B (DCDS-B), 18 (2013), 1189-1215. doi: 10.3934/dcdsb.2013.18.1189. Google Scholar

[3]

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

[4]

J. E. Avron, O. Gat and O. Kenneth, Optimal swimming at low reynolds numbers, Physical Review Letters, 93 (2004), 186001. doi: 10.1103/PhysRevLett.93.186001. Google Scholar

[5]

J. E. Avron and O. Raz, A geometric theory of swimming: Purcell's swimmer and its symmetrized cousin, New Journal of Physics, 10 (2008), 63016. doi: 10.1088/1367-2630/10/6/063016. Google Scholar

[6]

L. E. BeckerS. A. Koehler and H. A. Stone, On self-propulsion of micro-machines at low reynolds number: Purcell's three-link swimmer, Journal of Fluid Mechanics, 490 (2003), 15-35. doi: 10.1017/S0022112003005184. Google Scholar

[7]

A. DeSimoneF. Alouges and A. Lefebvre-Lepot, Biological fluid dynamics, non-linear partial differential equations, in Mathematics of Complexity and Dynamical Systems, SpringerVerlag, New York, (2012), 26-31. doi: 10.1007/978-1-4614-1806-1_3. Google Scholar

[8]

R. DreyfusJ. 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. doi: 10.1140/epjb/e2005-00302-5. Google Scholar

[9]

L. Giraldi, P. Martinon and M. Zoppello, Optimal design of purcell's three-link swimmer, Physical Review E, 91 (2015), 23012, 6pp. doi: 10.1103/PhysRevE.91.023012. Google Scholar

[10]

E. Gutman and Y. Or, Symmetries and gaits for Purcell's three-link microswimmer model, IEEE Transactions on Robotics, 32 (2016), 53-69. doi: 10.1109/TRO.2015.2500442. Google Scholar

[11]

E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms, Reports on Progress in Physics, 72 (2009), 96601, 36pp. doi: 10.1088/0034-4885/72/9/096601. Google Scholar

[12]

A. Lefebvre-Lepot and B. Merlet, A stokesian submarine, ESAIM: Proceedings, 28 (2009), 150-161. doi: 10.1051/proc/2009044. Google Scholar

[13]

M. J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers, Communications on Pure and Applied Mathematics, 5 (1952), 109-118. doi: 10.1002/cpa.3160050201. Google Scholar

[14]

A. Najafi and R. Golestanian, Simple swimmer at low reynolds number: Three linked spheres, Physical Review E, 69 (2004), 62901. doi: 10.1103/PhysRevE.69.062901. Google Scholar

[15]

E. M. Purcell, Life at low reynolds number, AIP Conference Proceedings, 28 (1976), p49. doi: 10.1063/1.30370. Google Scholar

[16]

L. Schwartz, Les Tenseurs, Herman, Paris, 1975. Google Scholar

[17]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low reynolds number, Journal of Fluid Mechanics, 198 (1989), 557-585. doi: 10.1017/S002211208900025X. Google Scholar

[18]

A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low reynolds number, Journal of Fluid Mechanics, 198 (1989), 587-599. doi: 10.1017/S0022112089000261. Google Scholar

[19]

D. Tam and A. E. Hosoi, Optimal stroke patterns for Purcell's three-link swimmer, Physical Review Letters, 98 (2007), 68105. doi: 10.1103/PhysRevLett.98.068105. Google Scholar

[20]

G. Taylor, Analysis of the swimming of microscopic organisms, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 209 (1951), 447-461. doi: 10.1098/rspa.1951.0218. Google Scholar

