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Koshiro Suzuki (Canon Inc.)Koshiro Suzuki (Canon Inc.)

Hisao Hayakawa (YITP)Hisao Hayakawa (YITP)

A rheological study of A rheological study of sheared granular flows by sheared granular flows by the Mode-Coupling Theorythe Mode-Coupling Theory

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Introduction Steady-state of the sheared granular flowSteady-state of the sheared granular flow

SettingSetting monodisperse monodisperse hard inelastic smooth hard inelastic smooth spheres spheres

(3D)(3D) uniform steadyuniform steady shear flow (bulk shearing) shear flow (bulk shearing)

Energy balance equation Energy balance equation (energy/volume/time)(energy/volume/time)

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y

vx

Control parameters volume fraction shear rate restitution coefficient

x

y

d

Granular temperatureT is determined

m

volume V

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Introduction Kinetic theoryKinetic theory

Shear stressShear stress

Energy dissipation rateEnergy dissipation rate

Consequence of the energy balanceConsequence of the energy balance

Bagnold scalingBagnold scaling

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[Garzo and Dufty, PRE 59, 5895 (1999)]

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Introduction Limitation of the kinetic theoryLimitation of the kinetic theory

It is known that kinetic theory works well up toIt is known that kinetic theory works well up to However, there is a large deviation from However, there is a large deviation from

simulationsimulationfor for

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Granular temperature Energy dissipation rate Shear stress

[Mitarai and Nakanishi, PRE 75, 031305 (2007)]

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Our work We aim to construct a theory viable for dense We aim to construct a theory viable for dense

sheared granular flows, sheared granular flows,

We attempt to apply the Mode-Coupling Theory We attempt to apply the Mode-Coupling Theory (MCT) :(MCT) : It is known that MCT captures the glass It is known that MCT captures the glass

transition,transition,

It incorporates It incorporates memory kernelsmemory kernels, which is , which is neglected in the kinetic theory.neglected in the kinetic theory.

We expect that the memory kernels play We expect that the memory kernels play significant roles.significant roles.

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Previous related works Driven granular fluids Driven granular fluids

Kranz, Sperl, Zippelius, PRL 104, 225701 (2010)Kranz, Sperl, Zippelius, PRL 104, 225701 (2010) Sperl, Kranz, Zippelius, EPL 98, 28001 (2012)Sperl, Kranz, Zippelius, EPL 98, 28001 (2012) Kranz, Sperl, Zippelius, PRE 87, 022207 (2013)Kranz, Sperl, Zippelius, PRE 87, 022207 (2013)

Sheared granular fluidsSheared granular fluids Hayakawa, Otsuki, PTP 119, 381 (2008)Hayakawa, Otsuki, PTP 119, 381 (2008) Suzuki, Hayakawa, AIP Conf. Proc., to be Suzuki, Hayakawa, AIP Conf. Proc., to be

published (2013) [arXiv:1301.0866]published (2013) [arXiv:1301.0866]

No work has been done for the rheology of sheared No work has been done for the rheology of sheared granular fluidsgranular fluids

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Results MCT shows better compatibility than to simulation MCT shows better compatibility than to simulation

than the kinetic theory for the than the kinetic theory for the shear stressshear stress.. However, MCT fails to obtain the However, MCT fails to obtain the energy dissipation energy dissipation

raterate, as is the case for the kinetic theory., as is the case for the kinetic theory.

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Shear stress

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A short review of MCT MCT describes (for MCT describes (for thermalthermal glassy systems) : glassy systems) :

two-step relaxation of the density time-two-step relaxation of the density time-correlation function (“cage effect”).correlation function (“cage effect”).

the nonlinear rheology (constitutive relation of the nonlinear rheology (constitutive relation of shear stress and shear rate).shear stress and shear rate).

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Newtonian

yield stress

high density

low density

[Suzuki and Hayakawa, PRE 87, 012304 (2013)]

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A short review of MCT

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Schematic pictureSchematic picture The red particle is “caged” by the surrounding The red particle is “caged” by the surrounding

particles above the critical density until the cage particles above the critical density until the cage is broken by shearing.is broken by shearing.

This picture does not hold for granular particles !This picture does not hold for granular particles !

