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 ? ESPCI (2005) Spintec Lab (2014)

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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. ...

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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.

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Modeling

◮ The fluid is governed by the Navier-Stokes equation

ρf (∂tu + (u · ∇) u) − µ∆u + ∇p = ρf g , div u = 0 in F. We add the boundary condition, u = ˙ q

• speed of the swimmer

+ ud

• speed of the deformation
• n ∂N,

◮ How does the swimmer move in response to the forces that the

surrounding medium applies to it ? ֒ → Equation of motion of the swimmer.

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Modeling

◮ The fluid is governed by the Navier-Stokes equation

ρf (∂tu + (u · ∇) u) − µ∆u + ∇p = ρf g , div u = 0 in F. We add the boundary condition, u = ˙ q

• speed of the swimmer

+ ud

• speed of the deformation
• n ∂N,

◮ with the Newton law

    

• ∂N

σ(u, p) · n ds = −m0 (g + ¨ q) ,

• ∂N

σ(u, p) · n × (x − q) ds = −m0q × g + ˙ Ω , where σ(u, p) = µ(∇u + ∇tu) − pId is the Cauchy tensor.

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Rescaling

The fluid is governed by the Navier-Stokes equation Re

• τ ∂u∗

∂t∗ + u∗ · ∇∗

• u∗ − ∆∗u∗ + ∇∗p∗ = Re

F g∗,

div u = 0 in F. , where, Re = ρf UL

µ , F = U2 LG , τ = TU L .

with the boundary condition, u = ˙ q + ud

• n ∂N,

with the Newton law

      

• ∂N

σ(u∗, p∗) · n ds = −ρm ρf Re

1

F g∗ + 1 τ 2 ¨ q∗

• ,
• ∂N

σ(u∗, p∗) · n × (x − q) ds = −ρm ρf Re

1

F q∗ × g∗ + 1 τ 2 ˙ Ω∗

• ,

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At micro scale Re := ρf UL

µ

∼ 10−6

The fluid is governed by the Navier-Stokes equation

✭✭✭✭✭✭✭✭✭✭✭

Re

• τ ∂u∗

∂t∗ + u∗ · ∇∗

• u∗ − ∆∗u∗ + ∇∗p∗ = ✟✟

✟

Re F g∗,

div u = 0 in F. , where, Re = ρf UL

µ , F = U2 LG , τ = TU L .

with the boundary condition, u = ˙ q + ud

• n ∂N,

with the Newton law

      

• ∂N

σ(u∗, p∗) · n ds =

✘✘✘✘✘✘✘✘✘✘✘ ✘

−ρm ρf Re

1

F g∗ + 1 τ 2 ¨ q∗

• ,
• ∂N

σ(u∗, p∗) · n × (x − q) ds =

✭✭✭✭✭✭✭✭✭✭✭✭✭✭

−ρm ρf Re

1

F q∗ × g∗ + 1 τ 2 ˙ Ω∗

• ,

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Fluid-Swimmer interaction

◮ q ∈ R3 × SO3 : position and orientation. ◮ ξ ∈ Rk

: shape.

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Equation of motion

◮ Newton Law,

    

• ∂N

σ(u, p) · n ds = 0 ,

• ∂N

σ(u, p) · n × (x − q) ds = 0 , where σ(u, p) = µ(∇u + ∇tu) − pId 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,

M(ξ, q) ˙ q + N(ξ, q)˙ ξ = 0 .

◮ Then,

˙ q =

k

• i=1

Fi(ξ, q) ˙ ξi .

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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 (ξ0, q0), could it reach (ξf , qf ) (i.e., a given position and orientation with a given shape) ?

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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.

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Purcell’s 3-link swimmer

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Non holonom

˙ q = ˙ ξ1F1(ξ, q) + ˙ ξ2F2(ξ, q), if (ξ(0), q(0)) = (ξ0, q0), (˙ ξ1, ˙ ξ2) = (1, 0)

• n

[0, ε[, (˙ ξ1, ˙ ξ2) = (0, 1)

• n

[ε, 2ε[, (˙ ξ1, ˙ ξ2) = (−1, 0)

• n

[2ε, 3ε[, (˙ ξ1, ˙ ξ2) = (0, −1)

• n

[3ε, 4ε[. Then, q(4ε) = q0 + ε2[F1, F2](ξ0, q0) + O(ε3), [F1, F2] = (F1 · ∇)F2 − (F2 · ∇)F1 is the Lie Bracket between F1 and F2.

