KINETIC THEORY OF GRANULAR SOME APPLICATIONS

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1 KINETIC THEORY OF GRANULAR BINARY MIXTURES: SOME APPLICATIONS Vicente Garzó Departamento t de Física, Universidad id dde Extremadura, Badajoz, Spain

2 OUTLINE 1. Introduction 2. Granular binary mixtures. Smooth inelastic hard spheres: Boltzmann kinetic equation 3. Navier-Stokes hydrodynamics: Some applications 4. Mixtures of inelastic rough hard spheres 5. Summary and conclusions

3 INTRODUCTION Granular systems are constituted by macroscopic grains that collide inelastically so that the total energy decreases with time. Behaviour of granular systems under many conditions exhibit a great similarity to ordinary fluids Rapid flow conditions: hydrodynamic-like y type equations.good example of a system which is inherently in non-equilibrium. Dominant transfer of momentum and energy is through binary inelastic collisions. Subtle modifications of the usual macroscopic balance equations To isolate collisional dissipation: idealized microscopic model

4 Smooth hard spheres with inelastic collisions V 12 = v 1 v 2 V 12 bσ = αv 12 bσ V 12 = v 1 v 2 Coefficient of restitution Direct collision 0 < α 1 v 1 = v (1+α) ( bσ V 12 ) bσ v 2 = v (1+α) ( bσ V 12 ) bσ

5 Momentum conservation v 1 + v = v v 2 Collisional energy change E = 1 2 m ³ v1 2 + v 2 2 v2 1 m v2 2 = 4 (1 α2 )(V 12 bσ) 2 Very simple model that captures many properties of granular flows, especially those associated with dissipation Nonequilibrium statistical mechanics tools for the (inelastic) hard sphere model

6 KINETIC DESCRIPTION One-particle velocity distribution ib ti function (vdf) f(r, v, t)drdv Average number of particles wich at t lie in r+dr and move with v+dv Boltzmann kinetic equation Dilute gas (binary collisions) Molecular chaos (no velocity correlations)

7 ( t + v ) f(v) =J [v f(t),f(t)] = σ d 1 Z Z J[f, f] dv 2 2 d bσ Θ( bσ g)( bσ g) h α 2 f(v 0 )f(v0 f(v 1)f(v 1 2 ) 1 2 ) i Scattering rules: v 0 1 = v ³ 1+α 1 ( bσ g) bσ V 12 g v 0 2 = v ³ 1+α 1 ( bσ g) bσ Differences with respect to the usual BE: Presence of α 2 in the gain term and collision rules

8 Binary granular mixtures. Boltzmann description Mechanical parameters: {m 1,m 2, σ 1, σ 2, α 11, α 22, α 12 } Extension of the BE to the multicomponent case: f i (r,, v,, t) ) ( t + v ) f i (v) = X h J ij v fi (t), f j (t) i j J ij[f = σ d 1 i, f j ] ij Z Z dv 2 2 d bσ Θ( ( bσ g)( bσ g) h α 2 i(v 0 j(v 0 i(v (v ij f 1 )f j ( 2 ) f i ( 1 )f j ( 2 ) i σ ij = σ i + σ j 2

9 Collision rules: v 0 = v ³ 1 α µ ji + ij ( bσ g) bσ v2 0 = v ³ 2 + µ ij 1+α 1 ij ( bσ g) bσ ij µ ij = m i /(m i + m j ) Hydrodynamic fields n i (r, t) = Z dvf i (r, v, t) Granular temperature X Z U(r, t) = 1 X Z (, ) dvm ivff i (r,, v,, t) ) ρ(r,t) i T (r,t)= 1 X Z dv m i (v U)2 f i (r, v,t) n(r, t) i d

10 Boltzmann collision operators J ij [f i,f j ] conserve the particle il number of each species and dh the total momentum Z dvj ij [v f i,f j ]=0, 2X 2X Z i=1 j=1 dvm i vj ij [v f i,f j ]=0 but the total energy is not conserved X2X X 2 Z mi i i=1 j=1 dvv 2 J ij[ [v f i, f j ] = dnt ζ V = v U ζ : fractional energy changes per unit time (cooling rate) )

11 Macroscopic balance equations Balance equation for the partial densities D t n i + n i U + m 1 i j i =0 Balance equation for the mean flow velocity ρdd t U + P = 0 Balance equation for the granular temperature d n (D + ζ) T +P : U+ q d 2 T X m 1 t q j =0 2 2 i j i i=1

12 D t = t + U Mass flux j i = m i Z j i i Z dvvf i (v) f i Pressure or stress tensor P = 2X i=1 m i Z dvvvf i (v) Heat flux q = 2X i=1 m i 2 Z dv V 2 V f i (v)

