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  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
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References:
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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