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Granular flows confined between flat, frictional walls

Patrick Richard (1,2), Alexandre Valance (2) and Renaud Delannay (2)

(1) Université Nantes-Angers-Le MansIFSTTARNantes, France

(2) Université de Rennes 1Institut de Physique de Rennes (IPR)UMR CNRS 6251Rennes, France

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Confined granular flows atop “static” heap

Q fixed → Steady and fully developed flows

Confined flows on a pile

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Complex flows•From quasi-static packing to ballistic flows (at the free surface)

•Interaction between liquid and “quasi-static” phase (erosion, accretion)

(PRL Taberlet 2003)

Sidewalls Stabilized Heap

qtan q = µI + µw h/W

h

q increases with Q

For large Q, q >> qrepose

effective friction coefficients (internal and with sidewalls resp.)

Numerical simulations• Discrete elements methods

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• Soft but stiff frictional spheres• Slightly polydisperse (d ± 20%) • Walls : spheres with infinite mass• Normal force : linear spring and dashpot

Fn = kd +g dd/dt• Tangential force :Coulomb law regularized by a

linear spring Ft = -min(kut,µ|Fn|)

• Solve motion equations

part. i

part. j

nijtij

ωi

δij

µ = 0.5, restitution coefficient e = 0.88

N = 48,000 grains (W = 30d) to N = 6,000 grains (W=5d)

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Full System (FS) Periodic Boundary Conditions (PBC)

Simulate the whole system

Input flow rate is a parameter,

the system chooses its angle

Simulate a periodic cell (stream wise)

The angle of inclination is a parameter

The system chooses its flow rate

Both give the same tan q .vs. Input flow rate

2 types of simulations

zx

y

xgg

n0 ≈ 0.6 : packing fraction in the quasi-static region, q.

Origin of z axis such that : n(z = 0) = n0/2

Profiles of n collapse on a single curve

n0

Packing fraction profiles

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n (z) = (n0 /2) [1+ tanh (z/ln)](PRL Richard 2008)

Velocity profiles

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Except close to jamming, Vx and n share the same characteristic length : ln

→ depth of the flowing Layer : h = 2ln

dzdVxgThe shear rate becomes Independent of q

for q > 40° and varies as W1/2

Characteristic length

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• The characteristic length ln scales with W and increases with inclination (as required ).

• Allows to obtain µI and µw

Effective friction coefficients

• The eff. Friction coefficients (especially mw) are more sensitive to the variation of mgw than to the variation of mgg

• The fact that mI varies with mgw is interesting (effect of the boundaries on the local rheology : mI =m(I))

Sidewall friction

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The resultant sidewall friction coefficient

•Also scales with ln

•In the flowing layer (y < ln), µ remains close to the microscopic friction mgw.

•µ decreases sharply at greater depths, but most grains slip on sidewalls.

(PRL Richard 2008)wyyw m

yx wyz

wxyw

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• Cage motion• jumps • Quick jumps become

less frequent deeper in the pile, increasing the residence time in cages.

• While trapped, grains describe a random oscillatory motion – with zero mean displacement – negligible contribution to the mean resultant wall friction force.

• As trapping duration grows with depth, the resultant wall friction weakens

ExperimentsParticle motion

Sidewall friction

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The grain-wall friction coefficient governs the value of the plateau reached close to the free surface

z / d

The effect of the grain-grain friction coefficient is weak : the dissipation at the sidewalls is crucial!

Viscoplastic rheology µ(I)

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Collapse for low values of I (< 0.5) or eq. Large packing fractions (0.35 - 0.6)The rheology based on a local friction law µ(I) breaks down in the quasi-static and the dilute zones

gmPdI

P

,

Viscosity• Effective viscosity (cf. Michel Louge talk) :

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gmP

Effective viscosity vs the rescaled depth z/lν

Viscosity

Effective viscosity vs the volume fraction

Seems adequate in the « liquid » and « quasi-static » zones.

Normalisation by T for the dilute part? (kinetic theory) 15

Scaling• Flow rate per unit width

Q* vs tanq for differents width W.

Q*sim W5/2

To compare with the experiments (cf. M. Louge) :

Q*exp W3/2

Question

Everything looks similar in the simulations and in the experiments (at least qualitatively).

BUT, the scaling in W is different, with qualitative effects :

the shear rate increases with W in the simulations, it decreases in the experiments.

Why???17

Wsim gW

1exp g