1A Experimental observations

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Multiscale modelling of complex fluids: a mathematical initiation.

Tony Leli `evre

CERMICS, Ecole des Ponts et projet MicMac, INRIA.

http://cermics.enpc.fr/∼lelievre

Reference (with Matlab programs, see Section 5):

C. Le Bris, TL, Multiscale modelling of complex fluids: A mathematical initiation,

in Multiscale Modeling and Simulation in Science Series, B. Engquist, P. L ¨otstedt, O. Runborg, eds., LNCSE 66, Springer, p. 49-138, (2009)

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Outline

1 Modeling

1A Experimental observations 1B Multiscale modeling

1C Microscopic models for polymer chains 1D Micro-macro models for polymeric fluids 1E Conclusion and discussion

2 Mathematics analysis 2A Generalities

2B Some existence results 2C Long-time behaviour

2D Free-energy for macro models

3 Numerical methods and numerical analysis 3A Generalities

3B Convergence of the CONNFFESSIT method

3C Dependency of the Brownian on the space variable 3D Free-energy dissipative schemes for macro models

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Outline

1 Modeling

1C Microscopic models for polymer chains 1D Micro-macro models for polymeric fluids

1E Conclusion and discussion

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1A Experimental observations

We are interesting in complex fluids, whose non-Newtonian behaviour is due to some microstructures.

Cover page of Science, may 1994 Journal of Statistical Physics, 29 (1982) 813-848

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1A Experimental observations

More precisely, we study the case when the microstructures are:

1. very numerous (statistical mechanics), 2. small and light (Brownian effects),

3. within a Newtonian solvent.

This is not the case for all non-Newtonian fluids with microstructures (granular materials).

A prototypical example is dilute solution of polymers.

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1A Experimental observations

Some examples of complex fluids:

• food industry: mayonnaise, egg white, jellies

• materials industry: plastic (especially during forming), polymeric fluids

• biology-medicine: blood, synovial liquid

• civil engineering: fresh concrete, paints

• environment: snow, muds, lava

• cosmetics: shaving cream, toothpaste, nail polish

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1A Experimental observations

Shearing experiments in a rheometer:

Rint

u = u(y) e_x V Rext

L x y

h

Ω, C

planar Couette flow

˙

γ = ^V_L = _R ^R^int^Ω

ext−Rint

(Ω, C) ⇐⇒ ( ˙γ, τ)

C

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1A Experimental observations

At stationary state:

γ

γ τ=η

τ Newtonian fluid

γ

τ Shear−thinning fluid

γ

τ

Shear−thickening fluid γ

σ_Y τ

Yield−stress fluid

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1A Experimental observations

A simple dynamics effect: the velocity overshoot for the start-up of shear flow.

Outflow

U=1

U=0 y

L 0

0 1

Re=0.1 Epsilon=0.9, T=1.

u

0 1

L 0

1 0

Re=0.1 Epsilon=0.9, We=0.5, T=1.

u

0 1

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1A Experimental observations

These are two typical non-Newtonian effects : the open syphon effect and the rod climbing effect.

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Outline

1 Modeling

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1B Multiscale modeling

Momentum equations (incompressible fluid):

ρ (∂_t + u.∇) u = −∇p + div(σ) + f_ext, div(u) = 0.

Newtonian fluids (Navier-Stokes equations):

σ = η

∇u + (∇u)^T ,

Non-Newtonian fluids:

σ = η

∇u + (∇u)^T

+ τ,

τ depends on the history of the deformation.

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1B Multiscale modeling

Methods of the integral

using the decreasing

Phenomenological modelling

Stochastic models Microscopic models (kinetic theory)

Macroscopic simulations

Integral models Differential models

Finite Element Discretization

memory function

FEM (fluid)

Monte Carlo (polymers)

Micro−macro simulations using principles of fluid mechanics

Differential models : ^D_Dt^τ = f(τ , ∇u),

Integral models : τ = R _t

m(t − t^′)S (t^′) dt^′.

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1B Multiscale modeling

?

