How do micro-swimmers swim ? CIMI Workshop, New trends in modeling, - PowerPoint PPT Presentation

How do micro-swimmers swim ? CIMI Workshop, New trends in modeling, control and inverse problems Laetitia Giraldi ENS de LYON 17 June 2014 Laetitia Giraldi Microswimming 1

Microswimming ◮ Well establish domain. ◮ Applications to Biology / Robotic. ◮ Emerging of artificial mechanisms : How to obtain self-propulsion controlled at micro-scale ? Spintec Lab (2014) ESPCI (2005) Laetitia Giraldi Microswimming 2

Swimming Swimming is the ability of moving in or under water with suitable movements (body deformation) and without external forces. Two main concepts : ◮ Swimming (and locomotion in general) is a control problem. ◮ Swimming at minimal time (or cost) is an optimal control problem. – See, for instance, the works of : F. Alouges and al., M. Tucsnak and al., T. Chambrion and al. ... Laetitia Giraldi Microswimming 3

Modeling Two conceptual ingredients ◮ How does the surrounding medium react (namely, which forces does it exert ?) to shape changes of the swimmer ? ֒ → Equation of motion of the surrounding medium. ◮ How does the swimmer move in response to the forces that the surrounding medium applies to it ? ֒ → Equation of motion of the swimmer. Laetitia Giraldi Microswimming 4

Modeling ◮ The fluid is governed by the Navier-Stokes equation ρ f ( ∂ t u + ( u · ∇ ) u ) − µ ∆ u + ∇ p = ρ f g , div u = 0 in F . We add the boundary condition, u = q ˙ + u d on ∂ N , ���� ���� speed of the swimmer speed of the deformation ◮ How does the swimmer move in response to the forces that the surrounding medium applies to it ? → Equation of motion of the swimmer. ֒ Laetitia Giraldi Microswimming 4

Modeling ◮ The fluid is governed by the Navier-Stokes equation ρ f ( ∂ t u + ( u · ∇ ) u ) − µ ∆ u + ∇ p = ρ f g , div u = 0 in F . We add the boundary condition, u = q ˙ + u d on ∂ N , ���� ���� speed of the swimmer speed of the deformation ◮ with the Newton law �  σ ( u , p ) · n d s = − m 0 ( g + ¨ q ) ,   ∂ N � σ ( u , p ) · n × ( x − q ) d s = − m 0 q × g + ˙  Ω ,  ∂ N where σ ( u , p ) = µ ( ∇ u + ∇ t u ) − p Id is the Cauchy tensor. Laetitia Giraldi Microswimming 4

Rescaling The fluid is governed by the Navier-Stokes equation � � τ ∂ u ∗ u ∗ − ∆ ∗ u ∗ + ∇ ∗ p ∗ = Re ∂ t ∗ + u ∗ · ∇ ∗ in F . , Re F g ∗ , div u = 0 µ , F = U 2 where, Re = ρ f UL LG , τ = TU L . with the boundary condition, u = ˙ q + u d on ∂ N , with the Newton law � 1  � � σ ( u ∗ , p ∗ ) · n d s = − ρ m F g ∗ + 1  Re τ 2 ¨ q ∗ ,   ρ f � 1 ∂ N � � σ ( u ∗ , p ∗ ) · n × ( x − q ) d s = − ρ m F q ∗ × g ∗ + 1 τ 2 ˙  Ω ∗ Re ,   ρ f ∂ N Laetitia Giraldi Microswimming 5

At micro scale Re := ρ f UL ∼ 10 − 6 µ The fluid is governed by the Navier-Stokes equation ✭✭✭✭✭✭✭✭✭✭✭ � � τ ∂ u ∗ Re ✟ Re ∂ t ∗ + u ∗ · ∇ ∗ u ∗ − ∆ ∗ u ∗ + ∇ ∗ p ∗ = ✟✟ F g ∗ , div u = 0 in F . , µ , F = U 2 where, Re = ρ f UL LG , τ = TU L . with the boundary condition, u = ˙ q + u d on ∂ N , with the Newton law � 1 ✘  ✘✘✘✘✘✘✘✘✘✘✘ � � − ρ m F g ∗ + 1  σ ( u ∗ , p ∗ ) · n d s = τ 2 ¨ q ∗ Re ,   ρ f � 1 ∂ N � ✭✭✭✭✭✭✭✭✭✭✭✭✭✭ � − ρ m F q ∗ × g ∗ + 1 τ 2 ˙  σ ( u ∗ , p ∗ ) · n × ( x − q ) d s = Ω ∗ Re ,   ρ f ∂ N Laetitia Giraldi Microswimming 6

