1. 1 Self-Propelled Motions of Solids in a Fluid: Mathematical Analysis, Simulation and Control Marius TUCSNAK Institut Elie Cartan de Nancy et INRIA Lorraine, projet CORIDA Collaborations with : Jorge SAN MARTIN (Santiago) Jean-François SCHEID (IECN) Takeo TAKAHASHI (IECN)

2. 2 Motivation: understanding locomotion of aquatic organisms, in particular to design swimming-robots

3. 3 Motion of solids in a fluid Incompressible Navier-Stokes (Euler, Stokes) equations for the fluid ODE’s or PDE’s system for the motion of the solids Continuity of the velocity field at the interface Homogenous Dirichlet boundary conditions on the exterior boundary

4. 4 Plan of the talk The model and the mathematical challenge An existence and uniqueness theorem Numerical method and simulations Self-propelling at low Reynolds number

5. 5 The model and the mathematical challenge

6. 6

7. 7 Kinematics of the creature (I) S*(T)

8. 8 Kinematics of the creature (II)

9. 9 The governing equations

10. 10 Existence theorems Theorem (J. San Martin, J. –F. Scheid, T. Takahashi, M.T., ARMA (2008)). If the given deformation is smooth enough than the system admits an unique strong solution. If no contact occurs in finite time then this solution is global. The Leray type mathematical theory has been initiated around year 2000. Early mathematical ref. for rigid-fluid interaction from a mathematical view-point : D. Serre ( 1987) K. H. Hoffmann and V. Starovoitov ( 2000) B. Desjardins and M. Esteban ( 1999,2000) C. Conca, J. San Martin and M. Tucsnak (2001) J. San Martin, V. Starovoitov and M. Tucsnak (2002) Control theoretic challenge: the input of the system is the geometry of the domain.

11. 11 Numerical Method and Simulations

12. 12 References Carling, Williams and G.~Bowtell, (1998) Liu and Kawachi (1999) Leroyer and M.Visonneau, (2005) J. San Martin, J.-F. Scheid and M. Tucsnak (SINUM 2005, ARMA 2008).

13. 13 Two bilinear forms

14. 14 Global weak formulation (J. San Martin, J.F –Scheid, T. Takahashi and M.T. in SINUM (2005) )

15. 15 Semi-discretization with respect to time

16. Choice of characteristics 16

17. 17 Finite element spaces Fixed mesh Rigidity matrix is (partially) re-calculated at each time step

18. 18 Convergence of solutions 18

19. 19 Straight-line swimming

20. 20 Turning

21. 21 Meeting

22. 22 Can they really touch? The answer is no for rigid balls (San Martin, Starovoitov and Tucsnak (2002), Hesla (2006), Hillairet (2006), but unknown in general.

23. 23 Mickey’s Reconstruction

24. 24 Kiss and go effect

25. 25 Perspectives

26. 26 Coupling of the existing model to elastodynamics type models for the fish Control problems Infirming or confirming « Gray’s paradox » Giving a rigorous proof of the existence of self-propelled motions

27. 27 A Control Theoretic Approach to the Swimming of Cilia Micro-Organisms

28. 28 The model : One rigid ball in the whole space

29. 29

30. 30 A simplified finite dimensional model The control is: with given « shape » functions.

31. The case of small deformations 31

32. 32 A controllability result By « freezing » the deformation (Blake’s « layer model »), we obtain a simplified model which can be written as a dynamical system Proposition. (J. San Martin, T. Takahashi and M.T., QAM (2008)) Generically (with respect to the shape functions), the above system is controllable in any time. With “standard” choices of the shape functions the controllabilty fails with less than 6 controls. Remark. For less symmetric shapes less than 6 controls suffice (Sigalotti and Vivalda, 2007)

33. Some perspectives Optimal control problems, in particular obtaining the motion of cilia by solving an optimization problems. Proving the existence of self-propelled motions for arbitrary Reynolds numbers Control at higher Reynolds numbers. 33

34. 34 The case of small deformations 34