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Self-Propelled Motions of Solids in a Fluid: Mathematical Analysis, Simulation and Control

Marius TUCSNAKInstitut Elie Cartan de Nancy et

INRIA Lorraine, projet CORIDA

Collaborations with :Jorge SAN MARTIN (Santiago) Jean-François SCHEID (IECN)Takeo TAKAHASHI (IECN)

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

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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

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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

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The model and the mathematical challenge

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Kinematics of the creature (I)

S*(T)

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Kinematics of the creature (II)

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The governing equations

0, ( ), 0 ,div 0, ( ), 0 ,

tu u u u p x F t t [ ,T]u x F t t [ ,T]

( )

( )

, ( ), [0, ],

, [0, ],

d , [0, ], d

S t

S t

w x S t t T

d t T

x

h x

h d t Tt

Mh

u

n

J n

h

0 0 1 0(0) ,( , '(0) , 0) ( (0,) ) i ih h hu u x hx

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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.

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Numerical Method and Simulations

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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)

.

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Two bilinear forms

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Global weak formulation(J. San Martin, J.F –Scheid, T. Takahashi and M.T.

in SINUM (2005) )

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Semi-discretization with respect to time

Choice of characteristics

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Finite element spaces

• Fixed mesh• Rigidity matrix is (partially) re-calculated at each time step

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Convergence of solutions

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Straight-line swimming

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Turning

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Meeting

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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.

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Mickey’s Reconstruction

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Kiss and go  effect

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Perspectives

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• 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

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

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The model :One rigid ball in the whole space

0, ( ), 0 ,div 0, ( ), 0 ,

tu u u u p x F t t [ ,T]u x F t t [ ,T]

( )

( )

, ( ), [0, ],

, [0, ],

d , [0, ], d

( )

R

B t

B t

V x B t t T

d t T

d

h x hu

n

t T

Mh

J x ht

n

( ).R RS

3( )) (\F t B tR

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Change of coordinates : ( )y x h t

( ) 0, y , 0 ,

div 0, y , 0 ,

tu u u u h t u p F t [ ,T]

u F t [ ,T]

, y , [0, ],d, , d [0, ], ) d(d

R

BB

V S t T

t Tt

u

n x

h

M n

x

h J

( )R RS

3 , ( ) \0 FB B B

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0, y , 0 ,div 0, y , 0 ,

u p F t [ ,T]u F t [ ,T]

( ( ))

, y , [0, ],

, , [0, ], 0 , , ) 0 , , )

,

( d ( dB B

B t Th xu

n x

u

l u l u

h l

t

R

n T

( ),RS

u

0 0 1 0(0) ,( , '(0) , 0) ( (0,) ) i ih h hu u y hy

A simplified finite dimensional model

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( )( , ) ( )ii iu y t v t y

The control is:

with given « shape » functions.i

The case of small deformations

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A controllability resultBy « freezing » the deformation (Blake’s « layer model »), we obtain a simplified model which can be written as a dynamical system

9+ in SO(3) .Z F BuZ R•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)

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.

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The case of small deformations

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