Granular flows confined between flat, frictional walls
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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 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
dzdVxThe shear rate becomes Independent of q
for > 40° q 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 l n
•In the flowing layer (y < l), µ remains close to the microscopic friction mgw.
•µ decreases sharply at greater depths, but most grains slip on sidewalls.
(PRL Richard 2008)
wyyw
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
P
dI
P
,
Viscosity• Effective viscosity (cf. Michel Louge talk) :
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P
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 W
1exp