Generalized Scallop Theorem for Linear Swimmers

Thomas Chambrion 1, 2 Alexandre Munnier 1, 2
1 CORIDA - Robust control of infinite dimensional systems and applications
IECN - Institut Élie Cartan de Nancy, LMAM - Laboratoire de Mathématiques et Applications de Metz, Inria Nancy - Grand Est
Abstract : In this article, we are interested in studying locomotion strategies for a class of shape-changing bodies swimming in a fluid. This class consists of swimmers subject to a particular linear dynamics, which includes the two most investigated limit models in the literature: swimmers at low and high Reynolds numbers. Our first contribution is to prove that although for these two models the locomotion is based on very different physical principles, their dynamics are similar under symmetry assumptions. Our second contribution is to derive for such swimmers a purely geometric criterion allowing to determine wether a given sequence of shape-changes can result in locomotion. This criterion can be seen as a generalization of Purcell's scallop theorem (stated in Purcell (1977)) in the sense that it deals with a larger class of swimmers and address the complete locomotion strategy, extending the usual formulation in which only periodic strokes for low Reynolds swimmers are considered.
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Thomas Chambrion, Alexandre Munnier. Generalized Scallop Theorem for Linear Swimmers. 2010. 〈inria-00508646〉

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