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

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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 (

PRL Taberlet 2003

) q increases with Q tan q = µ I + µ w h/W q For large Q, q >> q repose effective friction coefficients ( internal and with sidewalls resp.) 3

Numerical simulations

•

Discrete elements methods

t ij ω i n ij part. i part. j δ ij • Soft but stiff frictional spheres • Slightly polydisperse (d ± 20%) • Walls : spheres with infinite mass • Normal force : linear spring and dashpot

F n = k

d

+

g

d

d

/dt

• Tangential force :Coulomb law regularized by a linear spring Ft = -min(ku

t

,µ|F

n

|) • Solve motion equations µ = 0.5, restitution coefficient e = 0.88

N = 48,000 grains (W = 30d) to N = 6,000 grains (W=5d) 4

2 types of simulations

Full System (FS) Periodic Boundary Conditions (PBC) Both give the same tan q .vs. Input flow rate x y z g x Simulate the whole system Input flow rate is a parameter, the system chooses its angle g Simulate a periodic cell (stream wise) The angle of inclination is a parameter The system chooses its flow rate 5

n 0

Packing fraction profiles

n

0

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

Origin of

z

axis such that : n (

z

= 0) = n 0 /2 Profiles of n collapse on a single curve n (

z

)

=

( n

0

/2) [1+ tanh (

z

/

l

n )] 6 (

PRL Richard 2008

)

Velocity profiles

Except close to jamming, V x and the same characteristic length : l n n share → depth of the flowing Layer : h = 2l n The shear rate q > 40 g   dV x dz becomes Independent of q for and varies as W 1/2 7

Characteristic length

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

Allows to obtain µ I and µ w 8

Effective friction coefficients

• The eff. Friction coefficients (especially to the variation of m gw m w ) are more sensitive than to the variation of m gg • The fact that m I varies with m gw is interesting (effect of the boundaries on the local rheology : m I = m (I))

Sidewall friction

The resultant sidewall friction coefficient  

w

 

w xy



x

 

w yz



y

m    

w

(

PRL Richard 2008

) 

w yy

•Also scales with l n •In the flowing layer (y < l n ), µ  remains close to the microscopic friction m gw .

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

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Experiments

Particle motion

• • • Cage motion jumps Quick jumps become less frequent deeper in the pile, cages.

 increasing the residence time in • 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 11

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 13

Viscosity

•

Effective viscosity (cf. Michel Louge talk) :

  m

P

g 

Effective viscosity vs the rescaled depth z/l ν

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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) 15

Scaling

• Flow rate per unit width Q* vs tan q for differents width W.

Q* sim  W 5/2 To compare with the experiments (cf. M. Louge) : Q* exp  W 3/2

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 

sim



W

g  exp  1

W

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

Why???

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