Granular flows confined
between flat, frictional walls
Patrick Richard (1,2), Alexandre Valance (2) and Renaud Delannay (2)
(1) Université Nantes-Angers-Le Mans
IFSTTAR
Nantes, France
(2) Université de Rennes 1
Institut de Physique de Rennes (IPR)
UMR CNRS 6251
Rennes, France
1
Confined flows on a pile
Confined granular flows atop “static” heap
Q fixed → Steady and fully developed flows
2
Sidewalls Stabilized Heap
Complex flows
•From quasi-static packing to ballistic flows (at the free surface)
•Interaction between liquid and “quasi-static” phase (erosion,
accretion)
h
q increases with Q
(PRL Taberlet 2003)
tan q = µI + µw h/W
q
For large Q, q >> qrepose
effective friction coefficients (internal and with sidewalls resp.)
3
Numerical simulations
• Discrete elements methods
tij
nij
part. j
ωi
part. i
δij
•
•
•
•
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
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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2 types of simulations
Periodic Boundary
Conditions (PBC)
Full System (FS)
y
x
z
Both give the same tan q .vs. Input flow rate
x
g
g
Simulate the whole system
Simulate a periodic cell (stream wise)
Input flow rate is a parameter,
The angle of inclination is a parameter
the system chooses its angle
The system chooses its flow rate
5
Packing fraction profiles
n0
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 n (z) = (n0 /2) [1+ tanh (z/ln)]
(PRL Richard 2008)
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Velocity profiles
Except close to jamming, Vx and n share
the same characteristic length : ln
→ depth of the flowing Layer : h = 2ln
The shear rate g  dVx becomes Independent of q for
dz
q > 40 and varies as W1/2
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Characteristic length
• The characteristic length ln scales with W and increases
with inclination (as required ).
• Allows to obtain µI and µw
8
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
The resultant sidewall friction coefficient

(PRL Richard 2008)

m   w  yyw

w
 w   x   yz y
w
xy
•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.
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Particle motion
Experiments
• 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
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weakens
Sidewall friction
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!
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Viscoplastic rheology µ(I)
m

P
, I
gd
P

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
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Viscosity
mP
• Effective viscosity (cf. Michel Louge talk) :  
g
Effective viscosity vs the rescaled depth z/lν
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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)
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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 :
gexp  1
gsim  W
W
the shear rate increases with W in the simulations, it
decreases in the experiments.
Why???
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