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A Model Kinetic Equation for a A Model Kinetic Equation for a qqGranular Gas of Inelastic Granular Gas of Inelastic
Rough Hard Spheres Rough Hard Spheres
Andrés SantosAndrés SantosAndrés SantosAndrés Santos
Universidad de ExtremaduraUniversidad de Extremadura, , Badajoz Badajoz ((SpainSpain))
In In collaborationcollaboration withwithGilberto M. Gilberto M. KremerKremer andand Vicente Vicente GarzóGarzó
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MinimalMinimal modelmodel of a granular gas:of a granular gas:A gas ofA gas of identicalidentical smoothsmooth inelasticinelastic hardhardA gas of A gas of identicalidentical smoothsmooth inelasticinelastic hardhard
spheresspheres
Elastic collision Inelastic collision
http://demonstrations.wolfram.com/InelasticCollisionsOfTwoSpheres/
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ThisThis minimalminimal modelmodel ignoresignoresThisThis minimalminimal modelmodel ignores …ignores …Interstitial fluid
Caltech Granular Flows Group (http://www.its.caltech.edu/~granflow/)
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Non-constant coefficient of restitution
www.oxfordcroquet.com/tech/
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Non-spherical shape
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Polydispersity
http://www.cmt.york.ac.uk/~ajm143/nuts.html
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Roughness
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ModelModel of a granular gas:of a granular gas:A A mixturemixture of of inelasticinelastic roughrough hardhard spheresspheres
This model unveils an inherentbreakdown of energy equipartition ingranular fluids, even in homogeneous
d i iand isotropic states
Several circles(Kandinsky, 1926) Galatea of the Spheres
(Dalí 1952)Partículas y flujos. Pamplona, 20Partículas y flujos. Pamplona, 20--22 Mayo 201022 Mayo 2010
(Dalí, 1952)88
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OutlineOutline EnergyEnergy productionproduction ratesrates inin aa
OutlineOutline EnergyEnergy productionproduction ratesrates inin aa
mixturemixture ofof inelasticinelastic roughrough hardhardspheresspheres..
SimpleSimple kinetickinetic modelmodel forfor SimpleSimple kinetickinetic modelmodel forformonodispersemonodisperse systemssystems..
ApplicationApplication toto thethe simplesimple shearshearflowflowflowflow..
ConclusionsConclusions andand outlookoutlook..Partículas y flujos. Pamplona, 20Partículas y flujos. Pamplona, 20--22 Mayo 201022 Mayo 2010 99
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Mixture of Mixture of inelasticinelastic rough rough hardhard spheresspheres. . Material Material parametersparameters::
•• MassesMasses m•• MassesMasses mi•• DDiametersiameters σσii
MomentsMoments ofof inertiainertia II•• MomentsMoments ofof inertiainertia IIii•• CCoefficientsoefficients ofof normalnormal restitutionrestitution ααijij•• CoefficientsCoefficients ofof tangentialtangential restitutionrestitution βij• αij =1 forfor perfectlyperfectly elasticelastic particlesparticlesj
• βij=-1 forfor perfectlyperfectly smoothsmooth particlesparticles• βij=+1 forfor perfectlyperfectly roughrough particlesparticlesβij p yp y gg pp
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CollisionCollision rulesrulesωi
CollisionCollision rulesrules
ωj
σi2bσ
i j−σj2bσi j
vij
N t ti e (1 + ) eβ mijκij(1 + β )Notation: αij ≡ mij (1 + αij) , βij ≡ ij ij
1 + κij(1 + βij)
mij ≡mimj
, κij ≡ κiκjmi +mj
, κi ≡Ii( / )2
ijmi +mj
, ij i jκimi + κjmj
, imi(σi/2)2
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EnergyEnergy collisionalcollisional losslossEnergyEnergy collisionalcollisional lossloss
Eij =1
2miv
2i +
1
2mjv
2j +
1
2Iiω
2i +
1
2Ijω
2jj
2 i 2j j 2 i 2
j j
E0ij − Eij = −(1− α2ij)× · · ·Eij Eij (1 αij)×−(1− β2ij)× · · ·