Taille du système (nombre d’atomes)

Temps caractéristiques

10² 10⁶ 10²³

1 s

10 s⁻¹²

−9

10 s

moléculaire Echelle

macroscopique Echelle

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1B Multiscale modeling

à l’équilibre thermodynamique local Moyennes d’ensembles

1 s

10 s⁻¹²

−9

10 s

Systèmes "simples" ou "peu complexes"

moléculaire Echelle

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1B Multiscale modeling

Modèles hiérarchiques une multiplicité d’échelles

caractéristiques bien distinctes

10² 10⁶ 10²³

1 s

10 s⁻¹²

−9

10 s

moléculaire

Echelles

mésoscopiques

Echelle

Systèmes complexes présentant

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1B Multiscale modeling

Phénomènes hors d’équilibre

Multiplicité d’échelles imbriquées 1 s

10 s⁻¹²

−9

10 s

moléculaire

Echelle

Systèmes complexes

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1B Multiscale modeling

Phénomènes hors d’équilibre Modèles multi−échelles

pour traiter les couplages "forts"

Multiplicité d’échelles imbriquées

10² 10⁶ 10²³

1 s

10 s⁻¹²

−9

10 s

moléculaire

Echelle

Systèmes complexes

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Outline

1 Modeling

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1C Microscopic models for polymer chains

Micro-macro models require a microscopic model couped to a macroscopic description: difficulties wrt timescales and length scales.

The coupling requires some concepts from statistical mechanics: compute macroscopic quantities (stress, reaction rates, diffusion constants) from microscopic descriptions.

One needs a coarse description of the

microstructures. How to model a microstructure

evolving in a solvent ? Answer : molecular dynamics and the Langevin equations.

In Section 1C, we assume that the velocity field of the solvent is given (and is zero in a first stage).

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1C Microscopic models for polymer chains

Microscopic model: N particles (atoms, groups of

atoms) with positions (q₁, ..., q_N) = q ∈ R^3N, interacting through a potential V (q₁, ...,q_N). Typically,

V (q₁, ..., q_N) = X

i<j

V_paire(q_i, q_j)+ X

i<j<k

V_triplet(q_i, q_j, q_k)+. . .

For a polymer chain, for example, a fine description would be to model the conformation by the position of the carbon atoms (backbone atoms). The potential V

typically includes some terms function of the dihedral angles along the backbone.

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1C Microscopic models for polymer chains

Molecular dynamics (solvent at rest): Langevin dynamics

( dQ_t = M⁻¹P_t dt,

dP_t = −∇V (Q_t) dt−ζM⁻¹P_t dt + p

2ζβ⁻¹dW_t,

where P_t is the momentum, M is the mass tensor, ζ is a friction coefficient and β⁻¹ = kT.

Origin of the Langevin dynamics: description of a colloidal particle in a liquid (Brown).

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1C Microscopic models for polymer chains

The Langevin dynamics is a thermostated Newton dynamics: The fluctuation (^p2ζβ⁻¹dW_t) dissipation (₋ζM⁻¹P_t dt) terms are such that the

Boltzmann-Gibbs measure is left invariant:

ν(dp, dq) = Z⁻¹ exp

−β

p^TM⁻¹p

2 + V (q)

dpdq.

To explain this in a simpler context, let us make the following simplification M/ζ → 0:

dQ_t = −∇V (Q_t)ζ⁻¹ dt + p

2ζ⁻¹β⁻¹dW_t.

This dynamics leaves invariant the Boltzmann-Gibbs

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1C Micro models: some probabilistic background

The Stochastic Differential Equation

dQ_t = −∇V (Q_t)ζ⁻¹ dt + p

2ζ⁻¹β⁻¹dW_t

is discretized by the Euler scheme (with time step ∆t):

Q_n+1 − Q_n = −∇V (Q_n)ζ⁻¹ ∆t + p

2ζ⁻¹β⁻¹∆tG_n

where (Gⁱ_n)₁_≤_ile3,n_≥₀ are i.i.d. Gaussian random

variables with zero mean and variance one. Indeed

(W_(n+1)∆t − W_n∆t)_n_≥₀ =^L √

∆t(G_n)_n_≥₀.

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1C Micro models: some probabilistic background

The Itô formula. Let φ be a smooth test function. Then

dφ(Q_t) = ∇φ(Q_t) · dQ_t+∆φ(Q_t)ζ⁻¹β⁻¹ dt.

