El fluido de esferas penetrables. Teoría y simulación Andrés Santos Universidad de Extremadura, Badajoz, España Colaboradores: Luis Acedo (Universidad
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1 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 Físicas (UNAM) 25 Enero
2 El fluido de esferas penetrables. Teoría y simulación Andrés Santos Universidad de Extremadura, Badajoz, España Colaboradores: Luis Acedo (Universidad de Salamanca, España) Alexander Malijevský (Institute of Chemical Technology, Praga, Rep. Checa) Santos Bravo Yuste (Universidad de Extremadura, España) Instituto de Ciencias Físicas (UNAM), Cuernavaca, Mor., México 31 Enero
3 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 Físicas (UNAM) 25 Enero
4 What is a granular material? It is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the grains collide. The constituents must be large enough such that they are not subject to thermal motion fluctuations. Thus, the lower size limit for grains is about 1 µm. Instituto de Ciencias Físicas (UNAM) 25 Enero
5 What is a granular material? Examples of granular materials would include nuts, coal, sand, rice, coffee, corn flakes, fertilizer, ball bearings, Instituto de Ciencias Físicas (UNAM) 25 Enero
6 What is a granular material? and even Saturn s rings Instituto de Ciencias Físicas (UNAM) 25 Enero
7 What is a granular material? Granular materials are commercially important in applications as diverse as pharmaceutical industry, agriculture, and energy production. They are ubiquitous in nature and are the second-most manipulated material in industry (the first one is water). Instituto de Ciencias Físicas (UNAM) 25 Enero
8 What is a granular fluid? When the granular matter is driven and energy is fed into the system (e.g., by shaking) such that the grains are not in constant contact with each other, the granular material is said to fluidize. Instituto de Ciencias Físicas (UNAM) 25 Enero
9 Granular fluids (or gases) exhibit many interesting phenomena: Granular eruptions (from University of Twente s group) Instituto de Ciencias Físicas (UNAM) 25 Enero
10 Wave patterns in a vibrated container (from A. Kudrolli s group) (Simulations by D. C. Rapaport) Instituto de Ciencias Físicas (UNAM) 25 Enero
11 Segregation in sheared flow (Simulations by D. C. Rapaport) Segregation in a rotating cylinder (Simulations by D. C. Rapaport) Instituto de Ciencias Físicas (UNAM) 25 Enero
12 Granular jet hitting a plane Particles falling on an inclined heated plane Instituto de Ciencias Físicas (UNAM) 25 Enero
13 Minimal model of a granular gas: A gas of identical smooth inelastic hard spheres Elastic collision Inelastic collision Instituto de Ciencias Físicas (UNAM) 25 Enero
14 This minimal model ignores Interstitial fluid Caltech Granular Flows Group ( Instituto de Ciencias Físicas (UNAM) 25 Enero
15 Non-constant coefficient of restitution Instituto de Ciencias Físicas (UNAM) 25 Enero
16 Non-spherical shape Instituto de Ciencias Físicas (UNAM) 25 Enero
17 Polydispersity Instituto de Ciencias Físicas (UNAM) 25 Enero
18 Roughness Instituto de Ciencias Físicas (UNAM) 25 Enero
19 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) Instituto de Ciencias Físicas (UNAM) 25 Enero
20 Outline (Collisional Collisional) energy production rates in a mixture of inelastic rough hard spheres. Application to the homogeneous oge eous free cooling state. Simple kinetic model monodisperse systems. Application to the simple shear flow. Conclusions and outlook. for Instituto de Ciencias Físicas (UNAM) 25 Enero
