Concentrated non-Brownian suspensions in viscous fluids.

Numerical simulation results,

a “granular” point of view J.-N. Roux, with F. Chevoir, F. Lahmar, P.-E. Peyneau, S.

Khamseh

Laboratoire Navier, Université Paris-Est, FranceGranular materials : • quasistatic response : gradually apply shear stress or impose small shear rate on isotropic equilibrated state• steady shear flow with inertial effectsExtension to dense suspensions :Large hydrodynamic forces in narrow gaps between grains, role of contactsApproaches similar to dry granular case

Granular materials : initial density and critical stateGranular materials : initial density and critical state

Under large strain, critical state independent of initial state, characterized by a ‘’flow structure’’  (density + nb of contacts, anisotropy…)

Mon

od

isperse

sph

ere

asse

mb

ly, frictio

n co

effi

cien

t = 0

.3

Severa

l sam

ple

s with

40

00

bead

s, pre

pare

d a

t diff

ere

nt so

lid fra

ction

Triaxial test

Similar response in simple shear test !

Critical state of dry granular materials, 2D and3D results

Internal friction coefficient(actually somewhat different between shear tests and other load directions, e.g. triaxial, see e.g., Peyneau & Roux, PRE 2008)

Critical solid fraction

c = RCP for = 0

Compilation of simulation literature, in Lemaître, Roux & Chevoir, Rheologica Acta 2009

c and c do not depend on contact stiffness if large enough

Quasistatic rheology :

•interest of critical state concept (flow structure) c minimum solid fraction in flow

•Friction coefficient in contacts determines c and *c. Small rolling friction also quite influential

• using contact law and applied pressure P, define stiffness number (such that deflection is prop. to -1), assess approach to rigid grain limit

Inertial flows :

•Study shear flow under controlled normal stress rather than fixed density : non-singular quasistatic limit

• use internal friction * as an alternative to viscosity

What we know from simulations of dry grains (I)

23

(linear elasticity, 3D) (Hertz contacts, 3D) diameter ; NK Ea

aP P

Simulation of steady uniform shear flow

Fixed shear strain rate

Normal stress is imposed

22P

Material state in shear flow ruled by one dimensionless parameter, the inertial number

What we know from studies on dry grains (II) :

Inertial number and constitutive relations

2 3 D D

m mI I

P aP

Generalization of critical state to I-dependent states with inertial effects Useful constitutive law, applied to different geometries (Pouliquen, Jop,

Forterre…)

Monodisperse spheres, no intergranular

friction

(Peyneau & Roux, Phys Rev E 2008)

RCP

Steady shear flow of dry, frictionless beads at low I number : normal stress-controlled vs.

volume controlled(P.-E. Peyneau)

Same behavior, very large stress fluctuations if is imposed (4000 grains)

Ratio 12/22 = * expresses material rheology

• in simulation literature : nothing for > 0.6 (spheres), ad-hoc repulsive forces used to push grains apart, very few published results with N >1000…

• sharp contrast with simulations of dry grains!

• lubrication singularities believed to lead to some dynamic jamming phenomenon below RCP (related to possible origin of shear thickening)

Ball and Melrose, 1995-2004: no steady state (!?)

• Here : control normal stress rather than density, control lubrication cutoff and contact interaction stiffness

Simulations of dense suspensions

Experiments on dense suspensions

• Attention paid to possible density inhomogeneities local measurements

• Boyer et al. control of particle pressure !

Simulation of very dense suspensions

Simplified modeling approach, fluid limited to near-neighbor gaps and pairwise lubrication interactions (cf. Melrose & Ball)

Lubrication  singularity cut off at short distance. Contact forces (or short–range repulsion due to polymer layer)

Stokes régime : contact, external and viscous forces balance

Questions

Divergence of at < RCP ? Sensitivity to repulsive forces ? To alone ? Effective viscosity, non-Newtonian effects, normal stresses…

Lubrication and hydrodynamic resistance matrix

NNvN VhF )(

h

RhN

2

2

3)(

Normal hydrodynamic force :

For 2 spheres of radius R, with perfect lubrication :

No contact within finite time !

Cut-off for narrow interstices h<hmin

(asperities), contact becomes possible

Without cutoff: both physically irrelevant and computationally untractable

dominant forces transmitted by a network of ‘quasi-contacts’

Choice of systems and parameters

D systems of identical spheres, diameter a :

tobeads

hmax/a = 0.1 or 0.3

hmin/a = 10-4 ; = 105

Vi from 0.1 down to 5.10-4

= 0 in solid contacts

+ alternative systems with repulsive forces, no lubrication cutoff

D systems of polydisperse disks, diameter d,

disks

3D lubrication

hmax/a = 0.5

hmin/a = 10-4 or 10-2 ; = 104

Vi down to 10-4 or 10-6

= 0.3 in solid contacts

0.7a d a

• assemble hydrodynamic resistance matrix (similar to stiffness matrix in elastic contact network)

• non-singular tangential coefficient

• add up ‘ordinary’ contact forces (elasticity + friction) to viscous hydrodynamic ones when grains touch (simple approximation)

cF

Model, computation method

0 extvc FFF

VFv W

vF

with

W

Wand Fc depend on grain positions

Solve extc FFV

W

Lees-Edwards boundary conditions

+ variable height,ensuring constant yy

Measurements in steady state :

Check for stationarity of measurements

Obtain error bar from ‘blocking’ technique

Request long enough stationary intervals

10 at least ...t Regression of fluctuations

Some technical aspects about simulations

N -1/2N -1/2

Constant volume and/or

constant shear stress conditions should produce

same system state in large N limit

Choice of time step, integration

t such that matrix and r.h.s. do not change `too much’…

( ( )) ( ) ( ( ))

( ) ( ) . ( )c extX t V t F X t F

X t t X t t V t

W

(1)

(1) (2) (1)

(1) (2)

( ( )) ( ) ( ( ))

( ( ) ( ) ) ( ) ( ( ) ( ) )

( ) ( ) . ( ) ( )2

c ext

c ext

X t V t F X t F

X t V t t V t F X t V t t F

tX t t X t V t V t

W

W

Euler (explicit) :

(error ~t2)

‘Trapezoidal’ rule : (error ~ t3)

A crucial test on the numerical integration of equations of motion

Relative difference between variation of h and integration of normal relative velocity in various interstices, 2 different numerical schemes : Euler (dotted lines), trapezoidal (continuous lines)

Keep it below 0.05 !