Proof (dimension 1):

dX_t = b(X_t) dt + σ(X_t) dW_t X_n+1 − X_n = b(X_n)∆t + σ(X_n)√

∆tG_n

and thus

φ(X_n+1) = φ

X_n + b(X_n)∆t + σ(X_n)√

∆tG_n

= φ(X_n) + φ^′(X_n)(b(X_n)∆t + σ(X_n)√

∆tG_n)

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1C Micro models: some probabilistic background

Then, summing over n and in the limit ∆t → 0,

φ(X_t) = φ(X₀) +

Z _t

0

φ^′(X_s)(b(X_s)ds + σ(X_s) dW_s) +1

2

Z _t

0

σ²(X_s)φ^′′(X_s) ds,

= φ(X₀) +

Z t 0

φ^′(X_s)dX_s+1 2

Z t 0

σ²(X_s)φ^′′(X_s) ds,

which is exactly

dφ(X_t) = φ^′(X_t)dX_t + 1

2σ²(X_t)φ^′′(X_t) dt.

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1C Micro models: some probabilistic background

The Fokker-Planck equation. At fixed time t, ^Q_t has a density ψ(t, q). The function ψ satisfies the PDE:

ζ∂_tψ = div(∇V ψ + β⁻¹∇ψ).

Proof (dimension 1):

dX_t = b(X_t) dt + σ(X_t) dW_t,

and we show that X_t =^L ψ(t, x) dx with

∂_tψ = ∂_x (−bψ + ∂_x(σψ)) .

We recall the Itô formula:

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1C Micro models: some probabilistic background

By definition of ψ, IE(φ(X_t)) = R

φ(x)ψ(t, x) dx. Thus, we have

Z

φψ(t, ·) = Z

φψ(0, ·)+

Z _t

0

Z

φ^′bψ(s, ·)ds+1 2

Z _t

0

Z

σ²φ^′′ψ(s, ·)ds.

We have used the fact that

IE

Z _t

0

φ^′(X_s)dX_s = IE

Z _t

0

φ^′(X_s)b(X_s) ds + IE

Z _t

0

φ^′(X_s)σ(X_s) dW_s

=

Z _t

0

IE(φ^′(X_s)b(X_s)) ds

since

IE R t

0 φ^′(X_s)σ(X_s) dW_s ≃ IE Pn

k=0 φ^′(X_k)σ(X_k)√

∆tG_k = 0.

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1C Micro models: some probabilistic background

Thus the Boltzmann-Gibbs measure

µ(dq) = Z⁻¹ exp(−βV (q)) dq

is invariant for the dynamics

dQ_t = −∇V (Q_t)ζ⁻¹ dt + p

2ζ⁻¹β⁻¹dW_t.

Proof: We know that Q_t has a density ψ which satisfies:

ζ∂_tψ = div(∇V ψ + β⁻¹∇ψ).

If ψ(0, ·) = exp(−βV ), then _∀t ≥ 0, ψ(t, ·) = exp(−βV ). A similar derivation can be done for the Langevin

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1C Microscopic models for polymer chains

Back to polymers. Which description ? The fine

description is not suitable for micro-macro coupling

(computer cost, time scale). We need to coarse-grain.

Idea : consider blobs (1 blob _≃ 20 CH₂ groups).

The basic model (the dumbbell model): only two blobs.

The conformation is given by the “end-to-end vector”.

1 2

3

n

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1C Microscopic models for polymer chains

Coarse-graining at equilibrium: use the image of the Boltzmann-Gibbs measure by the end-to-end vector mapping (“collective variable”):

ξ :

( R^3N → R³

q = (q₁, . . . , q_N) 7→ x = q_N − q₁

namely:

ξ ∗ Z⁻¹ exp(−βV (q)) dq

= exp(−βΠ(x)) dx.

Thus

−1 Z

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1C Microscopic models for polymer chains

Typically, two forces ^F = ∇Π are used:

F(X) = HX Hookean dumbbell, F(X) = HX

1 − kXk²/(bkT /H) FENE dumbbell,

(FENE = Finite Extensible Nonlinear Elastic).

Notice that this effective potential Π (“free energy”) is correct wrt statistical properties at equilibrium:

R φ(x) exp(−βΠ(x)) dx = Z⁻¹ R

φ(ξ(q)) exp(−βV (q)) dq. We are now in position to write the basic model (the Rouse model).

