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, France

Granular materials :

• quasistatic response : gradually apply shear stress or impose small shear rate on isotropic equilibrated state • steady shear flow with inertial effects

Extension to dense suspensions :

Large hydrodynamic forces in narrow gaps between grains, role of contacts Approaches similar to dry granular case



Granular materials : initial density and critical state

nt solid fraction differe at red epa pr ads, 4000 be mples with l sa = 0.3

 ent coeffici embly, friction ass here e sp pers Monodis Severa Triaxial test

Similar response in simple shear test !

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

Critical state of dry granular materials, 2D and3D results

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



c and



c do not depend on contact stiffness if large enough

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

What we know from simulations of dry grains (I)

Quasistatic rheology : •interest of

critical state

concept (flow structure)  c minimum solid fraction in flow •Friction coefficient Small

rolling friction

 in contacts determines  c also quite influential and  * c . • using contact law and applied pressure P, define stiffness number  (

rigid grain limit

Inertial flows :  r

P

-1)  , assess approach to 2

E

(

P

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

quasistatic limit

• use internal friction  * as an alternative to viscosity

Simulation of steady uniform shear flow

Normal stress is imposed

P

  22

Fixed shear strain rate

 

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

Inertial number

and constitutive relations

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

inertial number

I



m P I m

 Monodisperse spheres, no intergranular friction (Peyneau & Roux, Phys Rev E 2008)  RCP  Generalization of critical state to I-dependent states with inertial effects  Useful constitutive law, applied to different geometries (Pouliquen, Jop, Forterre…)

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  imposed (4000 grains) is Ratio  12 /  22 =  * expresses material rheology

Simulations of dense suspensions

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

Experiments on dense suspensions

• Attention paid to possible density inhomogeneities  measurements local • 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  forces ? To  at  <  RCP effects, normal stresses… ?

Sensitivity to repulsive alone ? Effective viscosity , non-Newtonian

Lubrication and hydrodynamic resistance matrix

Normal hydrodynamic force :

v F N

 

N

(

h



N

For 2 spheres of radius R, with perfect lubrication : 

N

(

h

 3 2  2

h

dominant forces transmitted by a network of ‘quasi-contacts’ No contact within finite time !

Cut-off for narrow interstices h

Choice of systems and parameters

 D systems of identical spheres, diameter a :   to  beads  h max /a = 0.1 or 0.3

 h min /a = 10 -4 ;  = 10 5  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, 0.7

a d a

  disks  3D lubrication  h max /a = 0.5

 h min /a = 10 -4 or 10 -2 ;  = 10 4  Vi down to 10 -4 or 10 -6   = 0.3

in solid contacts

Model, computation method

  (similar to stiffness matrix in elastic contact network) • non-singular tangential coefficient approximation)

F F

with   Solve

F c



v



F



ext

and F c  

F v

    

V

depend on grain positions

V



F c



F ext

Some technical aspects about simulations Lees-Edwards boundary conditions

+ variable height, ensuring constant  yy Regression of fluctuations  

Measurements in steady state :

 Check for stationarity of measurements  Obtain error bar from ‘blocking’ technique  Request long enough stationary intervals 

t

1 t le t ...

Constant volume and/or constant shear stress conditions should produce same system state in large N limit N -1/2 N -1/2

Choice of time step, integration

 t such that matrix and r.h.s. do not change `too much’… Euler (explicit) : (error ~  t 2 ) (( ) (



((

c

) ‘Trapezoidal’ rule : (error ~  t 3 )

X



(( )

c

X

( )



c

X

 

)

t

)

Vt Vt

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

Vi

2

D Vi

3

D

     

a



P P

 Plays analogous role to inertia parameter defined for dry grains

I

2

D I

3

D

     

m P m aP

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  Ratio  12 /  22 =  * expresses material rheology is imposed (1372 grains).

3D results :



*and



as functions of Vi

N

 1372 identical spherical grains

Vi Vi

 = 0 : difficult case ! Approach to  * = 0.1,  =0.64 …

Back to more traditional (constant



) approach Effective viscosity

From *



and  0    one gets:    *

i



c

  if  *   * 0 (not satisfied in our case !)

Shear rate effects ?

Exponent : 2.5 to 3 ?

 

f

 

a



K

N

 0 In rigid grain limit replace 4 th 

No influence of on effective viscosity

Effective viscosity, as a fonction of solid fraction

(h c = h max ) 

Influence of repulsive force

Adsorbed polymer layer, short-range repulsion, as

(Fredrickson & Pincus, 1991) 0 [  

b

5 4 Polymer layer thickness l b = 0.01 or 0.001

(open/filled symbols ) Force F 0  ratio F 0 /a 2 P (0.1, 1, 10) = (diamond, square, circle)  * Shear-thinning within studied parameter range

Vi

 4.10

 3

Vi



Vi c Vi



Vi c

Vi c ~ 3.10

-2 

Ordered structure polydispersity < 20%

Pair correlations in plane yz

 *  

f

Shear thinning with repulsive forces

    ,

l

b

a



a

2

F

0     *  

a K

N

  

F

0

Vi

2

a 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 l b

Network of repulsive forces carry all shear stress as Vi decreases



xy



yy Vi

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

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

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

2D disk model

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 inertial case)

Viscous Dry, inertial   Viscous case h min /a = 10 -2 and h min /a = 10 -4 ,  =0,3 Dry inertial case (right) :  =0.3 and  =0 Coincidence I=0 / Vi=0

2D results : solid fraction versus Vi (in paste)

(analogous to



function of I in dry inertial case)

 Viscous  Dry, inertial Viscous case h min /a = 10 -2 and h min /a = 10 -4 (896 grains),  =0.3

Dry inertial case (right) :  =0.3 and  =0 Coincidence I=0 /Vi =0

Constitutive laws : dry grains (2D)



= 0 or 0.3

c cI

 1  1

c



eI



Constitutive laws, granular suspensions (2D),



= 0.3

 

cV

 1  1

c



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 approached is



(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 h min /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  or any specific singularity below RCP, except in small systems =0 in contacts, no jamming or viscosity divergence  If  is large enough,    (-frictional) beads (but… Re ? Brownian effects ?)  introduction of repulsive interaction with force scale  shear-thinning (or thickening)

Perspectives

Improve performance of numerical methods Continuum fluid !

(Keep separate treatment of lubrication singularities ?)

bridge the gap between real (and difficult) hydrodynamic calculations and ‘conceptual models’