Liquid money or hard cash? Drowning into granular material

Liquid&money&or&hard&cash?&! Drowning(into(granular(material.! ! Franco!Bagnoli,!! Dept.!of!Physics!and!Astronomy!and!Center!for!the!Study!of!Complex!Dynamics,!University!of! Florence,!Italy! Via!G.!Sansone,!1!50019!Sesto!Fiorentino!(FI)!Italy! franco.bagnoli@unifi.it! ! In!British!English,!the!term!“hard!cash”!refers!to!the!form!of!payment!using!coins!or!bill,! rather!than!cheques!or!credit!or!money!transfer.!In!American!English,!it!is!often!prefixed!by! the!adjective!“cold”.!On!the!contrary,!in!Italian!the!equivalent!expression!“denaro!liquido”!can! be!literary!translated!as!“liquid!money”.!In!French!the!expression!is!equivalent!with!the! additional!factor,!with!respect!to!the!rest!of!this!discussion,!that!money!becomes!“argent”.!! We!have!therefore!two!very!different!points!of!view:!Is!money!hard!and!cold,!or!rather!liquid! and!“jingling”!(“moneta!sonante”)?!As!usual,!we!shall!investigate!this!topic!starting!from!some! comics!about!the!duck!family.!We!all!know!that!Uncle!Scrooge!loves!to!dive!into!his!money! “like!a!porpoise”![1],!so!for!him,!money!is!indeed!liquid.!! ! Figure!1.!Swimming!into!a!sea!of!coins![1]! However,!near!the!end!of!the!same!story,!the!Beagle!Boys!succeeded!in!stealing!his!money,! but!they!are!lured!in!enjoying!it!“like!the!tightwad!used!to!do”.!!! ! Figure!2.!Luring!the!Beagle!Boys![1].! ! The!burglars!try!to!plunge!into!the!mass!of!coins!but,!as!expected,!they!crash!against!a!solid! body.! ! ! Figure!3.!Cold,!hard!money![1]! Indeed,!a!granular!material!can!behave!both!as!a!solid!and!as!a!liquid,!depending!on!the! density!of!its!kinetic!energy.!We!all!know!that!transferring!a!granular!material!like!rice,!sugar! or!coffee!from!a!container!to!another!is!not!a!trivial!task.!These!materials!behave!quite! differently!from!liquids:!if!the!container!is!inclined,!the!mass!of!grains!behaves!at!first!like!a! rigid!body!and!stays!compact.!Passing!a!critical!angle,!it!suddenly!(partially)!liquefies!and! originates!an!avalanche.!However,!even!without!an!academic!knowledge,!we!all!know!what!to! do!to!pour!a!granular!material!from!one!container!into!another:!fluidify!it!by!continuously! shaking.! The!cohesion!of!a!granular!material!is!due!to!the!same!kind!of!forces!that!originates!friction,! i.e.,!a!stickZandZslip!dynamics.!It!can!be!visualized!using!two!combs.! !! Figure!4.!StickBandBslip!dynamics!of!combs![2].! The!usual!distinction!from!static!and!dynamic!friction!law!is!sufficient!to!qualitatively!explain! the!behavior!of!a!granular!material.! ! Figure!5.!Static!and!kinetic!friction!experiment![3].! When!the!static!friction!is!overcome,!the!grain!start!moving,!and!the!dynamic!friction!is! unable!to!bring!the!system!at!rest,!when!the!external!force!is!kept!constant.!So,!if!one!provides! continuous!energy,!the!granular!material!behaves!like!a!sort!of!viscous!fluid,!although!the! viscosity!is!not!proportional!to!the!velocity,!and!has!a!nonlinear!transition!to!infinite!viscosity! for!low!enough!velocity.!The!system!dissipates!energy,!so!we!have!a!different!scenario!with! respect!to!a!molecular!fluid.!We!can!speak!of!two!different!temperatures:!the!microscopic!and! the!macroscopic!one,!the!latter!roughly!identified!by!the!macroscopic!kinetic!energy!of!grains,! even!though!the!velocity!distribution!of!a!granular!material!is!in!general!not!Gaussian,!due!to! the!continuous!conversion!of!the!macroscopic!energy!into!the!microscopic!one.!The!shape!of! the!velocity!distribution!is!not!universal:!it!depends!on!how!energy!is!injected!(for!instance,! by!vibrating!the!fluid!with!a!given!amplitude!and!velocity!shape)!and!how!it!is!dissipated!(by! collision,!by!viscous!drag!from!air,!and!so!on)!![4].!! Another!aspect!of!granular!materials!is!the!arching!phenomena.!The!distribution!of!stresses! inside!a!granular!system!is!not!isotropic.!Due!to!the!nonlinear!behavior!of!the!friction! coefficient,!the!material!selfZorganizes!so!that!all!stresses!above!threshold!are!removed,!and! what!is!left!are!percolating!chains!of!forces.! ! Figure!6.!Force!chains!in!a!discrete!model!of!a!sand!heap!–!arching!effect![5].! This!is!the!origin!of!!the!stability!of!sand!piles,!and!of!the!fact!that!granular!materials!do!not! obey!to!the!Stevino’s!law:!the!pressure!at!the!bottom!of!a!silo!is!not!what!can!be!expected:!the! pressure!stops!increasing!once!the!grain!is!higher!than!roughly!the!silo!diameter![6].!For! higher!levels,!the!load!is!redirected!to!the!walls,!and!this!can!make!the!silos!to!break.!Indeed,! when!loading!silos,!care!is!taken!to!let!the!granular!material!keep!the!liquid!state!long!enough,! using!appropriate!geometries.! The!arching!phenomenon!is!not!always!deleterious:!hourglasses!!behaves!differently!from! water!clocks!and!produce!a!stationary!flow!of!material!due!to!the!fact!that!the!arching!makes! the!pressure!at!bottom!independent!of!the!sand!load.!! However,!arching!can!also!clog!granular!material!in!silos.!! ! Figure!7.!Jamming!during!discharge!of!granular!material!due!to!arch!formation![7]. This!inconvenient!

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