Fluid-Swimmer interaction ◮ q ∈ R 3 × SO 3 : position and orientation. ◮ ξ ∈ R k : shape. Laetitia Giraldi Microswimming 7

Equation of motion ◮ Newton Law, �  σ ( u , p ) · n d s = 0 ,   ∂ N �  σ ( u , p ) · n × ( x − q ) d s = 0 ,  ∂ N where σ ( u , p ) = µ ( ∇ u + ∇ t u ) − p Id is the Cauchy tensor. ◮ on the boundary the speed u is linear on ˙ ξ and ˙ q . ◮ As the result of the linearity of the Stokes equation we get, q + N ( ξ , q )˙ M ( ξ , q ) ˙ ξ = 0 . ◮ Then, k � F i ( ξ , q ) ˙ q = ˙ ξ i . i = 1 Laetitia Giraldi Microswimming 8

Swimmer dynamics ◮ The dynamics is an ODE linear with respect to ( ˙ ξ ) and without drift. ◮ For each initial position and deformation, there exists an unique trajectory. Question If at beginning the swimmer is at the state ( ξ 0 , q 0 ) , could it reach ( ξ f , q f ) (i.e., a given position and orientation with a given shape) ? Laetitia Giraldi Microswimming 9

First answer by Purcell in 1976 The scallop theorem [Purcell, 1976] At low Reynolds number, a reciprocal shape change does not induce any net movement. Proof Reversibility property of a Stokes flow. Laetitia Giraldi Microswimming 10

Purcell’s 3-link swimmer Laetitia Giraldi Microswimming 11

Non holonom q = ˙ ξ 1 F 1 ( ξ , q ) + ˙ ˙ ξ 2 F 2 ( ξ , q ) , if ( ξ ( 0 ) , q ( 0 )) = ( ξ 0 , q 0 ) , (˙ ξ 1 , ˙ ξ 2 ) = ( 1 , 0 ) on [ 0 , ε [ , (˙ ξ 1 , ˙ ξ 2 ) = ( 0 , 1 ) on [ ε, 2 ε [ , (˙ ξ 1 , ˙ ξ 2 ) = ( − 1 , 0 ) on [ 2 ε, 3 ε [ , (˙ ξ 1 , ˙ ξ 2 ) = ( 0 , − 1 ) on [ 3 ε, 4 ε [ . Then, q ( 4 ε ) = q 0 + ε 2 [ F 1 , F 2 ]( ξ 0 , q 0 ) + O ( ε 3 ) , [ F 1 , F 2 ] = ( F 1 · ∇ ) F 2 − ( F 2 · ∇ ) F 1 is the Lie Bracket between F 1 and F 2 . Laetitia Giraldi Microswimming 12

Nonholonomic systems If [ F 1 , F 2 ] � = 0, then, the Purcell’s swimmer can move. Laetitia Giraldi Microswimming 13

Lie Algebra k � F i ( ξ , q ) ˙ q = ˙ ξ i i = 1 M = { ( ξ , q ) } ◮ Let F a family of vector fields (here F = ( F i ) ), Lie ( F ) is the smallest algebra generated by F . ◮ => Lie ( F ) is the smallest space which satisfies, ∀ ( F , G ) ∈ Lie ( F ) 2 , [ F , G ] ∈ Lie ( F ) ◮ For all ( ξ , q ) ∈ M , Lie ( ξ , q ) ( F ) := { G ( ξ , q ) G ∈ Lie ( F ) } . t . q . ◮ All the vector in Lie ( ξ , q ) ( F ) are reachable. ◮ Lie ( ξ , q ) ( F ) ⊆ T ( ξ , q ) M , is a finite-dimensional vector space. Laetitia Giraldi Microswimming 14

