Report

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) 4 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) 6 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 7 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. 10 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 11 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! 12 Viscoplastic rheology µ(I) m P , I gd 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 mP • Effective viscosity (cf. Michel Louge talk) : g Effective viscosity vs the rescaled depth z/lν 14 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 : gexp 1 gsim W W the shear rate increases with W in the simulations, it decreases in the experiments. Why??? 17