- PDF Free Download

Size: px

Start display at page:

Download ""

Ethan Hunt
5 years ago
Views:

1 Energy production rates in fluid mixtures of inelastic rough hard spheres Andrés Santos, 1 Gilberto M. Kremer, 2 and Vicente Garzó 1 1 Universidad de Extremadura, Badajoz (Spain) 2 Universidade id d Federal do Paraná, Curitiba (Brazil)

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 Our aim To derive manageable expressions for the (partial) energy production rates associated with each degree of freedom and each binary collision

10 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

11 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

12 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

13 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)

14 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

15 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

16 1. 1st BBGKY equation. Formally exact results ξ tr ij ξ rot ij m i σij 2 = 3ni Ti tr Z Z dv i Z dω i Z dv j Z dω j dbσ dσ Θ(v ij bσ)(v ij bσ) f (2) ij (r i, v i, ω i ; r i + σ ij bσ, v j, ω j ) (v 0 i u)2 (v i u) 2 = I iσij 2 3n i T rot i i Z Z Z Z Z dv i dω i dv j dω j f (2) ij (r i, v i, ω i ; r i + σ ij bσ, v j, ω j ) ³ ωi 0 2 ω 2 i dbσ Θ(v ij bσ)(v ij bσ) Collision rules Pre-collisional two-body correlation function (at contact)

17 2. Approximation (weak inhomogeneities and/or low density) f (2) j ij (r i, v i, ω i ; r i + σ ij bσ, v j, ω j ) (2) f j ij (r i, v i, ω i ; v j, ω j ) R dbσ (2) Θ(vij bσ)(v bσ)f (2) (r i, v i, ω i ; r σ bσ, v j, ω j ) ij (r ( ij ij i, i, i; i + ij, j, j) i, v i, ω i ; v j, ω j ) R dbσ Θ(vij bσ)(v ij bσ) f (2) ξ tr ξ rot ij, ξij rot hv ijv i v ij i h(σ i ω i + σ j ω j ) (v i v j )i = L.C. hv ij (σ i ω i + σ j ω j ) 2 i Two-body averages (at contact)

18 3. Information-theory theory estimates Pair correlation function (at contact) f (2) (v m i m j f ij i, i, ω i i; ; v j j,, ω j j) ) χ ij Ã4π 2 T tr 2 t i T j tr fi rot (ω i )fj rot (ω j )! 3/2 e m i (v i u)2 e 2T tr i m j (v j u) 2 2T tr j

19 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

20 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

21 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 21

22 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 =

23 Translational/Rotational Equipartition

24 Rotational/Rotational Equipartition

25 Translational/Translational 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)

26 Conclusions and outlook Collisional energy production rates obtained for mixtures of inelastic rough hard spheres. Interesting non-equipartition phenomena in the HCS ( 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.

27 Thanks for your attention! Viña del Mar

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

29 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

30 Translational/Rotational Equipartition Weak influence of inelasticity

31 Rotational/Rotational Equipartition Same qualitative behavior for different inelasticities

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

33 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

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

35 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

36 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

37 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

38 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

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

40 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

41 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

42 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