13 CHAPMAN-ENSKOG NORMAL SOLUTION Assumption: For long times (much longer than the mean free time) and far away from boundaries (bulk region) the system reaches a hydrodynamic regime. Normal solution f i (r, v; t) =f i (v {n i (r,t)}, U(r,t),T(r,t)}) (1) Small spatial gradients: f i = f (0) i + f (1) i +

14 Some controversy about the possibility of going from kinetic theory to hydrodynamics by using the CE method The time scale for T is set by the cooling rate instead of spatial gradients. This new time scale, T is much faster than in the usual hydrodynamic scale. Some hydrodynamic excitations decay much slower than T For large inelasticity (ζ 1 small), perhaps there were NO time scale separation between hydrodynamic and kinetic excitations: NO AGING to hydodynamics!! I assume the validity of a hydrodynamic description and compare with DSMC solutions of the BE

15 HOMOGENEOUS COOLING STATE (zeroth-order approximation) i Spatially homogeneous isotropic states Partial temperatures t f i (v, t) = X J ij [v f i (t),f j (t)] j n T = m i i i d Z dv v 2 f i (v)( ) Granular temperature T = X x i T i, x i = n i /n X i i i, i i/ Cooling rates for T i ζ = ln T = T 1 X i t i, ζ x it i ζ i i ζ = t ln T T (t) t 2 Haff s law (1983) X i iζ i

16 ζ 1 = ζ 11 + ζ 12 ζ 11 = m 1 dn 1 T 1 Z dvv 2 J 11 [f 1,f 1 ] Zero for elastic systems m 1 ζ 12 = m 1 dn 1 T 1 Z 2 Nonzero in general for elastic systems dvv 2 J 12 [f 1,f 2 ] If f i are Maxwellians at the same temperature, then ζ 12 =0 (elastic case). Detailed ed balance a whereby eby the energy e transfer between species es is balanced by energy conservation for this state

17 The analog of the balance detailed state for inelastic systems: Homogeneous cooling state (HCS) Assumption Hydrodynamic or normal state: all the time dependence of vdf occurs only through the temperature T(t) f i (v, t) =n i v d 0 (t)φ i(v/v 0 (t)) v 0 (t) T (t) v 2 0 Consequence: temperature ratio γ=t 1/T 2 must be constant (independent of time)

18 t ln γ = ζ 2 ζ 1 HCS condition: ζ1 = ζ 2 Elastic collisions: ζ 1 = ζ 2 =0, T 1 = T 2 = T Equipartition theorem for classical statistical mechanics What happens if the collisions are inelastic?

19 Well-posed mathematical problem: one has to solve the BE for the reduced distributions subject to ζ 1 =ζ 2 So far, an exact solution is not known. Approximate solution Φ i (V ) µ " Ã!# θi d/2 e θ iv 2 π 1+ c i 4 θ 2 i v 4 (d +2)v 2 + v = v/v 0, θ i = m j T, m i + m j T i j i 6= j d(d +2) 4 VG&Dufty PRE 60, 5706 (1999)

20 Time evolution of ftemperature t ratio γ. Comparison with Monte Carlo simulations Montanero&VG, Granular Matter 4, 17 (2002)

21 VG&Dufty PRE 60, 5706 (1999) Montanero&VG, Granular Matter 4, 17 (2002)

23 Comparison with molecular dynamics (MD) simulations Dahl, Hrenya,VG& Dufty, PRE 66, (2002)

24 Breakdown of energy equipartition Computer simulation studies: Barrat&Trizac GM 4, 57 (2002); Krouskop&Talbot, PRE 68, (2003); Wang et al. PRE 68, (2003); Brey et al. PRE 73,, (2006); Schroter et al. PRE 74, (2006);. Real experiments: Wildman&Parker, PRL 88, (2002); Feitosa&Menon, PRL 88, (2002). All these results confirm this new feature in granular mixtures!!

25 NAVIER-STOKES HYDRODYNAMIC EQUATIONS Previous studies: Jenkins et al. JAM 1987; PF 1989; Zamankhan, PRE 1995; Arnarson et al. PF 1998; PF 1999; PF 2004; Serero et al. JFM 2006 Limited to the quasielastic limit. They are based on the energy equipartition i i assumption Our motivation: Determination of the transport coefficients by using a kinetic theory which takes into account the effect of temperature t differences on them. NO limitationit ti to the degree of dissipation

26 Constitutive equations j 1 = m 1m 2 n D x 1 ρ D p p ρ T D T T ρ p P γβ = pδ γβ η µ U U 2 γ β + β γ U d q = T 2 D 00 x 1 L p λ T Seven transport coefficients: n D, Dp,D T, η,d 00 L, λ o

