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Julius Wells
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1 Free cooling of a binary granular fluid of inelastic rough hard spheres Andrés Santos Universidad de Extremadura, Badajoz (Spain) In collaboration with Gilberto M. Kremer and Vicente Garzó

2 Minimal model of a granular gas: A gas of identical smooth inelastic hard spheres Elastic collision Inelastic collision

3 This minimal model ignores Interstitial fluid Caltech Granular Flows Group (

4 Non-constant coefficient of restitution

5 Non-spherical shape

6 Polydispersity

7 Roughness

8 Model of a granular gas: A mixture of inelastic rough hard spheres This model unveils an inherent breakdown of energy equipartition in granular fluids, even in homogeneous and isotropici states Several circles (Kandinsky, 1926) Galatea of the Spheres (Dalí, 1952)

9 Two-fold aim To derive manageable expressions for the (partial) energy production rates associated with each degree of freedom and each binary collision. To apply the results to the free cooling state.

10 Material parameters: Masses m i Diameters σ i Moments of inertia I i Coefficients of normal restitution α ij Coefficients of tangential restitution β ij α ij =1 for perfectly elastic particles β ij =-1 for perfectly smooth particles β ij =+1 for perfectly rough particles

11 Collision rules ω i ω j i σ i 2 bσ σ j 2 bσ j v ij ij ij ti e e m ij κ ij Notation: α ij m ij (1 + α ij ), β ij (1 + β ij ) 1+κ ij m m im j m i + m j ij, κ ij κ iκ κ m i + m j, i j κ i m i + κ j m j I i 2 m i (σ i /2) 2

12 Energy collisional loss E ij = 1 m iv m jv I iω i j j i I jω j j E 0 = (1 ij) 2 ij E ij α ) (1 β 2 ij ) Energy is conserved only if the spheres are elastic (α ij =1) and either perfectly smooth (β ij =-1) or perfectly rough (β ij =+1)

13 Partial (granular) temperatures Translational temperatures: T tr = mi 2 i 3 h(v i u) i 2 i κ 2 2 Rotational temperatures: Ti rot = I i 3 hω2 i i = m iκ i 12 σ2 i hω2 i i i Total temperature: T = X n i 2n i T tr i + Ti rot

14 Collisional rates of change for temperatures Energy production rates: 1 µ T tr ξi tr i = T tr, ξi tr = X ξij tr i t coll j Binary collisions ξi rot = 1 µ T rot i Ti rot, ξi rot = X ξij rot t coll j Net cooling rate: ni ζ = 1 µ T, ζ = X n i T t coll 2nT i ξ tr i Ti tr + ξ rot i Ti rot

15 Scheme of the derivation (arxiv: ) Collision rules 1st BBGKY equation 1. Formally exact expressions f (2) f (2) (2) ij f ij hf ij i Ω 2. Two-body averages Information-theory estimate of f (2) ij 3. Final results

16 Final results. Energy production rates " ξij tr = ν ij m i Ti tr 2 ³eα ij + β ij e µ β e T ij 2 rot i + T j rot m i κ i m j κ j T tr i # ³ eα ij 2 + β e µt ij 2 tr i + T j tr m i m j ξij rot = " µ ν ij e β ij 2Ti rot β e T tr i ij + T j tr + T i rot + T j rot m i κ i T rot m i m j m i κ i m j κ j i # s ν ij 4 2π χ ij n j σij 2 Ti tr + T j tr 3 m i m j Effective collision frequencies

17 Final results. Net cooling rate ζ = X i n i 2nT ξ tr i Ti tr + ξ rot i Ti rot ζ = X n i ν ij m i m j 4nT m i + m j ij κ ij + (1 β 2 ij ) 1+κij µ T (1 αij) 2 tr i + T j tr m i m j µ T tr i T j tr T i rot T j rot m i mj mi κ i mj κ j

18 Decomposition Energy production rates = Equipartition rates + Cooling rates Net cooling rate = Σ Cooling rates Equip. Equip. Equip. T tr T tr T rot j Ti tr Ti rot j Cooling Cooling

19 Simple application: Homogeneous Cooling State (HCS) The HCS is Spatially homogeneous Isotropic Undriven Freely cooling T t = ζt ζ Ti tr t T = ξi tr ζ Ti tr T, Ti rot t T = ξi rot ζ Ti rot T t ξ tr 1 = ξ tr 2 = = ξ rot 1 = ξ rot 2 =

20 Rotational/Rotational. Asymptotic values (t ) Equipartition

21 Translational/Rotational. Asymptotic values (t ) Equipartition

22 Translational/Translational. Asymptotic values (t ) Pure smooth spheres (Garzó&Dufty, 1999) Equipartition Ghost effect: A tiny amount of roughness has dramatic effects on the temperature ratio (enhancement of non-equipartition)

