Optimal strokes for driftless swimmers: A general geometric approach

Abstract

Swimming consists by definition in propelling through a fluid by means of bodily movements. Thus, from a mathematical point of view, swimming turns into a control problem for which the controls are the deformations of the swimmer. The aim of this paper is to present a unified geometric approach for the optimization of the body deformations of so-called driftless swimmers. The class of driftless swimmers includes, among other, swimmers in a 3D Stokes flow (case of micro-swimmers in viscous fluids) or swimmers in a 2D or 3D potential flow. A general framework is introduced, allowing the complete analysis of five usual nonlinear optimization problems to be carried out. The results are illustrated with examples coming from the literature and with an in-depth study of a swimmer in a 2D potential flow. Numerical tests are also provided.

Controllability of low Reynolds numbers swimmers of ciliate type

“…Based on the above theorem, we can also derive the existence of optimal controls. We refer to [13] for similar optimal control problems.…”

“…Such a model is considered in [24,25]. Let us also mention some other related works: the case where the fluid is inviscid and potential leads to a very close theory see [11][12][13]29].…”

Optimal Strokes at Low Reynolds Number: A Geometric and Numerical Study of Copepod and Purcell Swimmers

“…[26]), and have been widely studied since they are of great interest for applications (cf. [2,14,29], and see also [22] for earlier literature). The concept of optimal stroke in swimming modeling conveys the idea of periodic 'best deformation strategy' (minimizing the mechanical energy dissipated by the drag forces, for instance) performed by the swimmer body interacting with a fluid, and can be contextualized within the periodic optimal control framework (see for instance [2,14] and the references therein).…”

Optimal Strokes of Low Reynolds Number Linked-Sphere Swimmers

“…The linear and driftless properties of Stokes flows make the LRN swimming problem a classic driftless controllable system [35,43,31,32,28,25,44]. Consider a cyclic stroke γ(t) of a LRN swimmer, where t ∈ [0, 1], γ(0) = γ(1).…”