Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Hydro-kinetic approach to

multi-fluid flows

Sergio ChibbaroInstitut Jean Le Rond D’alembert

Université Pierre et Marie Curie, Paris, France

Institut Jean Le Rond D’Alembert

With

R. Benzi, L. Biferale, M. Sbragaglia (University of Rome, Italy);

M. Bernaschi, F. Toschi and S. Succi (IAC, CNR, Rome Italy)

K. Binder, D. Dimitrov and A. Miltchev (University of Mainz)

sergio.chibbaro@upmc.fr

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Moving Contact Lines….Why a Challenge ?

1) Multiscale problem (inner/outer) (from molecular to millimetric)

2) Singular problem (inner) (Divergence of viscous stress)

Huh & Scriven (1971), Dussan et al. (1974), Voinov (1979), Cox (1986), Quéré (1991), Podgorski (2001), Eggers (2004), Snoeijer (2007)

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MESOSCOPIC APPROACH: Lattice Boltzmann Equations

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

fluid 1

fluid 2

Basic Equations

Bulk

Interface

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Ideal discontinuous interface: drawbacksa)Interface close to the critical point (fat)

b)Coalescence of two air bubbles (singular at the merging)

c)Nucleation of a second phase

d)Moving boundary conditionsVan der Waals approach to diffuse

interface.

Korteweg stress

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

1)Take a diffuse interface model with a given interface width

2)Develop an algorithm to simulate it

3)If macroscopic/mesoscopic physics-> check the robustness at changing

4)If nanoscopic physics-> check the consistency of taking statistical equilibrium

Recipes

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

MACROSCOPIC

NAVIER-STOKES

KINETIC

(LATTICE BOLTZMANN)

MICROSCOPIC

MOLECULAR DYNAMICS

Chapman-Enskog

(particle-particle interactions)

(Particles p.d.f.)

(continuum description)

The 10 orders of magnitude hierarchy

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Streaming collide

Lattice Boltzmann for SINGLE-phase flows

Density

Momentum

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Streaming collide

Intermolecular forces Pseudo potential

Shan and Chen (1993,1994)

Non ideal gas effects Surface tension

Lattice Boltzmann for MULTI-phase flows

NON-IDEAL PRESSURE TENSORKorteweg stress

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Modelling wetting properties in lattice Boltzmann

Mechanical equilibrium

Liquid (l)

gas (g)

solid (s)

Boundary condition

Benzi R., Biferale L.,Sbragaglia M., Succi S. and Toschi F Phys. Rev. E 74, 021509 (2006)..

θ−π

M. Sbragaglia, R. Benzi, L. Biferale, S. Succi and F. ToschiPhys. Rev. Lett. 97, 204503 (2006).

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Periodic boundary conditions

H

Liquid

Gas

z

ρl

ρgvf

ρw

Effective channel

Capillary Filling

AsymptoticWashburn’s Law

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

EFFECTS OF EVAPORATION/CONDENSATION

EFFECST OF FINITE INTERFACE WIDTH

velocity profiles

liquidgas

Capillary Filling

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

EFFECTS OF EVAPORATION/CONDENSATION

EFFECTS OF FINITE DENSITY RATIO

velocity profiles

liquid gas

Capillary Filling

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

INLET dynamics: vena contracta

effects.

Capillary Filling

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Conclusions 1.

Need to reach high density ratioNeed to reach high separation of scalesImportant effects from the inlet.

a)Smooth walls: the role of density ratio and of interface width.

European journal of physics ST 171, 237-243 (2009) arXiv : 0801.4223Lattice Boltzmann simulations of capillary filling: finite vapour density effectsAuthors: F. Diotallevi, L.Biferale, S. Chibbaro, G. Pontrelli, F. Toschi, S. Succi

European journal of physics ST 166, 111 (2009) arXiv:0707.0945Title: Capillary filling using Lattice Boltzmann Equations: the case of multi-phase flowsAuthors: F. Diotallevi, L. Biferale, S. Chibbaro, A. Lamura, G. Pontrelli, M. Sbragaglia, S. Succi, F. Toschi

Capillary Filling

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

b) Perfectly wetting walls: the dynamics of precursor film.

Capillary Filling

Lennard-Jones Molecular Dynamics

ULJ (r) =4ε σ / r( )12 − σ / r( )6⎡⎣

⎤⎦

ε =1.4 σ =1.0

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

interface profiles

de Gennes RMP 1985

Capillary Filling

Contact angle

meniscus

Precursor

1.Washburn’s Law still holds at nanoscopic scales2.Wetting can be quantitatively described by hydro-kinetic methods like LB

S. Chibbaro, L. Biferale, F. Diotallevi, S. Succi, K. Binder, D. Dimitrov,  A. Milchev Europhysics Letters, 84 (2008) 44003,

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Obstacle and roughness: Concus-Finn/Gibbs criterionCapillary Filling

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40°

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50°

LBCF does not hold

Fluctuations are important at the critical point at these

scales

MD

S. Chibbaro, L. Biferale, K. Binder, D. Dimitrov,  F. Diotallevi,  A. Milchev, S. Succi JSAT: Theory and expermients (2009) P06007

S. Chibbaro, E. Costa, D. I. Dimitrov, F. Diotallevi, A. Milchev, D. Palmieri, S. SucciLangmuir, 2009, 25 (21), pp 12653–12660

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

State of the art and point of view

Microscopic simulations : molecular dynamics and spin glass models.

MD is computational demanding (available number of particles, short time-scale)

Spin glass models are “reference state” to achieve theoretical understanding

no microscopic model aimed at investigating the “hydrodynamic-like” behavior of soft glasses.

New approach kinetic/mesoscopic dynamics of soft-glassy materials

Diffuse interface/Lattice Boltzmann Equation for a two fluid system + “frustration”

Model scale

Micro-emulsions

A B

Two fluids : A and B

repulsive interactions A-B

“non ideal” (Shan-Chen) self attraction interaction (A-A, B-B)

σAB surface tension between the two fluids

• No frustration

• Phase separation between A and B (spinodal decomposition)

Standard multi-component model

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

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Next-to-next neighbour

BA

Soft-glassy materialsIntroducing frustration : Surfactant effect

Under Shear Non ergodic behaviour and structural arrest

Emergent Herschel-Bulkley rheologyR. Benzi, S. Chibbaro & S. Succi, Phys. Rev. Lett. 102, 026002 (2009)R. Benzi, M. Sbragaglia, S. Succi, M. Bernaschi & S.Chibbaro, Jour. Chem. Phys. 131, 104903 (2009)

Institut Jean Le Rond D’Alembertsergio.chibbaro@upmc.fr

Aging leads the system to be trapped in a deep valleyStrong enough drive can force to leave the valley

A small extra drive can keep the system above the treshold

Ludovic Berthier, Jean-Louis Barrat and Jorge Kurchan (2000)

Decreasing shear

Yield-fluid

τw = 2 ×105

Aging under shear

Micro-emulsions

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Aging properties in glassesMolecular dynamics simulations of binary mixture

N = number of different initial conditions

In our case:

U0 =0.02

τw = 3 ×105 ; U0 = 0.03

1.Validation of diffuse-interface LB method•Need to reach high density ratio

•Need to reach high separation of scales

•Important effects from the inlet.

2. Quantitative description (MD/LB) of capillary filling at nanoscopic scales

•Dynamic law of precursors

•Washburn’s law holds

3. Study of geometrical roughness•Analysis of the effect of roughness on front dynamics

•Concus-Finn criterion not valid

4. Development of a new kinetic approach to flowing soft-glasses

CONCLUSIONS