23 Smooth spheres (β=-1) 1+α T1 rot T2 rot 1 α 2 T1 tr T2 tr

24 Quasi-smooth smooth spheres (β.-1) 1+α T1 rot T2 rot 1 α 2 T1 tr T2 tr 1 + β 1 β 2

25 Rotational/Rotational. Time evolution

26 Rotational/Rotational. Time evolution

27 Translational/Translational. Time evolution

28 Conclusions and outlook Collisional energy production rates obtained for mixtures of inelastic rough hard spheres. Interesting non-equipartition phenomena in the HCS (paradoxical ghost effect). Simulations planned to test the theoretical predictions. Proposal of asimple model kinetic equation for the single-component case. Solution of the above model in the uniform shear flow. Simulations planned. Derivation of the Navier-Stokes constitutive equations.

29 Thanks for your attention!

30 Translational/Translational Equipartition Pure smooth spheres Ghost effect: A tiny amount of roughness has dramatic effects on the temperature ratio

31 Simple application: White-noise heating (steady state) T tr 1 ξ tr 1 = T tr 2 ξ tr 2 = ξ rot 1 = ξ2 rot = = 0

32 Translational/Rotational Equipartition Weak influence of inelasticity

33 Rotational/Rotational Equipartition Same qualitative behavior for different inelasticities

34 Translational/Translational Pure smooth spheres (Barrat&Trizac, 2002) Equipartition No ghost effect! (steady state)

35 Locus of equipartition: Under which conditions does equipartition hold? Coefficients of normal restitution α 11 = α 12 = α 22 = α Coefficients of tangential restitution β 11 = β 12 = β 22 = β Inertia-moment parameters κ 1 = κ 2 = κ Size ratio σ 1 /σ 2 = free Mass ratio m 1 /m 2 =free Mole fraction n 1 /(n 1 + n 2 )=free

36 First condition: ( 1 α 2 = 1 κ 1+κ (1 β2 HCS β = ±1 White noise

37 Second condition: n 1 n 2 = q m1 +m 2 q q m 2 m1 +m σ 2 m2 2 12q m 1 σ2 2 2m 2 q q σ12q 2 m1 m 2 σ1 2 m1 +m 2 2m 1

38 Simple kinetic model for monodisperse inelastic rough hard spheres Three key ingredients we want to keep: 1. ( t T tr ) coll = ξ tr T tr 2. ( T rot = ξ rot T rot t ) coll ξ 3. R dvi R dωi v i J ij [v i, ω i f i,f j ] = 1+α ij +β ij κ ij /(1+κ ij ) 2 R R dv i dωi v i J ij [v i, ω i f i,f j ] α αij = 1 β ij = 1 Elastic smooth spheres

39 t f(r, v, ω,t)+v f(r, v, ω,t)=j[r, v, ω,t f,f] J[f,f] λν 0 (f f 0 ) + ξtr 2 v [(v u)f]+ξrot 2 ω (ωf) λ 1+α + κ 1+β, ν 0 = 16 π nσ 2p T tr /m κ 2 5 µ 3/2 mi m(v f u)2 Iω2 0 = n 4π 2 T tr T rot exp 2T tr 2T rot

40 An even simpler version t f tr (r, v,t)+v f tr (r, v,t) = λν 0 f tr (r, v,t) f tr 0 (r, v,t) + ξtr 2 v (v u)f tr (r, v,t) t T rot + u T rot = ξ rot T rot

41 Application to simple shear flow y =+L/2 y = L/ u x = ay - ¾ n =const ¾ ¾ T = 0 ¾ ¾

42 Application to simple shear flow Translational/Rotational temperature ratio ξ rot =0 T rot T tr = κ(1 + β) 2κ +1 β Independent of α

43 Application to simple shear flow Shear stress P q3ξ btr /2 xy nt = tr 1+ ξ btr bξ tr = α 2 +2κ(1 β 2 )/(2κ +1 β) 1+α + κ(1 + β)/(1 + κ) Scaled thermal rate

44 Application to simple shear flow Anisotropic translational temperatures T tr x T tr = 1+3b ξ tr 1+ b ξ tr T tr y T z tr 1 T = tr T = tr 1+ ξ btr

El fluido de esferas penetrables. Teoría y simulación Andrés Santos Universidad de Extremadura, Badajoz, España Colaboradores: Luis Acedo (Universidad

El fluido de esferas penetrables. Teoría y simulación Andrés Santos Universidad de Extremadura, Badajoz, España Colaboradores: Luis Acedo (Universidad Influence of particle roughness on some properties of granular gases Andrés Santos Universidad de Extremadura, Badajoz (Spain) In collaboration with Gilberto M. Kremer and Vicente Garzó Instituto de Ciencias