Figure 1.  The swimmer sPr3 is composed of three spheres of equal radii. The three spheres can move along three horizontal axes that mutually meet at $c$ with angle $2 \pi / 3$. The spheres do not rotate around their axes so that the shape of the swimmer is characterized by the three lengths $\xi_1, \xi_2, \xi_3$ of its arms, measured from the origin to the center of each ball. However, the swimmer may freely rotate around $c$ in the horizontal plane.
Figure 2.  An observer watching the dynamics $\gamma(p_0, \xi)$ of sPr3 projected on a glass, imposes to another observer, lying on the other side of the glass, to watch the dynamics $\gamma (p_0, L \xi)$ of a micro-swimmer obtained from sPr3 by inverting arms ${\left. \right\|_\boldsymbol{1}}$ and ${\left. \right\|_\boldsymbol{2}}$.
Figure 3.  (left) With respect to the standard euclidean space $\left( \mathbb{R}^3, (\cdot, \cdot)_2 \right)$, the energy minimizing strokes able to reach a prescribed displacement $\delta p$ are ellipses of $\mathbb{R}^3$ centered at the origin and contained in the plane spanned by the vectors $u : = U_{} a_1$ and $v : = U_{} b_1$. (right) With respect to the inner-product space $\left( \mathbb{R}^3, (\cdot, \cdot)_{\mathfrak{g}}\right)$, the energy minimizing strokes describe circles of radius $\sqrt{|\omega |}$.
[1]

Qixuan Wang, Hans G. Othmer. The performance of discrete models of low reynolds number swimmers. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1303-1320. doi: 10.3934/mbe.2015.12.1303

[2]

Jorge San Martín, Takéo Takahashi, Marius Tucsnak. An optimal control approach to ciliary locomotion. Mathematical Control & Related Fields, 2016, 6 (2) : 293-334. doi: 10.3934/mcrf.2016005

[3]

Giulio G. Giusteri, Alfredo Marzocchi, Alessandro Musesti. Steady free fall of one-dimensional bodies in a hyperviscous fluid at low Reynolds number. Evolution Equations & Control Theory, 2014, 3 (3) : 429-445. doi: 10.3934/eect.2014.3.429

[4]

Serge Nicaise, Simon Stingelin, Fredi Tröltzsch. Optimal control of magnetic fields in flow measurement. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 579-605. doi: 10.3934/dcdss.2015.8.579

[5]

Tehuan Chen, Chao Xu, Zhigang Ren. Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1251-1269. doi: 10.3934/jimo.2018052

[6]

Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545

[7]

Lino J. Alvarez-Vázquez, Néstor García-Chan, Aurea Martínez, Miguel E. Vázquez-Méndez. Optimal control of urban air pollution related to traffic flow in road networks. Mathematical Control & Related Fields, 2018, 8 (1) : 177-193. doi: 10.3934/mcrf.2018008

[8]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006

[9]

Wilfrid Gangbo, Andrzej Świech. Optimal transport and large number of particles. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1397-1441. doi: 10.3934/dcds.2014.34.1397

[10]

Alex Bombrun, Jean-Baptiste Pomet. Asymptotic behavior of time optimal orbital transfer for low thrust 2-body control system. Conference Publications, 2007, 2007 (Special) : 122-129. doi: 10.3934/proc.2007.2007.122

[11]

Clara Rojas, Juan Belmonte-Beitia, Víctor M. Pérez-García, Helmut Maurer. Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1895-1915. doi: 10.3934/dcdsb.2016028

[12]

Jingang Zhai, Guangmao Jiang, Jianxiong Ye. Optimal dilution strategy for a microbial continuous culture based on the biological robustness. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 59-69. doi: 10.3934/naco.2015.5.59

[13]

Rafael Vázquez, Emmanuel Trélat, Jean-Michel Coron. Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 925-956. doi: 10.3934/dcdsb.2008.10.925

[14]

Thalya Burden, Jon Ernstberger, K. Renee Fister. Optimal control applied to immunotherapy. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 135-146. doi: 10.3934/dcdsb.2004.4.135

[15]

Ellina Grigorieva, Evgenii Khailov. Optimal control of pollution stock. Conference Publications, 2011, 2011 (Special) : 578-588. doi: 10.3934/proc.2011.2011.578

[16]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

[17]

Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1

[18]

Changjie Fang, Weimin Han. Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5369-5386. doi: 10.3934/dcds.2016036

[19]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495

[20]

Aurea Martínez, Francisco J. Fernández, Lino J. Alvarez-Vázquez. Water artificial circulation for eutrophication control. Mathematical Control & Related Fields, 2018, 8 (1) : 277-313. doi: 10.3934/mcrf.2018012

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (35)
  • HTML views (198)
  • Cited by (0)

Other articles
by authors

[Back to Top]