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) 1

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

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 3

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 4

The model and the mathematical challenge

5

6

Kinematics of the creature (I)

S*(T)

7

Kinematics of the creature (II)

8

The governing equations



u t

div

u

   

u

  

u

0,

x

 0,

x

 ( ),

t



[

0

,T]

, ( ),

t



[

0

,T]

,

u



h

  

x



h

  

w

,

x

  ( ),

t

 [0,

T

],

Mh

   

S t



n d

 ,

t

 [0,

T

], d d

t



J

      

x



h

   

n d

 ,

t

 [0,

T

],

u

0) 

u

0 (

x

) ,

h

(0) 

h

0 ,

h i

'(0) 

h

1

i

,  (0 )   0 9

Existence theorems

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) 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. Control theoretic challenge: the input of the system is the geometry of the domain.

10

Numerical Method and Simulations

11

.

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) 12

Two bilinear forms

13

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

14

Semi-discretization with respect to time

15

Choice of characteristics

16

Finite element spaces

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

Convergence of solutions

18 18

Straight-line swimming 19

Turning 20

Meeting

21

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.

22

Mickey’s Reconstruction

Kiss and go effect

Perspectives

25

• 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 26

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

27

The model : One rigid ball in the whole space



u t

div

u

   

u

  

u

0,

x

 0,

x

 ( ),

t



[

0

,T]

, ( ),

t



[

0

,T]

,

u



h

   

x



h

 

V R

,

x

  ( ),

t

 [0,

T

],

Mh

   

B t



n d

 ,

t

 [0,

T

], d d

t

(

J

 )     

x



h

  

n d

 ,

t

 [0,

T

],

R



RS

(  ).

) 

R

3 \ ) 28

Change of coordinates :

y u t

div

u

  

u

  

u

    

u

0, y 

F

,

t



[

0

,T]

, 0, y 

F

,

t



[

0

,T]

,

u



h

 

M h V R

, y  

S

,

t

 [0,

T

],    

B



n

d  , d d

t

(

J

 )    

B x

 

n

d  ,

t

 [0,

T

],

R



RS

(  )

B



B

,

F

 3 \

B

29

A simplified finite dimensional model

 div

u



p

0, y 

F

,

t



[

0

,T]

, 0, y 

F

,

t



[

0

,T]

,

u



h

  0  

B

(   )  (

l

, 

u

, y  

B

,

t

 [0,

T

],

n

d  , 0  

B

(   )

x

  (

l

, 

h

  

l

,

R

 

u RS

(  ),

n

d  ,

t

 [0,

T

],

u

0) 

u

0 (

y

) ,

h

(0) 

h

0 ,

h i

'(0) 

h

1

i

,  (0 )   0 The control is:  

i

6  1

i



i

 30

The case of small deformations

31

A controllability result

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

Z



F

+

Bu

in

R

9  SO(3) .

• 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) 32

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

The case of small deformations