Controllability Results Under the hypothesis that ( F i ) are analytics. Theorem [Chow(1939) - Rashevski 1938)] if Lie ( ξ , q ) ( F ) = T ( ξ , q ) M , then the system is locally controllable around the state ( ξ , q ) . Theorem [Hermann (1963) - Nagano (1966) - (Lobry (1970))] ◮ Each orbit is an analytic manifold. ◮ Its tangent space is the set Lie ( ξ , q ) ( F ) , for all ( ξ , q ) . ◮ In particular, the dimension of Lie ( ξ , q ) ( F ) remains constant along an orbit. Laetitia Giraldi Microswimming 15

What is missing to prove a controllability theorem ? ◮ The vector fields ( F i ) are expressed by the Dirichet-to-Neumann map of the associated stokes problem. => In general, ( F i ) are not explicits ! ◮ How to prove the regularity of the vector fields ( F i ) ? ◮ How to compute the dimension of the Lie algebra generated by ( F i ) ? Laetitia Giraldi Microswimming 16

Approximate the Dirichlet-to-Neumann map Resistive Force Theory For a sitck with a speed v then the associated distribution of forces, called F , is given by F = d 1 ( v . e θ ) e θ + d 2 ( v . f θ ) f θ J. Gray and J. Hancock The propulsion of sea-urchin spermatozoa , Journal of Experimental Biology, 1955. Laetitia Giraldi Microswimming 17

Proof ξ 2 • ξ 1 ( x , y ) θ 1 Here the variables are q = ( x , y , θ 1 ) and ξ = ( ξ 1 , ξ 2 ) ◮ The Resistive Force Theory gives an explicit equation of motion. ◮ However, the expressions of F i are huge. ◮ Analyticity of the vector fields F 1 , F 2 are deduced by its expression. ◮ Formal calculation leads to get : ∃ ξ 0 ∈ [ 0 , 2 π ] 2 and ∀ θ 1 ∈ [ 0 , 2 π ] , det ( F 1 , F 2 , [ F 1 , F 2 ] , [ F 1 , F 2 ] , F 1 ] , [ F 1 , F 2 ] , F 2 ])( ξ 0 , θ 1 ) � = 0 ◮ Orbit theorem and Chow theorem lead to get the result. Laetitia Giraldi Microswimming 18

Controllability of the Purcell’s swimmer Theorem [L.G., P. Martinon, M. Zoppello (2013)] The Purcell’s swimmer is globally controllable. From any state ( ξ 0 , q 0 ) , one can reach any other state ( ξ 1 , q 1 ) with a suitable force law ( f i ( t )) i such that � fi ( t ) = 0 (or equivalently suitable functions ˙ ξ i ). This proof gives a general framework (see [T. Chambrion and A. Munnier], [M. Tucsnak, J. Loheac, J. F. Scheid], [F. Alouges, A. DeSimone, A. Lefevbre]...) Laetitia Giraldi Microswimming 19

By adding sticks Theorem [L.G., P. Martinon, M. Zoppello (2013)] The N-link swimmer is globally controllable. α N − 1 • xN • q = ( x 1 , θ 1 ) • ξ = ( α 1 , ..., α N − 1 ) α 1 • θ 1 • x 1 N − 1 � q = ˙ F i ( ξ , q ) ˙ α i i = 2 Laetitia Giraldi Microswimming 20

By adding sticks Theorem [L.G., P. Martinon, M. Zoppello (2013)] The N-link swimmer is globally controllable. Laetitia Giraldi Microswimming 20

By adding sticks Theorem [L.G., P. Martinon, M. Zoppello (2013)] The N-link swimmer is globally controllable. N − 1 � ˙ q = F i ( ξ , q ) ˙ α i , N=25, avec une tête. i = 2 Laetitia Giraldi Microswimming 20

What happens when a boundary is added in the fluid domain ? Photo by Stephen C. Jacobson L. Rothschild Non-random distribution of bull spermatozoa in a drop of sperm suspension , Nature, 1963. Laetitia Giraldi Microswimming 21