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Nonholonomic systems

If [F1, F2] = 0, then, the Purcell’s swimmer can move.

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Lie Algebra

˙ q =

k

• i=1

Fi(ξ, q) ˙ ξi M = {(ξ, q)}

◮ Let F a family of vector fields (here F = (Fi)), 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) t.q. G ∈ Lie(F)} .

◮ All the vector in Lie(ξ,q)(F) are reachable. ◮ Lie(ξ,q)(F) ⊆ T(ξ,q)M, is a finite-dimensional vector space.

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Controllability Results

Under the hypothesis that (Fi) 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.

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What is missing to prove a controllability theorem ?

◮ The vector fields (Fi) are expressed by the

Dirichet-to-Neumann map of the associated stokes problem. => In general, (Fi) are not explicits !

◮ How to prove the regularity of the vector fields (Fi) ? ◮ How to compute the dimension of the Lie algebra generated

by (Fi) ?

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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 = d1(v.eθ)eθ + d2(v.fθ)fθ

• J. Gray and J. Hancock

The propulsion of sea-urchin spermatozoa, Journal of Experimental Biology, 1955.

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Proof

• (x, y)

ξ1 ξ2 θ1 Here the variables are q = (x, y, θ1) and ξ = (ξ1, ξ2)

◮ The Resistive Force Theory gives an explicit equation of

motion.

◮ However, the expressions of Fi are huge. ◮ Analyticity of the vector fields F1, F2 are deduced by its

expression.

◮ Formal calculation leads to get :

∃ξ0 ∈ [0, 2π]2 and ∀θ1 ∈ [0, 2π], det(F1, F2, [F1, F2], [F1, F2], F1], [F1, F2], F2])(ξ0, θ1) = 0

◮ Orbit theorem and Chow theorem lead to get the result.

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Controllability of the Purcell’s swimmer

Theorem [L.G., P. Martinon, M. Zoppello (2013)]

The Purcell’s swimmer is globally controllable. From any state (ξ0, q0), one can reach any other state (ξ1, q1) with a suitable force law (fi(t))i such that fi(t) = 0 (or equivalently suitable functions ˙ ξi). 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]...)

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By adding sticks

Theorem [L.G., P. Martinon, M. Zoppello (2013)]

The N-link swimmer is globally controllable.

• θ1

α1 x1 xN αN−1

q = (x1, θ1) ξ = (α1, ..., αN−1) ˙ q =

N−1

• i=2

Fi(ξ, q) ˙ αi

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By adding sticks

Theorem [L.G., P. Martinon, M. Zoppello (2013)]

The N-link swimmer is globally controllable.

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By adding sticks

Theorem [L.G., P. Martinon, M. Zoppello (2013)]

The N-link swimmer is globally controllable. ˙ q =

N−1

• i=2

Fi(ξ, q) ˙ αi, N=25, avec une tête.

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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.

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3-sphere swimmer / 4-sphere swimmer

The swimmers consist of N = 3, 4 balls connected by jacks. The swimmer changes its shape by changing the length of its arms (ξi).

x y z (xc, yc, zc) ξ1 ξ2 φ θ

◮ Control functions : ˙

ξk with k ∈ {1, · · · , 4}

◮ Position : xc := (xc, yc, zc) ∈ R3 ◮ Orientation : R ∈ SO3 or (θ, φ) ∈ [0, 2π] × [0, 2π]

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Why spheres ?

The Dirichlet-to-Neumann map is defined by DN :

N

• l=1

H1/2(∂Bl) →

N

• l=1

H−1/2(∂Bl) (ul) → (fl := σ(u, p)n|∂Bl) , where (u, p) is the solution of the Stokes equation −∆u + ∇p = 0, div u = 0 in F, u|∂O = 0, u|∂Bl = ul. We have the explicit representation, ND(f)i

=ui

(x) =

N

• l=1
• ∂Bl

G(x, y)fl(y)dy, ∀i, ∀x ∈ ∂Bi.