Energy is conserved only if the spheres are • elastic (αij=1) and
either• either• perfectly smooth (βij=-1) or• perfectly rough (βij=+1)
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perfectly rough (βij 1)1212
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Elastic & smooth
http://demonstrations wolfram com/InelasticCollisionsOfTwoRoughSpheres/
Elastic & smooth
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http://demonstrations.wolfram.com/InelasticCollisionsOfTwoRoughSpheres/ 1313
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Inelastic & smooth
http://demonstrations wolfram com/InelasticCollisionsOfTwoRoughSpheres/
Inelastic & smooth
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http://demonstrations.wolfram.com/InelasticCollisionsOfTwoRoughSpheres/ 1414
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Elastic & (perfectly) rough
http://demonstrations wolfram com/InelasticCollisionsOfTwoRoughSpheres/
Elastic & (perfectly) rough
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http://demonstrations.wolfram.com/InelasticCollisionsOfTwoRoughSpheres/ 1515
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Inelastic & (perfectly) rough
http://demonstrations wolfram com/InelasticCollisionsOfTwoRoughSpheres/
Inelastic & (perfectly) rough
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http://demonstrations.wolfram.com/InelasticCollisionsOfTwoRoughSpheres/ 1616
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http://demonstrations.wolfram.com/InelasticCollisionsOfTwoRoughSpheres/
Elastic & smooth Inelastic & smooth
Elastic & (perfectly) rough Inelastic & (perfectly) rough
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PartialPartial (granular)(granular) temperaturestemperaturesPartialPartial (granular) (granular) temperaturestemperatures
Translational temperatures: T tri =mi h(vi − u)2i
t Ii 2 miκi 2 2
Translational temperatures: Ti 3h(vi u) i
Rotational temperatures: T roti =Ii3hω2i i =
miκi12
σ2i hω2i i
Total temperature: T =X ni
2n
¡T tri + T
roti
¢Xi2n
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CollisionalCollisional ratesrates of of changechange forfortemperaturestemperatures
Energy production rates:
ξtr1µ∂T tri
¶ξtr
Xξtrξtri = −T tri
µi
∂t
¶coll
, ξtri =Xj
ξtrij
1µ∂T rot
¶ Binary collisions
ξroti = − 1
T roti
µ∂T roti
∂t
¶coll
, ξroti =Xj
ξrotij
Net cooling rate:
1µ∂T¶ X ni
ζ = − 1T
µ∂T
∂t
¶coll
, ζ =Xi
ni2nT
¡ξtri T
tri + ξroti T roti
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Energy production rates. Scheme of the derivation(arXiv:0910 5614)
Collision rules(arXiv:0910.5614)
1st BBGKY equation
1. Formally exact expressions
f(2) → f
(2) ≡ hf (2)iΩ
2. Two-body averages
fij → fij ≡ hfij iΩ
2. Two body averages
Information-theory estimate of f(2)ij
3. Final results
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Final Final resultsresults. . EnergyEnergy productionproduction ratesrates
ξtrij =νijmiT tri
"2³eαij + eβij´T tri −
³eα2ij + eβ2ij´µT trimi+T trjmj
¶
−eβ2ij µ T roti
miκi+T rotj
mjκj
¶#
ξrotij =νij
m κ T roteβij"2T roti − eβij µT tri
m+T trjm
+T roti
m κ+T rotj
m κ
¶#j miκiT roti
j
"j
µmi mj miκi mjκj
¶#
4√2
sT tr T tr
νij ≡4√2π
3χijnjσ
2ij
sT trimi
+T trjmj
Effective collision frequencies
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Final Final resultsresults. . Net Net coolingcooling raterateζ =
Xi
ni2nT
¡ξtri T
tri + ξroti T roti
¢i
ζ =X niνij
4nT
mimj
mi +mj
·(1− α2ij)
µT trimi
+T trjmj
¶i,j
4nT mi +mj
· µmi mj
¶
+κij
(1 β2 )
µT tri +
T trj+T roti +
T rotj
¶¸+1 + κij
(1− βij)
µmi
+mj
+miκi
+mjκj
¶¸
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DecompositionDecompositionDecompositionDecompositionE d ti t E i titi t + C li tEnergy production rates = Equipartition rates + Cooling rates
Net cooling rate = Σ Cooling rates
T tr T tr T rotT rotEquip. Equip. Equip.