References: R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Hassager, Dynamic of Polymeric Liquids, Wiley / M. Doi, S.F. Edwards, The theory of polymer dynamics, Oxford Science

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1C Microscopic models for polymer chains

Forces on bead i (i = 1 or 2) of coordinate vector Xⁱ_t in a velocity field u(t, x) of the solvent (Langevin equation with negligible mass):

• Drag force:

−ζ

dXⁱ_t

dt − u(t, Xⁱ_t)

,

• Entropic force between beads 1 and 2

(X = X² − X¹

):

F(X) = HX Hookean dumbbell, F(X) = HX

2 FENE dumbbell,

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1C Microscopic models for polymer chains

• “Brownian force”: Fⁱ_b(t) such that

Z _t

0

Fⁱ_b(s) ds = p

2kT ζ Bⁱ_t

with Bⁱ_t a Brownian motion.

We introduce the end-to-end vector X_t = X²_t − X¹_t

and the position of the center of mass R_t = ¹₂ X¹_t + X²_t

. We have:

( dX¹_t = u(t, X¹_t ) dt + ζ⁻¹F(X_t) dt + p

2kT ζ⁻¹dB¹_t dX²_t = u(t, X²_t ) dt − ζ⁻¹F(X_t) dt + p

2kT ζ⁻¹dB²_t

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1C Microscopic models for polymer chains

By linear combinations of the two Langevin equations on X¹ and X², one obtains:











dX_t = u(t, X²_t ) − u(t, X¹_t)

dt − 2

ζ F(X_t) dt + 2

skT

ζ dW_t¹, dR_t = 1

2 u(t, X¹_t ) + u(t, X²_t )

dt +

skT

ζ dW²_t ,

where W¹_t = ^√¹

2 B²_t − B¹_t

and W ²_t = ^√¹

2 B¹_t + B²_t

. Approximations:

• u(t, Xⁱ_t) ≃ u(t, R_t) + ∇u(t, R_t)(Xⁱ_t − R_t),

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1C Microscopic models for polymer chains

We finally get







dX_t = ∇u(t, R_t)X_t dt − 2

ζ F(X_t) dt +

s4kT

ζ dW_t, dR_t = u(t, R_t) dt.

Eulerian version:

dX_t(x) + u(t, x).∇X_t(x) dt =

∇u(t, x)X_t(x) dt − 2

ζ F(X_t(x)) dt +

s4kT

ζ dW_t.

X_t(x) is a function of time t, position x, and probability variable _ω.

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1C Microscopic models for polymer chains

Discussion of the modelling (1/2).

Discussion of the coarse-graining procedure:

• The construction of Π has been done for zero

velocity field (u = 0). How do the two operations :

u 6= 0 and “coarse-graining” commute ?

• Imagine u = 0. The dynamics

dX_t = −2

ζ F(X_t) dt +

s4kT

ζ dW_t

is certainly correct wrt the sampled measure (exp(−βΠ)). But what can be said about the

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1C Microscopic models for polymer chains

Discussion of the modelling (2/2).

Discussion of the approximations:

• The expansion used on the velocity requires some regularity on u: the term _∇u leads to some

mathematical difficulties in the mathematical analysis.

• If the noise on ^R_t is not neglected, a diffusion term in space (x-variable) in the Fokker-Planck equation gives more regularity.

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1C Microscopic models for polymer chains

We have presented a suitable model for dilute solution of polymers.

Similar descriptions (kinetic theory) have been used to model:

• rod-like polymers and liquid crystals (Onsager, Maier-Saupe),

• polymer melts (de Gennes, Doi-Edwards),

• concentrated suspensions (Hébraud-Lequeux),

• blood (Owens).

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Outline

1 Modeling

(41)

1D Micro-macro models for polymeric fluids

To close the system, an expression of the stress

tensor τ in terms of the polymer chain configuration is needed. This is the Kramers expression (assuming

homogeneous system):

n

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1D Micro-macro models for polymeric fluids

How to derive this formula ? One approach is to use the principle of virtual work. Another idea is to go back to the definition of stress:

τ n dS = IE sgn(X_t · n)F(X_t)1_{_X_t intersects plane_}

.

Since the system is assumed to be homogeneous,

given ^X_t, the probability that ^X_t intersects the plane is

N_p ^dS^|^X_V^t^·ⁿ^|.

n

X_t

|X · n|

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1D Micro-macro models for polymeric fluids

Thus we have:

τ n dS = IE sgn(X_t · n)F(X_t)1_{_X_t intersects plane_}

= IE (sgn(X_t · n)F(X_t)IP(X_t intersects plane_|X_t))

= n_pIE (sgn(X_t · n)F(X_t)|X_t · n|) dS

= n_pIE (X_t ⊗ F(X_t)) n dS,

where n_p = N_p/V .