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In the case of a whole space

If O = R3, G(r) = 1 8πµ

Id

|r| + r ⊗ r |r|3

• .

– This implies the analyticity of the vector fields Fi’s with respect to the state of the swimmer.

Theorem [F. Alouges et al. (2010)]

In R3,

◮ the 4-sphere swimmer is controllable and ◮ the 3-sphere swimmer is controllable in a straight line (the

• ne which contains the swimmer).

– The proof fits the same framework than the Purcell’s swimmer

• ne.

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Controllability results

Theorem [F. Alouges and L.G (2012)]

In the presence of a flat wall,

◮ the 4-sphere swimmer is “almost everywhere” locally

controllable and

◮ the 3-sphere swimmer is “almost everywhere” locally

controllable in a plane (the one which contains the swimmer).

Theorem rough wall [D. Gérard-Varet and L.G. (2013)]

◮ In the presence of a rough wall, the 4-sphere swimmer is

“almost everywhere” locally controllable.

◮ There exists a rough wall such that the 3-sphere swimmer be

“almost everywhere” locally controllable.

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Some remarks on these results

Definition

We call a swimmer “almost everywhere” locally controllable, if for almost all initial configurations it can reach all final state in a neighborhood. Moral of this story :

◮ A boundary does not affect a controllable swimmer. ◮ However, the boundary enhances the reachable set of a

non-controllable swimmer.

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Proof

Notation : ε amplitude of roughness (z = εh(x, y) with |h|∞ = 1) a radius of the spheres A := {(ε, a, ξ, q) ∈ R × R∗

+ × (R∗ +)k × (R3 × SO(3)) :

Bi ∩ Bj = ∅ ∀i = j, and Bi ∩ ∂O = ∅ ∀i}.

Theorem (D. Gérard-Varet and L.G.)

For all i = 1 . . . k, the field Fi(ξ, p) (which depends also implicitly

• n ε and a) is an analytic function of Y := (ε, a, ξ, q) over A.

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Analyticity of the Dirichlet-to-Neumann map

I :=

N

• l=1
• ∂Bl

( 1

x ) ⊗ (σ(u, p)n) ds

Lemma

For fixed balls (Bl) and fixed boundary ∂O, there exists a diffeomorphism ϕ such that ϕ(Bl) = Bl and ϕ(∂O) = ∂O. ϕ(x) := x +

• l

χ(x − xl) (ϕl(x) − x) + (ε − ε)χh(x)(0, 0, h(x1, x2)) with,

◮ χ ∈ C∞ c (R3), χ = 1 near B(0, a), ◮ χh ∈ C∞ c (R3), χh = 1 near x3 = ε h(x1, x2). ◮ ϕl(x) := a a(x − xl) + xl.

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Introducing U := u ◦ ϕ and P := p ◦ ϕ, it remains to prove the following proposition.

Proposition

Y → (U, P) is analytic from B(¯ Y, δ) to V0 × L2(F), where

V0 :=

• U ∈ D′(F, R3) | ∇U ∈ L2(F),

U(r)

• 1 + |r|2 ∈ L2(F),

U|∂ ¯

O = 0

• .

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Proof

First write the system satisfied by U, P. A simple computation yields

      

− div (A∇U) + B∇P = 0 in F, div (BtU) = 0 in F, U = 0 at ∂O, U = Ul at ∂Bl, (1) where A and B depend on Y. Then apply the implicit function theorem.

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The 4-sphere swimmer

Consequence : controllability of the “4-sphere swimmer”.

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The 3-sphere swimmer

Asymptotic expansion for the Dirichlet-to-Neumann map.

Proposition

DN = T 0 + εT 1 + O(ε2) in L(H−1/2

N

, H1/2

N )

where T 0 and T 1 are defined as T 0(f)i(r) :=

• j
• ∂B

K0(xi + ar, xj + as)fj(s)ds and T 1(f)i(r) :=

• j
• ∂B

K1(xi + ar, xj + as)fj(s)ds. The Green functions K0(x, x0) and K1(x, x0) can be expressed.

