# American Institute of Mathematical Sciences

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  June 2018 Early access  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 and Continuous Dynamical Systems - B, 2018, 23 (4) : 1797-1817. doi: 10.3934/dcdsb.2018085
##### References:
 [1] F. Alouges, A. 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. [2] F. Alouges, A. DeSimone, L. Heltai, A. 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. [3] F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymmetric microswimmers, The European Physical Journal E, 28 (2009), 279-284. [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. [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. [6] L. E. Becker, S. 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. [7] A. DeSimone, F. 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. [8] 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.  doi: 10.1140/epjb/e2005-00302-5. [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. [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. [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. [12] A. Lefebvre-Lepot and B. Merlet, A stokesian submarine, ESAIM: Proceedings, 28 (2009), 150-161.  doi: 10.1051/proc/2009044. [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. [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. [15] E. M. Purcell, Life at low reynolds number, AIP Conference Proceedings, 28 (1976), p49. doi: 10.1063/1.30370. [16] L. Schwartz, Les Tenseurs, Herman, Paris, 1975. [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. [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. [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. [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.

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##### References:
 [1] F. Alouges, A. 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. [2] F. Alouges, A. DeSimone, L. Heltai, A. 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. [3] F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymmetric microswimmers, The European Physical Journal E, 28 (2009), 279-284. [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. [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. [6] L. E. Becker, S. 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. [7] A. DeSimone, F. 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. [8] 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.  doi: 10.1140/epjb/e2005-00302-5. [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. [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. [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. [12] A. Lefebvre-Lepot and B. Merlet, A stokesian submarine, ESAIM: Proceedings, 28 (2009), 150-161.  doi: 10.1051/proc/2009044. [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. [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. [15] E. M. Purcell, Life at low reynolds number, AIP Conference Proceedings, 28 (1976), p49. doi: 10.1063/1.30370. [16] L. Schwartz, Les Tenseurs, Herman, Paris, 1975. [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. [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. [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. [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.
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.
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 ${\left. \right\|_\boldsymbol{2}}$.
(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 $\sqrt{|\omega |}$.
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