Dry and wet granular flows Diego Berzi
Outline 2 What? Why? How? When? Who? Where? Then?
What? Granular flows many solid moving particles 3 particle mass is large (at least 10 20 molecular masses). Hence, Brownian motion is irrelevant particles dissipate energy whenever they interact If we measure it by tons, the material most manipulated by man is water; the second-most-manipulated is granular matter. P.-G. de Gennes. 1999. Granular matter: A tentative view., Rev. Mod. Phys. 71, S374. Nobel Prize in Physics 1991 for discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and polymers.
Why? 4 Astrophysics Industrial engineering GRANULAR FLOWS Civil engineering Bio-engineering
Why? Civil Engineering 5
Why? Industrial and Bio-Engineering 6
Why? Exotic Engineering 7
Why? Exotic Engineering 8
Why? Exotic Engineering 9
How? 10 Continuum mechanics Hydrodynamic fields: mass density velocity v p (solid volume fraction) v t t v 0 Mass balance vv f Momentum balance Closure for the stress tensor (rheology)?
When? Who? Grand-grand-fathers 11 J.C. Maxwell. 1860. Illustrations of the dynamical theory of gases. 1. on the motions and collisions of perfectly elastic spheres. Phil.Mag.,19(4):19 32. L. Boltzmann. 1872. Weitere studien uber das warmegleichgewicht unter gasmolekulen.wiener Berichte, 66:275 370. Origin of stresses for molecular gases: transfer of momentum associated with velocity fluctuations (kinetic theory)
When? Who? Grand-father R.A. Bagnold. 1954. Experiments on a Gravity- Free Dispersion of Large Solid Spheres in a Newtonian Fluid under Shear. Proc. R. Soc. Lond. A.,225(1160):49-63. 12 Neutrally buoyant spheres in a rheometer pressure momentum frequency md 2 shear stress area d d 2 2 Intensity of fluctuations?
When? Who? Fathers S. Ogawa. 1978. Multitemperature theory of granular materials. In Cowin & Satake, 208-217. 13 Granular temperature T: one third of the mean square of velocity fluctuations Kinetic theory of granular gases S.B. Savage and D.J. Jeffrey. 1981. The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255-272. pressure 1/2 1/2 momentum frequency mt T / d T 2 area d shear stress 1/2 momentum frequency md T / d 2 area d dt 1/2
When? Who? Fathers 14 J.T. Jenkins and S.B. Savage 1983. A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 187-202. Coefficient of restitution e: Negative ratio of post to preimpact relative velocity between two colliding grains Balance of fluctuation energy with dissipation rate governs the evolution of T (completely different from molecular gases)
So far (mid 1980s), we have a theory for 15 identical, cohesionless, frictionless, nearly elastic spheres undergoing random, binary, instantaneous, uncorrelated (molecular chaos) collisions, in absence of interstitial fluid.
Inelasticity 16 V. Garzo and J.W. Dufty 1999. Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895-5911. Any value of the coefficient of restitution!
s/p Where? Simple Shearing 17 Rigid, frictionless spheres with e=0.7 0.7 0.6 0.5 0.4 0.3 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Update 18 identical, cohesionless, frictionless, nearly elastic spheres undergoing random, binary, instantaneous, uncorrelated (molecular chaos) collisions, in absence of interstitial fluid.
Correlated motion Correlation length in the dissipation rate N. Mitarai and H. Nakanishi 2005. Bagnold Scaling, Density Plateau, and Kinetic Theory Analysis of Dense Granular Flow. Phys. Rev. Lett. 94, 128001. J.T. Jenkins 2007. 19 Breaking of molecular chaos Dense inclined flows of inelastic spheres. Granul. Matter 10, 47-52. J.T. Jenkins and 2010. Dense inclined flows of inelastic spheres: tests of an extension of kinetic theory. Granul. Matter 12, 151-158. D. Vescovi,, P. Richard and N. Brodu 2014. Plane shear flows of frictionless spheres: Kinetic theory and 3D softsphere discrete element method simulations. Phys. Fluids 26, 053305. Influence of boundaries
s/p Where? Simple Shearing 20 Rigid, frictionless spheres with e=0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Update 21 identical, cohesionless, frictionless, nearly elastic spheres undergoing random, binary, instantaneous, uncorrelated (molecular chaos) collisions, in absence of interstitial fluid.
Friction Solve for rotational momentum and energy (nightmare!) J.T. Jenkins and C. Zhang 2002. Kinetic theory for identical, frictional, nearly elastic spheres. Phys. Fluids 14, 1228-1235. Pressure, shear stress and correlation length are singular at different volume fractions C.K.K. Lun. 1991. and D. Vescovi 2015. 22 Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. J. Fluid Mech.233, 539-559. Different singularities in the functions of extended kinetic theory at the origin of the yield stress in granular flows. Phys. Fluids 27, 013302. Rotation and sliding imply additional dissipation: effective coefficient of restitution
h/w Where? Simple Shearing and inclined flows 23 6 5 4 3 2 1 0 10-4 10-3 10-2 10-1 10 0 q/w 5/2
Update 24 identical, cohesionless, frictionless, nearly elastic spheres undergoing random, binary, instantaneous, uncorrelated (molecular chaos) collisions, in absence of interstitial fluid.
Stiffness 25 H. Hwang and K. Hutter 1995. A new kinetic model for rapid granular flow. Cont. Mech. Thermodyn. 7, 357-384. Frequency of collisions decreases for soft particles (contact duration). Rate-independent, components of the stresses proportional to the stiffness., D. Vescovi and C.G. di Prisco 2011. Constitutive relations for steady, dense granular flows. Phys. Rev. E 84, 031301. and J.T. Jenkins. 2015. Steady shearing flows of deformable, inelastic spheres. Soft Matter 11, 4799-4808. Constitutive relations for soft spheres.