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1D Micro-macro models for polymeric fluids

This is the complete coupled system:











ρ (∂_t + u.∇) u = −∇p + η∆u + div(τ ) + f_ext, div(u) = 0,

τ = n_p

− kT I + IE (X_t ⊗ F(X_t)) , dX_t + u.∇^xX_t dt =

∇uX_t − ²_ζ F(X_t)

dt + q

4kT

ζ dW_t.

The S(P)DE is posed at each macroscopic point x. The random process X_t is space-dependent: X_t(x).

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1D Micro-macro models for polymeric fluids

One can replace the SDE by the Fokker-Planck equation, which rules the evolution of the density probability function ψ(t, x, X) of X_t(x):

∂ψ

∂t + u · ∇^xψ = − div_X

(∇u X − 2

ζ F(X))ψ

+ 2kT

ζ ∆_Xψ,

and then:

τ(t, x) = −n_p k TI + n_p Z

R^d

(X ⊗ F(X)) ψ(t, x, X) dX.

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1D Micro-macro models for polymeric fluids

Once non-dimensionalized, we obtain:











Re (∂_t + u · ∇) u = −∇p + (1 − ǫ)∆u + div(τ ) + f_ext,

div(u) = 0^,

τ = _We^ǫ (µIE(X_t ⊗ F(X_t)) − I)^,

dX_t + u · ∇^xX_t dt = ∇u · X_t − _2We¹ F(X_t)

dt + ^√_We¹ _µdW_t,

with the following non-dimensional numbers:

Re = ρU L

η ^, We = λU

L ^, ǫ = η_p

η ^, µ = L²H k_bT ^,

and λ = _4H^ζ : a relaxation time of the polymers,

η_p = n_pkT λ: the viscosity associated to the polymers,

U and L: characteristic velocity and length. Usually, L

is chosen so that µ = 1.

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1D Micro-macro models for polymeric fluids

Link with macroscopic models. the Hookean dumbbell model is equivalent to the Oldroyd-B model: if

F(X) = X, τ satisfies:

∂τ

∂t + u.∇τ = ∇uτ + τ(∇u)^T + ǫ

We (∇u + (∇u)^T) − 1

We τ .

There is no macroscopic equivalent to the FENE model. However, using the closure approximation

F(X) = HX

1 − kXk²/(bkT /H) ≃ HX

1 − IEkXk²/(bkT /H)

one ends up with the FENE-P model.

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1D Micro-macro models for polymeric fluids

The FENE-P model:







 λ

∂τ

∂t + u · ∇τ − ∇uτ − τ(∇u)^T

+ Z(tr(τ))τ

−λ

τ + η_p

λ I ∂

∂t + u · ∇

ln (Z(tr(τ )))

= η_p(∇u + (∇u)^T),

with

Z(tr(τ)) = 1 + d b

1 + λtr(τ) d η_p

,

where d is the dimension.

Remark: The derivative ^∂_∂t^τ + u · ∇τ − ∇uτ − τ (∇u)^T is called the Upper Convected derivative.

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Outline

1 Modeling

(50)

1E Conclusion and discussion

This system coupling a PDE and a SDE can be solved by adapted numerical methods. The interests of this micro-macro approach are:

• Kinetic modelling is reliable and based on some clear assumptions (macroscopic models usually derive from kinetic models (e.g. Oldroyd B), sometimes via closure approximations, but some

microscopic models have no macroscopic equivalent (e.g. FENE)),

• It enables numerical explorations of the link

between microscopic properties and macroscopic behaviour,

• The parameters of these models have a physical meaning and can be evaluated,

• It seems that the numerical methods based on this

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1E Conclusion and discussion

However, micro-macro approaches are not the solution:

• One of the main difficulties for the computation of viscoelastic fluid is the High Weissenberg Number Problem (HWNP). This problem is still present in micro-macro models (highly refined meshes would be needed ?).

• The computational cost is very high. Discretization of the Fokker-Planck equation rather than the set of SDEs may help, but this is restrained to

low-dimensional space for the microscopic variables.

