Mathematisches Kolloquium

Freitag, 30.11.2018, 14:30 Uhr

Optimally swimming mechanisms at low Reynolds number

François Alouges (Ecole Polytechnique Paris)

Abstract:
Like any locomotion problem, driving micro mechanisms that can swim can be studied using tools of control theory. Nevertheless, this problem possesses some peculiarities that are consequences of the negligible inertia at that scale, the most famous being probably the obstruction formalized by E. M. Purcell as the "scallop theorem". We will show how Lie brackets computations unveils the set of attainable directions that the swimmer may take, permitting to circumvent the obstruction. When energy minimizing conditions are furthermore required, as for optimizing Lighthill's efficiency, the problem becomes an optimal control problem and optimal gaits can be geometrically expressed as suitable geodesics in the space of  shape deformations. Finding periodic optimal gaits is a problem which is only tractable under numerical studies, and very little intuition can be extracted from the optimal strategies that are found in this way. However, if one considers furthermore the small amplitude stroke regime, it is possible to characterize optimal strokes in a way that produces naturally common features that are well known such as the bending wave along Taylor sheet. Other Mechanisms such as a swimmer with 3 arms that can be elongated will be also studied. Results shown in the talk are joint works with A. DeSimone, G. DiFratta,  L. Giraldi, Y. Or, O. Wiezel and M. Zoppello.

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