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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 Mans IFSTTAR Nantes, France (2) Université de Rennes 1 Institut de Physique de Rennes (IPR) UMR CNRS 6251 Rennes, France. Granular flows confined between flat, frictional walls.

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Granular flows confined between flat, frictional walls

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  1. 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 Granular flows confined between flat, frictional walls

  2. Confinedflows on a pile Confined granular flows atop “static” heap Q fixed → Steady and fully developed flows

  3. SidewallsStabilizedHeap • 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) • increaseswithQ • For large Q, q>> qrepose h tan q = µI + µw h/W q effective friction coefficients (internal and with sidewalls resp.)

  4. nij δij part. j part. i tij ωi Numerical simulations • Discrete elements methods • Soft but stiff frictional spheres • Slightly polydisperse (d ± 20%) • Walls : spheres with infinite mass • Normal force : linear spring and dashpot • Fn = kd +gdd/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)

  5. g g 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 Simulate a periodic cell (stream wise) The angle of inclination is a parameter The system chooses its flow rate Simulate the whole system Input flow rate is a parameter, the system chooses its angle

  6. Packing fraction profiles n0 n0≈ 0.6 : packing fraction in the quasi-staticregion, q. Origin ofzaxis suchthat :n(z = 0) = n0/2 Profiles of  collapse on a single curve n (z) = (n0 /2) [1+ tanh (z/ln)] (PRL Richard 2008)

  7. Velocity profiles Except close to jamming, Vx and n share the same characteristic length : ln → depthof the flowingLayer : h= 2ln • The shear rate becomes Independent of q for q > 40° and varies as W1/2

  8. Characteristiclength • The characteristic length ln scales with W and increases with inclination (as required ). • Allows to obtainµIand µw

  9. Effective friction coefficients • The eff. Friction coefficients (especiallymw) are more sensitive to the variation of mgwthan to the variation of mgg • The factthatmI varies withmgwisinteresting (effect of the boundaries on the local rheology : mI =m(I))

  10. Sidewall friction (PRL Richard 2008) The resultant sidewall friction coefficient • Also scales with ln • 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.

  11. 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 weakens

  12. 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!

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

  14. Viscosity • Effective viscosity (cf. Michel Louge talk) : Effective viscosity vs the rescaled depth z/lν

  15. Viscosity Effective viscosity vs the volume fraction Seemsadequate in the « liquid » and « quasi-static » zones. Normalisation by T for the dilute part? (kinetictheory)

  16. Scaling • Flow rate per unit width Q* vs tanq for differentswidth W. Q*sim W5/2 To compare with the experiments (cf. M. Louge) : Q*exp W3/2

  17. Question Everything looks similar in the simulations and in the experiments (at least qualitatively). BUT, the scaling in W isdifferent, with qualitative effects : the shear rate increaseswith W in the simulations, itdecreases in the experiments. Why???

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