g g
T trj T tri T rotjT roti
q p q p
Cooling Coolingg g
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Simple Simple applicationapplication::HomogeneousHomogeneous Free Free CoolingCooling StateState
∂T
∂t= −ζT
∂t
∂ T tri =¡ξtr ζ
¢ T tri ∂ T roti =¡ξrot ζ
¢ T roti
∂t T= −
¡ξi − ζ
¢T,
∂t T= −
¡ξi − ζ
¢T
t→∞⇒ ξtr1 = ξtr2 = · · · = ξrot1 = ξrot2 = · · ·
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InelasticInelastic smoothsmooth spheresspheres ((ββ==--1)1)InelasticInelastic smoothsmooth spheresspheres ((ββ==--1)1)
T rot1 T rot2
T tr T trEquip.
T tr1 T tr2
CoolingCooling
q p
gg
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InelasticInelastic quasiquasi--smoothsmooth spheresspheres((ββ..--1)1)
T rot1 T rot2Equip.
T tr T trEquip.
Cooling Cooling
T tr1 T tr2q p
Cooling Coolingg g
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RotationalRotational//RotationalRotational..Time Time evolutionevolution
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RotationalRotational//RotationalRotational..Time Time evolutionevolution
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TranslationalTranslational//TranslationalTranslational..Time Time evolutionevolution
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A simpleA simple kinetickinetic modelmodelA simple A simple kinetickinetic modelmodelforfor monodispersemonodisperseforfor monodispersemonodisperse
inelasticinelastic roughrough hardhardinelasticinelastic rough rough hardhardspheresspheresspheresspheres
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(Cartoon by BernhardReischl, University of
Vienna)
(1844-1906)
∂tfi(r,vi,ωi, t) + vi ·∇fi(r,vi,ωi, t) =X
Jij [r,vi,ωi, t|fi, fj ]
Boltzmann equation:
∂tfi(r,vi,ωi, t) + vi ∇fi(r,vi,ωi, t)Xj
Jij [r,vi,ωi, t|fi, fj ]
Binary collisions
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AntecedentsAntecedents forfor smoothsmooth particlesparticlesAntecedentsAntecedents forfor smoothsmooth particlesparticlesBoltzmann eq : ∂tf(r v t) + v ·∇f(r v t) = J [r v t|f f ]
Elastic collisions:
Boltzmann eq.: ∂tf(r,v, t) + v ∇f(r,v, t) J [r,v, t|f, f ]
[Bhatnagar-Gross-Krook (BGK) & Welander, 1954]
J [f, f ]→ −ν (f − f0) , f0 = n³ m ´3/2
exp
·−m(v− u)
2¸
J [f, f ]→ ν (f f0) , f0 n³2πT
´exp
·2T
¸
Inelastic collisions:[Brey, Dufty, Santos, 1999]
J [f, f ]→ −ν(α) (f − f0) +ζ(α)
2
∂
∂v· [(v − u)f ]
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Simple Simple kinetickinetic modelmodel forfordidi i l tii l ti hhmonodispersemonodisperse inelasticinelastic rough rough
hardhard spheressphereshardhard spheresspheresThree key ingredients we want to keep:
1. (∂tTtr)coll = −ξtrT tr
2. (∂tTrot)coll = −ξrotT rot2. (∂tT )coll ξ T
3.Rdvi
Rdωi viJij [vi,ωi|fi, fj ] = λij
Rdvi
Rdωi viJij [vi,ωi|fi, fj ]
¯αij = 1¯ij
βij = −1
λij ≡ 1+αij2 +
κij1+κij
1+βij2
Elastic smooth spheresElastic smooth spheres
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TheThe kinetickinetic modelmodel.. JointJoint distributiondistribution∂tf(r,v,ω, t) + v ·∇f(r,v,ω, t) = J [r,v,ω, t|f, f ]
TheThe kinetickinetic modelmodel. . JointJoint distributiondistribution
J [f, f ] → −λν0 (f − f0)t t
+ ξtr
2∂∂v · [(v − u)f ] +
ξrot
2∂∂ω · (ωf)
λ ≡ 1 + α
2+
κ
1 + κ
1 + β
2, ν0 =
16√π
5nσ2
pT tr/m
2 1 + κ 2 5
f0 = n
µmI
¶3/2exp
·−m(v − u)
2
− Iω2¸
f0 = n
µ4π2T trT rot
¶exp
·2T tr 2T rot
¸
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A A simplersimpler versionversion. . Marginal Marginal distributionsdistributions
f tr(r,v, t) =
Zdω f(r,v,ω, t), f rot(r,ω, t) =
1
n