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Asymptotics of the Dirichlet-to-Neumann map

◮ T 0 and T 1 are the sum of maps Ti,j.

Ti,j : H−1/2(∂B) → H1/2(∂B) fj →

• ∂B

K(xi + a·, xj + as) fj(s) ds , with the Green kernel K given by K(r, r′) := G(r − r′) + K1(r, r′) + K2(r, r′) + K3(r, r′)

• K0(r,r′)

+ K4(r, r′)

• K1(r,r′)

.

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Structure of Green’s functions

We call T G : H−1/2(∂B) → H1/2(∂B) fj →

• ∂B

G(a(· − s)) f(s) ds .

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◮ For every f ∈ H−1/2 N

, for all (x, ξ), (DN f)i (r) = T Gfi + 4

l=1 Kl(xi, xi)fi, Id∂B

+

• j=i

K(xi, xj)fj, Id∂B + Ri(f), with RiL(H−1/2

N

,H1/2

N

) = O

• a + ε2

, and i = 1...N.

◮ For every u ∈ H1/2 N , for all (q, ξ),

DN−1u

• i =

(T G)−1

• ui −

4

• k=1

Kk(xi, xi)(T G)−1ui, Id∂B

• −(T G)−1

 

j=i

K(xi, xj)(T G)−1uj, Id∂B

 

+ ˜ Ri(u) with ˜ RiL(H1/2

N

,H−1/2

N

) = O

• a3 + a2ε2

, i = 1...N.

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Asymptotic expansion of the dynamics

By plugging the asymptotic expansion of fl into the self-propulsion constraint

        

• l
• ∂B

fl(r) = 0 ,

• l
• ∂B

(xl − x2 + ar) × fl(r) = 0 , we get an asymptotic expansion of the dynamics of the swimmer for small radius of the spheres and small roughness.

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Controllability fo the 3-sphere swimmer

◮ A formal computation of the determinant of the Lie brackets

which generate the Lie algebra. (technical computation)

◮ Chow’s theorem.

Remark : Here we cannot apply the orbit theorem to get the global controllability (in that case there are several orbits). Remarks [Yuri L. Sachkov (2003)] :

◮ In the case of the whole space, the distribution generated by

the Fi’s corresponds to the Heisenberg case.

◮ In the case of the flat wall, the distribution generated by the

Fi’s corresponds to the Cartan case.

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What is happening ?

“break the symmetry of the system enhances the Lie algebra”

◮ dimM = 8 ◮ By symmetry, the dimension of the Lie

algebra is ≤ 3 and actually = 3 (Theorem [Alouges et al.]).

◮ By symmetry, the dimension of the Lie

algebra is ≤ 5 and actually = 5 generically (Theorem [Alouges and L.G.]).

◮ The dimension of the Lie algebra is

generically = 7 (Theorem [Gérard-Varet, L.G.]).

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Conclusion

– “The more complex the domain around the swimmer is, the more controllable is the swimmer”. – In real life, all micro-organisms are controllable. – To explain the biologist’s observations, we have to interested in an optimal control problem. – Controllability results in Navier-Stokes ? (Muriel Boulakia, Sergio Guerrero, Jean-Pierre Raymond, ...)

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Publications

◮ Journaux à comité de lecture [1] Enhanced controllability of low Reynolds number swimmers in the presence of a wall. Avec F. Alouges. Acta Applicandae Mathematicae, 2013. [2] Self-propulsion of slender micro-swimmers by curvature control : N-link

• swimmers. Avec F. Alouges, A. DeSimone, M. Zoppello. International

Journal of Non-Linear Mechanics, 2013. ◮ Conférences internationales [3] Controllability and optimal strokes for N-link micro-swimmer. Avec P. Martinon, M. Zoppello. 52th IEEE Conf. on Decision and Control (2013). ◮ Articles soumis à un journal à comité de lecture [4] Rough wall on micro-swimmers. Avec D. Gérard-Varet. Soumis. [5] Optimal Strokes for driftless swimmers : A general geometric approach. Avec T. Chambrion, A. Munnier. Soumis. ◮ Journal à comité de lecture national [6] Comment les spermatozoïdes nagent-ils ?. Accepté pour publication par la maison d’édition Nouveau monde.

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