21 I. Collisional energy production rates in a mixture of inelastic rough hard spheres Instituto de Ciencias Físicas (UNAM) 25 Enero
22 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 Instituto de Ciencias Físicas (UNAM) 25 Enero
23 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 ij, κ i m i + m j I i 2, κ ij κ i κ j m i + m j m i (σ i /2), 2 ij i j κ i m i + κ j m j 23 Instituto de Ciencias Físicas (UNAM) 25 Enero 2011
24 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) Instituto de Ciencias Físicas (UNAM) 25 Enero
25 Elastic & smooth Instituto de Ciencias Físicas (UNAM) 25 Enero
26 Inelastic & smooth Instituto de Ciencias Físicas (UNAM) 25 Enero
27 Elastic &(perfectly) rough Instituto de Ciencias Físicas (UNAM) 25 Enero
28 Inelastic &(perfectly) rough Instituto de Ciencias Físicas (UNAM) 25 Enero
29 Elastic & smooth Inelastic & smooth Elastic & (perfectly) rough Inelastic & (perfectly) rough Instituto de Ciencias Físicas (UNAM) 25 Enero
30 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 Instituto de Ciencias Físicas (UNAM) 25 Enero
31 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 Instituto de Ciencias Físicas (UNAM) 25 Enero
32 Energy production rates. Scheme of the derivation [A.S., G.M. Kremer, V. Garzó (2010)] 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 Instituto de Ciencias Físicas (UNAM) 25 Enero
33 Two-body velocity distribution function: Ã f ij (v i, ω i ; v j, ω j ) n i n j fi rot! 3/2 (v m m mi i u) 2 i j 2T e i tr 4π 2 T tr i T tr j f (ω rot i i )f j (ω j ) (v m j u)2 j 2T j tr Molecular chaos+maxwellian approx. for translational distribution Instituto de Ciencias Físicas (UNAM) 25 Enero
34 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 Instituto de Ciencias Físicas (UNAM) 25 Enero
35 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 i,j 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 Instituto de Ciencias Físicas (UNAM) 25 Enero
36 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 Instituto de Ciencias Físicas (UNAM) 25 Enero
37 Simple application: Homogeneous Free Cooling State T t = ζt Ti tr tr Ti tr Ti rot rot Ti rot t T = ξ i ζ T, t T = ξ i ζ T t ξ tr 1 = ξ tr 2 = = ξ rot 1 = ξ rot 2 = Instituto de Ciencias Físicas (UNAM) 25 Enero
38 Inelastic smooth spheres (β=-1) T1 rot T2 rot Equip. T1 tr T2 tr Cooling Cooling Instituto de Ciencias Físicas (UNAM) 25 Enero
39 Inelastic quasi-smooth smooth spheres (β.-1) Equip. T1 rot T2 rot Cooling Equip. T1 tr T2 tr Cooling Cooling Cooling Instituto de Ciencias Físicas (UNAM) 25 Enero
40 Paradoxical ghost effect in the limit β -1 Stationary values A.S. (2011) n 2 = n 1, m 2 /m 1 =8, σ 2 /σ 1 =2, α =0.5 Instituto de Ciencias Físicas (UNAM) 25 Enero
41 Paradoxical ghost effect in the limit β -1 Time evolution A.S. (2011) n 2 = n 1, m 2 /m 1 =8, σ 2 /σ 1 =2, α =0.5 Instituto de Ciencias Físicas (UNAM) 25 Enero
42 II. A simple kinetic model for monodisperse inelastic rough hard spheres Instituto de Ciencias Físicas (UNAM) 25 Enero
43 (Cartoon by Bernhard Reischl, University of Vienna) ( ) Boltzmann equation: = X t f i (r, v i, ω i, t) + v i f i (r, v i, ω i, t) J ij [r, v i, ω i, t f i, f j ] j Inelastic+Rough collisions Instituto de Ciencias Físicas (UNAM) 25 Enero
