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 Under large strain, critical state independent of initial state, characterized by a ‘’flow structure’’ (density + nb of contacts, anisotropy…) Monodisperse sphere assembly, friction coefficient  = 0.3 Several samples with 4000 beads, prepared at different solid fraction  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)

Simulation of steady uniform shear flow Fixed shear strain rate Normal stress is imposed

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 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, : 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 Normal hydrodynamic force : For 2 spheres of radius R, with perfect lubrication : No contact within finite time ! Cut-off for narrow interstices h<h min (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 :   to  beads  h max /a = 0.1 or 0.3  h min /a = ;  = 10 5  Vi from 0.1 down to   = 0 in solid contacts  + alternative systems with repulsive forces, no lubrication cutoff  D systems of polydisperse disks, diameter d,   disks  3D lubrication  h max /a = 0.5  h min /a = or ;  = 10 4  Vi down to or   = 0.3 in solid contacts

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) Model, computation method with and F c depend on grain positions Solve

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 Regression of fluctuations Some technical aspects about simulations N -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’… Euler (explicit) : (error ~  t 2 ) ‘Trapezoidal’ rule : (error ~  t 3 )

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

3D results :  *and  as functions of Vi  = 0 : difficult  case ! Approach to  * = 0.1,  =0.64 … Vi identical spherical grains

Back to more traditional (constant  ) approach Effective viscosity Fromandone gets: if (not satisfied in our case !) Shear rate effects ? In rigid grain limit replace 4 th argument by zero. No influence of on effective viscosity Exponent : 2.5 to 3 ?

Effective viscosity, as a fonction of solid fraction  (h c = h max )

Polymer layer thickness l b = 0.01 or (open/filled symbols ) Force F 0  ratio F 0 /a 2 P (0.1, 1, 10) = (diamond, square, circle) Influence of repulsive force Adsorbed polymer layer, short-range repulsion, as Ordered structure polydispersity < 20% Shear-thinning within studied parameter range Pair correlations in plane yz (Fredrickson & Pincus, 1991) Vi c ~

Shear thinning with repulsive forces Shear thinning due to change in reduced shear rate Results correspond to varying from to No shear thinning for smaller l b

Network of repulsive forces carry all shear stress as Vi decreases Other ‘granular’ features: force distribution, coordination number… Vi

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

2D results : solid fraction versus Vi (in paste) (analogous to  function of I in dry inertial case) Viscous case h min /a = and h min /a = (896 grains),  =0.3 Dry inertial case  (right) :  =0.3 and  =0 Coincidence I=0 /Vi =0 Viscous Dry, inertial  

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 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  =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 ?)