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1E Conclusion and discussion

Macro-macro approach:









Du

Dt = F(τ_p, u), Dτ_p

Dt = G(τ_p, u).

Multiscale, or micro-macro approach:











Du

Dt = F(τ_p, u),

τ_p = average over Σ, DΣ

Dt = G^µ(Σ, u).

(53)

1E Conclusion and discussion

Pros and cons for the macro-macro and micro-macro approaches:

MACRO MICRO-MACRO

modelling capabilities low high

current utilization industry laboratories

discretization discretization by Monte Carlo of Fokker-Planck

computational cost low high moderate

computational bottleneck HWNP variance, HWNP dimension, HWNP

(54)

Outline

2 Mathematics analysis 2A Generalities

2B Some existence results 2C Long-time behaviour

2D Free-energy for macro models

(55)

2A Generalities

The main difficulties for mathematical analysis:

transport and (nonlinear) coupling.











Re

∂u

∂t + u · ∇u

= (1 − ǫ)∆u − ∇p + div(τ) ,

div(u) = 0 ^, τ = ǫ

We (IE(X ⊗ F(X)) − I) ,

dX + u · ∇Xdt =

∇uX − 1

2We F(X)

dt + 1

√We dW_t.

Similar difficulties with macro models (Oldroyd-B):

∂τ ǫ 1

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2A Generalities

The state-of-the-art mathematical well-posedness analysis is local-in-time existence and uniqueness results, both for macro-macro and micro-macro models.

One exception (P.L Lions, N. Masmoudi) concerns models with co-rotational derivatives rather than upper-convected derivatives, for which global-in-time existence results have been obtained. It consists in replacing

∂τ

∂t + u.∇τ − ∇uτ − τ(∇u)^T

by

∂τ

∂t + u.∇τ − W(u)τ − τW(u)^T,

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2A Generalities

These better results come from additional a priori estimates based on the fact that

W(u)τ + τ W(u)^T

: τ = 0.

For micro-macro models, it consists in using the SDE:

dX_t+u·∇X_tdt =

∇u − ∇u^T

2 X_t − 1

2We F(X_t)

dt+ 1

√We dW_t.

However, these models are not considered as good models. For example, ψ ∝ exp(−Π) is a stationary solution to the Fokker Planck equation whatever ^u.

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2A Generalities

Well-posedness results for micro-macro models:

• The uncoupled problem: SDE or FP.

• SDE in the FENE case (B. Jourdain, TL: OK for b ≥ 2),

• the case of non smooth velocity field, transport term in the SDE or FP (C. Le Bris, P.L Lions).

• The coupled problem: PDE + SDE or PDE + FP.

• PDE+SDE: shear flow for Hookean or FENE

(C. Le Bris, B. Jourdain, TL / W. E, P. Zhang),

• PDE+FP: FENE case (M. Renardy / J.W. Barrett, C. Schwab, E. Süli: (mollification) OK for b ≥ 10 / N. Masmoudi, P.L. Lions).

Another interesting (not only) theoretical issue is the long-time behaviour.

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2A Generalities

Two simplifications: (i) the case of a plane shear flow.

00000 00000 00000 00000 11111 11111 11111 11111

y

velocity profile

Inflow Outflow

u

We keep the coupling, but we get rid of the transport

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2A Generalities

The equations in this case read (0 ≤ t ≤ T, y ∈ O = (0, 1)):











∂_tu(t, y) − ∂_yyu(t, y) = ∂_yτ(t, y) + f_ext(t, y),

τ(t, y) = IE (X_t(y) F₂(X_t(y), Y_t(y))) = IE (Y_t(y) F₁(X_t(y), Y_t(y))) dX_t(y) = −¹₂F₁(X_t(y), Y_t(y)) + ∂_yu(t, y)Y_t(y)

dt + dV_t, dY_t(y) = −¹₂F₂(X_t(y), Y_t(y))

dt + dW_t,

• F(X_t) = X_t = (X_t, Y_t) (Hookean), or

• F(X_t) = ^X^t

1−^k^X_b^t^k² =

Xt

1−^X^t²^+Y_b ^t² , ^Y^t

1−^X^t²^+Y_b ^t²

(FENE),

where u(t, x, y) = (u(t, y), 0), τ =

∗ τ τ ∗

, and F(X ) = (F (X , Y ), F (X , Y )).