Where? Simple Shearing 26 Soft, frictional spheres with e=0.7 and m=0.5 1 mm glass spheres basketballs red blood cells soccer balls Yoga balls
Update 27 identical, cohesionless, frictionless, nearly elastic spheres undergoing random, binary, instantaneous, uncorrelated (molecular chaos) collisions, in absence of interstitial fluid.
Interstitial fluid F. Yang and M.L. Hunt 2006. Dynamics of particleparticle collisions in a viscous liquid. Phys. Fluids 18, 121506. Kinetic theory with local coefficient of restitution. and L. Fraccarollo. 2013. Inclined, collisional sediment transport. Phys. Fluids 25, 106601. Role of density ratio (transport on exoplanets). 28 Lubrication forces induce damping: coefficient of restitution depends on relative velocity 2011. Analytical Solution of Collisional Sheet Flows. J. Hydr. Engrg. ASCE 137, 1200-1207. Role of gravity (from sediment transport to debris flow)., J.T. Jenkins and A. Valance. 2015. Periodic saltation over hydrodynamically rough beds: Aeolian to aquatic. J. Fluid Mech.
Where? Sediment transport and debris flow 29
Update 30 identical, cohesionless, frictionless, nearly elastic spheres undergoing random, binary, instantaneous, uncorrelated (molecular chaos) collisions, in absence of interstitial fluid.
Then? identical, cohesionless, 31 mixing/segregation (not yet) attractive potential (not yet) frictionless, nearly elastic spheres undergoing random, binary, cylinders (with J. Curtis) ordered (liquid crystals) multiple collisions (with S. Luding) instantaneous, uncorrelated (molecular chaos) collisions, in absence of interstitial fluid.
The end? 32 Matter and energy seem granular in structure, and so does life, but not so mind. E. Schrödinger. 1956. In Tarner Lecture, at Trinity College, Cambridge 'The Arithmetical Paradox: The Oneness of Mind', printed in Mind and Matter (1958), 61. To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour. W. Blake. 1863. Auguries of Innocence
More details in 33 Correlated motion Berzi, D. 2014. Extended kinetic theory applied to dense, granular, simple shear flows. Acta Mech., 225(8), 2191-2198. Jenkins, J.T. & Berzi, D. 2012. Kinetic Theory applied to Inclined Flows. Granul. Matter, 14(2), 79 84. Berzi, D. & Jenkins, J.T. 2011. Surface Flows of Inelastic Spheres. Phys. Fluids, 23(1), 013303. Jenkins, J.T. & Berzi, D. 2010. Dense Inclined Flows of Inelastic Spheres: Tests of an Extension of Kinetic Theory. Granul. Matter, 12 (2), 151 158. Friction Berzi D. & Vescovi D. 2015. Different singularities in the functions of extended kinetic theory at the origin of the yield stress in granular flows. Phys. Fluids, 27(1), 013302. Stiffness Berzi D. & Jenkins J.T. 2015. Steady shearing flows of deformable, inelastic spheres. Soft Matter, 11(24), 4799-4808. Vescovi D., Di Prisco C.G. & Berzi D. 2013. From solid to granular gases: the steady state for granular materials. Int. J. Numer. Anal. Met., 37, 2937 2951. Berzi, D., Di Prisco, C.G. & Vescovi, D. 2011. Constitutive relations for steady, dense granular flows. Phys. Rev. E, 84 (3), 031301. Boundaries Berzi D. & Jenkins J.T. 2015. Inertial shear bands in granular materials. Phys. Fluids, 27(3), 033303. Vescovi D., Berzi D., Richard P. and Brodu N. 2014. Plane shear flows of frictionless spheres: Kinetic theory and 3D soft-sphere discrete element method simulations. Phys. Fluids, 26(5), 053305.
More details in 34 Collisional suspension Berzi, D., and Fraccarollo, L. (2013). Inclined, collisional sediment transport. Phys. Fluids, 25(10), 106601. Berzi, D. (2013). Simple shear flow of collisional granular-fluid mixtures. J. Hydraul. Eng.-ASCE, 139(5), 547 549. Berzi D. (2011). Analytical solution of collisional sheet flows. J. Hydraul. Eng.-ASCE, 137(10), 1200 1207. Turbulent/collisional suspension Berzi, D. (2013). Transport formula for collisional sheet flows with turbulent suspension. J. Hydraul. Eng.-ASCE, 139(4), 359 363. Debris flow Berzi, D., and Larcan, E. (2013). Flow resistance of inertial debris flows. J. Hydraul. Eng.-ASCE, 139(2), 187 194. Berzi, D., Bossi, F.C., and Larcan, E. (2012). Collapse of granular-liquid mixtures over rigid, inclined beds. Phys. Rev. E, 85 (5), 051308. Berzi, D., Jenkins, J.T., and Larcher, M. (2010). Debris Flows: Recent Advances in Experiments and Modeling. Adv. Geophys., 52, 103 138. Berzi, D., and Jenkins, J.T. (2009). Steady inclined flows of granular-fluid mixtures. J. Fluid Mech., 641, 359 387. Berzi, D., and Jenkins, J.T. (2008). Approximate analytical solutions in a model for highly concentrated granular-fluid flows. Phys. Rev. E, 78, 011304. Berzi, D., and Jenkins, J.T. (2008). A theoretical analysis of free-surface flows of saturated granularliquid mixtures. J. Fluid Mech., 608, 393 410.