Zdv f(r,v,ω, t)
∂tftr + v ·∇f tr = −λν0
¡f tr − f tr0
¢+
ξtr
2
∂
∂v·£(v − u)f tr
¤
t
µ1
n
Zdv vf(r,v,ω, t)→ uf rot(r,ω, t)
¶∂tf
rot + u ·∇f rot = −λν0¡f rot − f rot0
¢+
ξrot
2
∂
∂ω·¡ωf rot
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AnAn eveneven simplersimpler versionversion. . TranslationalTranslational distributiondistribution
f tr(r,v, t) =
Zdω f(r,v,ω, t), T rot(r, t) =
I
3n
Zdv
Zdω ω2f(r,v,ω, t)
∂tftr + v ·∇f tr = −λν0
¡f tr − f tr0
¢+
ξtr
2
∂
∂v·£(v − u)f tr
¤
µI
3n
Zdv v
Zdω ω2f(r,v,ω, t)→ uT rot
¶∂tT
rot + u ·∇T rot = −ξrotT rot
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ApplicationApplication toto simplesimple shearshear flowflowApplicationApplication toto simple simple shearshear flowflow((steadysteady statestate))(( yy ))
¡¡y = +L/2 -
¡¡¡¡
ux = ay----
¡¡¡ x y
n = const
∇T 0
-¾
¡¡¡¡ ∇T = 0¾
¾¾¾¡y = −L/2¾
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ApplicationApplication toto simple simple shearshear flowflowpppp ppTranslationalTranslational//RotationalRotational temperaturetemperature ratioratio
ξrot = 0⇒ T tr
T rot=2κ+ 1− β
κ(1 + β)
Independent of α
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ApplicationApplication toto simple simple shearshear flowflowpppp ppShearShear stressstress
Pxy
q3bξtr/2
xy
nT tr= −
q/
1 + bξtrbξtr = 5
6
1− α2 + 2κ(1− β2)/(2κ+ 1− β)
1 + α+ κ(1 + β)/(1 + κ)
S l dScaled energyproduction rate
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ApplicationApplication toto simple simple shearshear flowflowpppp ppAnisotropicAnisotropic translationaltranslational temperaturestemperatures
T tryT tr
=T trzT tr
=1
1 + bξtrT trxT t
= 3− 2Ttry
T tT tr T tr
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ApplicationApplication toto simplesimple shearshear flowflowApplicationApplication toto simple simple shearshear flowflow
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ApplicationApplication toto simple simple shearshear flowflow“Universal” “Universal” relationshiprelationship
µPxy
¶2=3 T try
µ1
T try¶µ
nT tr
¶=2 T tr
µ1−
T tr
¶
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Conclusions and Conclusions and outlookoutlook CollisionalCollisional energyenergy productionproduction ratesrates obtainedobtained forfor
mixturesmixtures ofof inelasticinelastic roughrough hardhard spheresspheresmixturesmixtures ofof inelasticinelastic roughrough hardhard spheresspheres.. InterestingInteresting nonnon--equipartitionequipartition phenomenaphenomena inin thethe
homogeneoushomogeneous freefree coolingcooling statestate.. ParadoxicalParadoxical effecteffecthomogeneoushomogeneous freefree coolingcooling statestate.. ParadoxicalParadoxical effecteffectinin thethe quasiquasi--smoothsmooth limitlimit..
SimulationsSimulations plannedplanned toto testtest thethe theoreticaltheoreticalpppredictionspredictions..
ProposalProposal ofof aa simplesimple modelmodel kinetickinetic equationequation forfor thethesinglesingle--componentcomponent casecase..
SolutionSolution ofof thethe aboveabove modelmodel inin thethe simplesimple shearshearflfl Si l tiSi l ti l dl dflowflow.. SimulationsSimulations plannedplanned..
DerivationDerivation ofof thethe NavierNavier--StokesStokes constitutiveconstitutiveequationsequations
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equationsequations..4343
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Thanks for your attention!Thanks for your attention!Thanks for your attention!Thanks for your attention!
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