Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid
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Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid Šárka Neˇcasová · Takéo Takahashi · Marius Tucsnak
Received: 23 January 2011 / Accepted: 14 September 2011 / Published online: 1 October 2011 © Springer Science+Business Media B.V. 2011
Abstract We consider the three-dimensional motion of a self-propelled deformable structure into a viscous incompressible fluid. The deformation of the solid is given whereas its position is unknown. Such a system could model the propulsion of fish-like swimmers. The equations of motion of the fluid are the Navier-Stokes equations and the equations for the structure are deduced from Newton’s laws. The corresponding system is a free boundary problem and the main result of the paper is the existence of weak solutions for this problem. Keywords Navier-Stokes equations · Fluid-structure interaction · Weak solutions · Free boundary
1 Introduction The understanding of swimming or flying is one of the main challenges in fluid dynamics. This problem has been considered by many scientists for a long time: for instance around
The research of S.N. was supported by Grant Agency of Czech Republic n. P201/11/1304 and by Institutional Reasearch Plan n. AVOZ10190503, of the Academy of Sciences of the Czech Republic. The work of the second and third authors was partially supported by ANR Grant BLAN07-2 202879 and by ANR Grant 09-BLAN-0213-02. Š. Neˇcasová Mathematical Institute, Academy of Sciences, Zitna 25, Prague 1, 11567, Czech Republic e-mail: [email protected] T. Takahashi () · M. Tucsnak Institut Élie Cartan, UMR 7502, INRIA, Nancy-Université, CNRS, BP239, 54506 Vandœuvre-lès-Nancy Cedex, France e-mail: [email protected] M. Tucsnak e-mail: [email protected] T. Takahashi · M. Tucsnak Team-Project CORIDA, INRIA Nancy - Grand Est, 615, rue du Jardin Botanique, 54600 Villers-lès-Nancy, France
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350 BC, Aristotle was already writing observations on fish and cephalod locomotion. Much later, during the 17th century, Borelli [2] started the study of swimming and flying by using mathematics to confirm his theories. In the 20th century, a zoologist, James Gray, introduced (see [13]) a paradox—Gray’s paradox—suggesting that the undulating way of swimming of dolphin is much more efficient than a conventional propeller for underwater motion. Even if this paper is controversial, it has led to many studies in order to contradict or understand this paradox. Many other works were dedicated to the understanding of fish locomotion: Taylor [28], Lighthill [18], Childress [7], Wu [30], Sparenberg [26], etc. A possible common starting point for studying how fishes or micro-organisms swim or for studying how birds or insects fly is to consider the classical Navier–Stokes system for the fluid dynamics. For the dynamics of the creature, which swims or flies, there are many possibilities but it is reasonable to consider it as a deformable structure. However, this deformation, which allows the creature to swim or fly, depends
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