Control parameter for dense suspensions : viscous number Vi

PVi

P

aVi

D

D

3

2

aP

mI

P

mI

D

D

3

2Plays analogous role to inertia parameter

defined for dry grains

Vi = (decay time of h(t) in compressed layer within gap) / (shear time)

I = (acceleration time) / (shear time)

Acceleration (inertial) time replaced by a squeezing time in viscous layer

Cassar, Nicolas, Pouliquen, Phys. of Fluids 2005 (similar Vi, with drainage time)

Steady shear flow of lubricated beads at low Vi number : normal stress-controlled vs. volume

controlled

Vi = 10-3

Same behavior, large stress fluctuations if is imposed (1372 grains). Ratio 12/22 = * expresses material rheology

3D results : *and as functions of Vi

= 0 : difficultcase ! Approach to * = 0.1, =0.64 …

ViVi

identical spherical grains1372N

Back to more traditional (constant ) approach

Effective viscosity

*min, , ,

N

h af

a K

* *

12 22 Vi

From and

0 Vi one

gets:

* *

1

cVi

if* *

0 (not satisfied in our case !)

Shear rate effects ?

In rigid grain limit replace 4th argument by zero. No influence

of on effective viscosity 0

Exponent : 2.5 to 3 ?

Effective viscosity, as a fonction of solid fraction

(hc = hmax)

Polymer layer thicknesslb = 0.01 or 0.001(open/filled symbols )

Force F0 ratio F0/a2P (0.1, 1, 10) = (diamond, square, circle)

Influence of repulsive force

Adsorbed polymer layer, short-range repulsion, as

*

cVi Vi

cVi Vi

Ordered structurepolydispersity < 20%

34.10Vi

Shear-thinning within studied parameter range

Pair correlations in plane yz

5

40[ 1]bF l h

(Fredrickson & Pincus, 1991)

Vic ~ 3.10-2

Shear thinning with repulsive forces

2*

0

*

02

, , ,b

N

l af

a F

a ViFKa P

Shear thinning due to

change in reduced shear rate

Results correspond to

varying from 5.10-5 to 10-1

*

No shear thinning for smaller lb

Network of repulsive forces carry all shear stress as Vi decreases

Other ‘granular’ features: force distribution, coordination number…

Vi

xy

yy

Viscosity of random, isotropic suspensions

•Assume ideal hard sphere particle distribution at given

•Measure instantaneous shear viscosity

Comparison with Stokesian Dynamics results: encouraging agreement at large densities, although treatment of subdominant terms not entirely innocuous

2D disk model

3D frictionless spherical beads :

Analogy with dry granular flow

Difficulty to approach quasistatic limit

No singularity below RCP density, a steady-state can be reached

Importance of non-hydrodynamic interactions

Purely hydrodynamic model approached with stiff interactions

Effects of additional forces: 2-parameter space to be explored

Easier system, introduction of tangential forces, friction, faster approach to quasistatic limit

Show coincidence of quasistatic limits for dry grains and dense suspension

2D results : internal friction coefficient versus Vi

(analogous to function of I in dry inertialcase)

Viscous case hmin/a = 10-2 and hmin/a = 10-4 , =0,3

Dry inertial case (right) : =0.3 and =0

Coincidence I=0 / Vi=0

Viscous Dry, inertial

2D results : solid fraction versus Vi (in paste)(analogous to function of I in dry inertial case)

Viscous case hmin/a = 10-2 and hmin/a = 10-4 (896 grains), =0.3

Dry inertial case(right) : =0.3 and =0

Coincidence I=0 /Vi =0

ViscousDry,

inertial

* *c cI

* *c cVi 1 1

( )c

e Vi

1 1

c

eI

Constitutive laws : dry grains (2D) = 0 or 0.3

Constitutive laws, granular suspensions (2D), = 0.3

Same constant terms (quasistatic limit), different power laws

Effective viscosity, as a function of solid fraction

Divergence of effective viscosity as critical solid fraction c is approached

(once again) no accurate determination of exponent (2 ? 2.5 ?)

Little influence of roughness length scale

Importance of direct contact interactions

Pressure due to solid contact forces / total pressure

(Case hmin/a = 10-2)

Some conclusions on dense suspensions

interesting to study dense suspensions under controlled normal stress

importance of contact interactions : solid friction, asperities…) Contact or static forces dominate at low Vi

relevance of critical state as initially introduced in soil mechanics. Viscosity diverges as approaches c

With =0 in contacts, no jamming or viscosity divergence or any specific singularity below RCP, except in small systems

If is large enough, * independent of with elastic (-frictional) beads (but… Re ? Brownian effects ?)

introduction of repulsive interaction with force scale shear-thinning (or thickening)

Perspectives

bridge the gap between real (and difficult) hydrodynamic

calculations and ‘conceptual models’

Improve performance of numerical methods

Continuum fluid !(Keep separate treatment of lubrication singularities ?)