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Microscopic dynamics of grains SLLOD equation (Newtonian eq. for uniform shear)SLLOD equation (Newtonian eq. for uniform shear)

No external noisesNo external noises Viscous dissipation of Viscous dissipation of softsoft, , smoothsmooth grains grains

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ζ : viscous coefficient(mass/time)

t=0Equilibrium Relaxation

to a steady state

steady shear &dissipation

hard-core limitis taken later

k

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Liouville equation (1) Time evolution of physical quantities A(q(t),p(t))

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Formal solution

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Liouville equation (1) Physical quantities of interestPhysical quantities of interest

Shear stressShear stress

Energy dissipation rateEnergy dissipation rate

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Liouville equation (2) Time evolution of distribution function

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Formal solution

: Phase volume contraction

(non-Hermitian)

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Formula (MCT) : shear stress

It is written in terms of It is written in terms of time/wavenumbertime/wavenumber integration of integration of time-correlation functionstime-correlation functions , ,

Dissipative vertex functionDissipative vertex function

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dominantterm

(functions of equilibrium RDF at contact)

(hard-core limit)

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Formula (MCT) : energy dissipation rate

The first term coincides with the kinetic theoryThe first term coincides with the kinetic theory⇒⇒ Maxwellian contributionMaxwellian contribution

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n.b. kinetic theory result

non-Maxwelliancontribution

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Time correlation functions

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Slow modesSlow modes

Time correlation functionsTime correlation functions

Isotropic approximationIsotropic approximation

current density fluctuation

density fluctuation

scalar functions ,

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MCT equations Isotropic, weak-shear approximationIsotropic, weak-shear approximation

Effective friction coefficientsEffective friction coefficients

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g(d) : equilibrium RDF at contact

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Memory kernels for Density-Density correlation function for Density-Density correlation function ΦΦ

Dissipative kernels are Dissipative kernels are negativenegative

⇒⇒ They can cancel the glassy kernelThey can cancel the glassy kernel

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glassy kernel

dissipative

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Disappearance of the plateau Parametrical study (varying Parametrical study (varying ee))

Glassy plateau of the density-density correlation Glassy plateau of the density-density correlation function disappears due to dissipative memory function disappears due to dissipative memory kernels.kernels.

Plateau appears complementarily in the density-Plateau appears complementarily in the density-current correlation function.current correlation function.

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Density-Density correlation

= 0.521 = 1.0e-2 T = 1.0 qd = 7.0

conditions

Density-Current correlation

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Disappearance of the plateau

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Schematic pictureSchematic picture The red particle loses its kinetic energy due to

inelastic collisions ⇒ density correlations disappear.

Eventually the cage is destructed by the shear.

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Scheme of the analysis Temperature is not well predicted neither for MCT Temperature is not well predicted neither for MCT

nor kinetic theorynor kinetic theory ⇒　⇒　 adopt the result of the simulation for adopt the result of the simulation for T T ((energy energy balance equation isbalance equation is not not solvedsolved))..

Compare the values of and Compare the values of and separatelyseparately for for given sets of (given sets of (T, , eT, , e);); is chosen as the unit of timeis chosen as the unit of time

5 conditions for 5 conditions for (0.55, 0.57, 0.58, 0.59, 0.60) (0.55, 0.57, 0.58, 0.59, 0.60) 3 conditions for 3 conditions for ee (0.98, 0.92, 0.70) (0.98, 0.92, 0.70)

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Inputs Static structure factor (Static structure factor (equilibriumequilibrium))

FactFact : : MCTMCT = 0.516 corresponds to = 0.516 corresponds to gg ～～ 0.60.6

⇒⇒ we adopt the following *, we adopt the following *, * = – * = – gg + + MCTMCT,,　　　　 gg =0.595, =0.595, MCTMCT = 0.516. = 0.516.

Radial distribution function (Radial distribution function (equilibriumequilibrium, at contact), at contact)

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Interpolation formula (Torquato)

volume fraction 0.55 0.57 0.58 0.59 0.60g(r) at contact 9.486 12.196 14.229 17.075 21.344

ν: volume fraction

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Result : shear stress e=0.98/0.92/0.70e=0.98/0.92/0.70

aa

Kinetic theory under-(over-)estimates for Kinetic theory under-(over-)estimates for e=0.98(0.70)e=0.98(0.70)

MCT works well for e=0.98, 0.92, 0.70MCT works well for e=0.98, 0.92, 0.70

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Why does MCT work well ? 　⇒　 time-correlation function

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Result : time-correlation function e=0.98e=0.98

Dissipation is weak; glassy plateau appears in the density-density correlation.

The plateau is crucial for the evaluation of the The plateau is crucial for the evaluation of the stress.stress.

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Density-Density correlation Density-Current correlation

・ ・

・

qd = 7.0

qd = 7.0

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Result : time-correlation function e=0.92e=0.92

No clear plateau appears in the density-density correlation.