44 Antecedents for smooth particles Boltzmann eq.: t f(r, v, t) + v f(r, v, t) = J[v f, f] Elastic collisions: [Bhatnagar-Gross-Krook (BGK) & Welander, 1954] ³ m 3/2 m(v u)2 J[v f, f] ν ν (f f 0 ), f 0 = n exp 2πT 2T Inelastic collisions: [Brey, Dufty, Santos, 1999] J[v f,f] λ(α)ν (f f 0 )+ ζ(α) 2 v [(v u)f] Instituto de Ciencias Físicas (UNAM) 25 Enero
45 Simple kinetic model for monodisperse inelastic rough Four key ingredients we want to keep: hard spheres de-spinning rate 1. ( t Ω) coll = ζ Ω Ω, Ω hωi 2. ( T tr = ξ tr T tr T tr m t ) h(v coll ξ, u)2 3 ) i Energy production rates 3. ( t T rot ) coll = ξ rot T rot, T rot I 3 hω2 i 4. R R dv 1 dω1 v 1 J 12 [v 1, ω 1 f 1,f 2 ] λ R R dv 1 dω1 v 1 J 12 [v 1, ω 1 f 1,f 2 ] λ(α, β) 1+α 2 + κ 1+κ 1+β 2, κ 4I mσ 2 Elastic smooth spheres α =1 β = 1 Instituto de Ciencias Físicas (UNAM) 25 Enero
46 Collisional rates of change ζ Ω = β 6 1+κ ν ξ tr = α 2 + κ 1+κ 1 β 2 + µ κ (1 + β)2 1 T rot (1 + X) (1 + κ) 2 T tr ν rot = 5 1+β T tr ξ κ T rot (1 T rot (1 + X) κ µ (1 1 T rot (1 + X) β) ν T tr 1 + κ + β) T tr 2 X κmσ2 Ω 2 12T rot, ν 16 5 σ2 n p πt tr /m Instituto de Ciencias Físicas (UNAM) 25 Enero
47 The kinetic model. Joint distribution t f(r, v, ω,t)+v f(r, v, ω,t)=j[v, ω f,f] J[f,f] λν (f f 0 ) ξtr v [(v u)f]+ 1 2 ω nh i o 2ζ Ω Ω + ξ rot (ω Ω) f µ mi f 0 = n 4π 2 T tr T rot 3/2 m(v u)2 exp 2T tr Iω2 2T rot rot I rot ξrot 2ζ X T h(ω Ω) 2 i = T rot (1 X), ξ = ξ ζ Ω 3 1 X Instituto de Ciencias Físicas (UNAM) 25 Enero
48 f tr (r, v,t)= A simpler version. Marginal distributions Z dω f(r, v, ω,t), f rot (r, ω,t)= 1 n Z dv f(r, v, ω,t) t f tr + v f tr = λν f tr f0 tr ξtr v (v u)f tr µ 1 n Z dvvf(r, v, ω,t) uf rot (r, ω,t) t f rot + u f rot = λν 0 f rot f0 rot + 1 nh i 2ζ Ω Ω + rot (ω Ω) roto o ζ ξ ) f 2 ω Instituto de Ciencias Físicas (UNAM) 25 Enero
49 An even simpler version. f tr = Z Translational distribution dω f, Ω = 1 n Z Z dv dωωf, T rot = I 3n Z Z dv dω ω 2 f t f tr + v f tr = λν 0 f tr f0 tr ξtr v (v u)f tr µz Z dv dω vωf nuω, I 3 Z Z dv dω vω 2 f nut rot t Ω + u Ω = ζ Ω Ω, t T rot + u T rot = ξ rot T rot Instituto de Ciencias Físicas (UNAM) 25 Enero
50 Application to simple shear flow (steady state) y =+L/2 y = L/ u x = ay - ¾ n =const ¾ ¾ T = 0 ¾ ¾ Instituto de Ciencias Físicas (UNAM) 25 Enero
51 Application to simple shear flow Translational/Rotational temperature ratio ξ rot =0 T tr 2κ +1 β = T rot κ(1 + β) Independent of α Instituto de Ciencias Físicas (UNAM) 25 Enero
52 Application to simple shear flow Shear stress Anisotropic translational temperature Scaled energy production rate bξ tr = α 2 +2κ(1 β 2 )/(2κ +1 β) 1+α + κ(1 + β)/(1 + κ) Instituto de Ciencias Físicas (UNAM) 25 Enero
53 Application to simple shear flow Universal relationship µ 2 Pxy 3 Ty tr = 2 T tr nt tr µ Ty tr 1 T tr Instituto de Ciencias Físicas (UNAM) 25 Enero
54 Application to simple shear flow Velocity distribution function R(V x,v y ) R dv z f tr (V) R dv z f tr 0 (V) Instituto de Ciencias Físicas (UNAM) 25 Enero
55 Conclusions and outlook Collisional energy production rates obtained for mixtures of inelastic rough hard spheres. Interesting non-equipartition phenomena in the homogeneous free cooling state. Paradoxical effect in the quasi-smooth smooth limit. 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 simple shear flow. Simulations planned. Derivation of the Navier-Stokes constitutive equations. Instituto de Ciencias Físicas (UNAM) 25 Enero
56 Thanks for your attention! Instituto de Ciencias Físicas (UNAM) 25 Enero
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