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Density-Density correlation Density-Current correlation

・ ・

・

qd = 7.0 qd = 7.0

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Result : time-correlation function e=0.70e=0.70

Dependence on density decreases as eDependence on density decreases as e↓↓ The origin of the density dependence resides in The origin of the density dependence resides in

the memory kernel, which dissapears as ethe memory kernel, which dissapears as e↓↓

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Density-Density correlation Density-Current correlation

・ ・

・

qd = 7.0

qd = 7.0

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Significance of the plateau e=0.98e=0.98

The result of the kinetic theory is close to the The result of the kinetic theory is close to the MCT without the memory kernelMCT without the memory kernel..

Precise evaluation of the relaxation of the time-Precise evaluation of the relaxation of the time-correlation function is crucial for the shear stress.correlation function is crucial for the shear stress.

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・

Density-Density correlation

qd = 7.0

Shear stress

e=0.70e=0.70

The result of MCT is coincident to The result of MCT is coincident to MCT without MCT without memory kernelsmemory kernels..

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Significance of the plateau

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・

Density-Density correlation

qd = 7.0

Shear stress

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Summary of the results MCT shows better compatibility with the simulation MCT shows better compatibility with the simulation

than the kinetic theory for the than the kinetic theory for the shear stressshear stress.. Precise evaluation of the Precise evaluation of the relaxation timerelaxation time of the of the

time-correlation functiontime-correlation function is crucial. is crucial. The relaxation of the time-correlation function is The relaxation of the time-correlation function is

determined by the determined by the dissipative memory kerneldissipative memory kernel..

However, MCT fails to explain the However, MCT fails to explain the energy energy dissipation ratedissipation rate⇒⇒ we will retry !we will retry !

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Discussions Nonlinear rheologyNonlinear rheology

In MCT, deviation from the Bagnold scaling In MCT, deviation from the Bagnold scaling is is notnot observed. observed.

This is due to the hard-core limit.This is due to the hard-core limit. Proper treatment of the soft spheres is Proper treatment of the soft spheres is

necessary.necessary.

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[Hatano, Otsuki, Sasa, JPSJ 76, 023001 (2007)]

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Discussions Jamming transitionJamming transition

Precise evaluation of the Precise evaluation of the vibrational frequency of vibrational frequency of the contact networksthe contact networks (instead of the collision (instead of the collision frequency) is necessary.frequency) is necessary.

The replica theory might be of help for this issue.The replica theory might be of help for this issue.

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Appendix

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Hard-core limit Relation of ζ and eRelation of ζ and e

Collision frequencyCollision frequency

It is difficult to derive in MCT.It is difficult to derive in MCT. We adopt its expression for the kinetic theory.We adopt its expression for the kinetic theory.

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(contact duration time)

k : linear spring coefficient(1)

hard-core limit

effective collision frequency(1)

[Otsuki, Hayakawa, Luding, PTP Suppl. 184, 110 (2010)]

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Hard-core limit Collision frequency (kinetic theory)Collision frequency (kinetic theory)

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g(φ, d) : equilibrium RDF at contact

The step function is formally replaced by the delta function.

viscous coefficient(hard-core limit)

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Interpretation of the granular temperature Normalization of current projectionNormalization of current projection

Requirement 1 : Requirement 1 : TT should be treated as constant should be treated as constant TT is not included as a basis of the projected space is not included as a basis of the projected space

Requirement 2 : Requirement 2 : TT should satisfy the energy balance should satisfy the energy balance eq.eq.

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T : granular temperature

We simply assume T = TSS in this work.

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Projection operator formalism Basis of the projected spaceBasis of the projected space

Projection operatorsProjection operators Mori equationMori equation

Mode-Coupling ApproximationMode-Coupling Approximation

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density fluctuation

current density fluctuation

specific for shearedgranular systems

: static structure factor

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Time correlation functions

Additional functions are required from Additional functions are required from Initial conditionsInitial conditions

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additional correlation functionsfor granular systems

translational invariance in the sheared frame

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Mori equation

ExactExact equation for the time correlation functions equation for the time correlation functions Coefficients , are Coefficients , are equilibriumequilibrium quantities quantities

(n.b. (n.b. TT is exceptional) is exceptional)

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Approximations Isotropic approximation

Dissipation is almost isotropic For reducing the load of calculation （ 3D⇒1D ）

Weak-shear approximationWeak-shear approximation Second (and higher) order terms in and/or Second (and higher) order terms in and/or

are neglected. are neglected. e.g. , , , etc.e.g. , , , etc.

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e.g.

reduces to two scalar functions ,