27 Transport coefficients given in terms of solutions of coupled linear integral equations. Complex mathematical problem Sonine polynomial approximation. Only leading terms are usually considered To test the accuracy of the Sonine solution: comparison with numerical solutions of the BE (DSMC) Parameter space: {m 1 /m 2, σ 1 /σ 2,x 1, α 11, α 22, α 12 }

28 What is the influence of energy nonequipartition on transport properties of the granular mixture? In particular, the pressure diffusion coefficient VG&Dufty, Phys.Fluids 14, 1476 (2002)

29 COMPARISON WITH MONTE CARLO SIMULATIONS Tracer diffusion coefficient: Diffusion of impurities in granular gas (x 1 0) VG&Montanero, PRE 69, (2004)

30 1 1 m 1 /m 2 = 1 8, σ 1/σ 2 = 1 2 VG&Vega Reyes, PRE 79, (2009) VG, Vega Reyes&Montanero, J.Fluid Mech. 623, 387 (2009)

31 Shear viscosity coefficient Montanero&VG, PRE 67, (2003)

32 Previous results support the validity of hydrodynamics y to describe granular fluids. However, some care is warranted in extending properties of ordinary fluids to those with inelastic collisions i For example, For elastic collisions, response to an external force of an impurity immersed in a granular gas (mobility coefficient) is proportional to the diffusion coefficient Einstein relation (fluctuation-dissipation theorem)

33 Einstein relation in a driven granular gas Granular gas is heated by an external driving force (thermostat) that does work on the system to compensate for the collisional loss of energy: NESS MD simulations (Srebro&Levine PRL, 2004; Barrat et al. Physica A, 2004; Shokef et al. PRE, 2006; Puglisi et al. JSTAT, 2007) have proposed a modified Einstein relation ² = D T 0 χ =1

34 MD simulations do not detect deviations of ε from unity (validity of the modified Einstein form) Kinetic theory calculations ² =1 c 0 2 ν 2 ν 4 Non-Maxwellian behavior of the reference state Second Sonine approximation

35 Modified Einstein relation is not exactly verified but. VG, JSTAT P05007 (2008) Deviations from unity are smaller than 2%!! The Einstein relation holds exactly for IMM (VG&Astillero, J. Stat. Phys. 118, 935 (2003))

36 Onsager s reciprocal relations in granular gases Language of flit j i = X j L ij ( µ j /T ) T L iq ( T/T 2 ) C p p, J q = L qq T X i L qi ( µ j /T ) T C 0 p p Ordinary fluids: L ij = L ji,l iq = L qi,c p = C 0 p =0 (Onsager s theorem)

37 NO time reversal invariance for granular fluids: Onsager s relations are not expected to apply Quantitative extent of violation: L 12 = L 21,L 1q 6= L q1,c p 6= C 0 p 6= 0 VG&Montanero&Dufty, Phys. Fluids 18, (2006)

38 Segregation g of an intruder in a granular gas Segregation driven by gravity and thermal gradients: thermal (Soret) diffusion

39 Mechanical parameters of the system {m, m 0, σ, σ 0, α, α 0 } We assume that σ 0 > σ C lli i l di i ti Collisional dissipation Experimental conditions: inhomogeneous steady state without convection (zero mass flux) and gradients along the z direction Λ z ln T = z ln(n 0 /n) Λ > 0 Λ < 0 Intruder rises with respect fluid (BNE) Intruder falls with respect fluid (RBNE)

40 Hydrodynamic description to evaluate thermal diffusion Λ a) Momentum balance equation: p = ρg p = nt z b) Constitutive equation for the mass flux of intruder: j z = m 0 D zx 0 ρ D p z p ρ T D T z T ρ p

41 Non-convecting steady state j z = 0 Mole fraction gradient in terms of gravity and thermal gradient Λ = ρ D T D p g x 0 m 0 D mg g = z T < 0

42 Order parameter for the transition BNE/RBNE θ = mt 0 m 0 T Brey, Ruiz-Montero& Moreno PRL 95, (2005); VG EPL 75, 521 (2006) If θ>1 (θ<1), then BNE (RBNE) Due to the lack of energy equipartition, criterion is rather complicated since it involves all the parameter space If T 0 =T, segregation is predicted for particles that differ only in mass!!

43 Comparison with experiments of Schroter et al. (PRE 74, (2006)) 3 m 1 /m 2 = x 2 /x 1 = (σ 1 /σ 2 ) 3, α =0.78 VG, EPL 75, 521 (2006); VG, EPJ E 29, 261 (2009) Consistent with experiments and simulations

44 MIXTURES OF INELASTIC ROUGH HARD SPHERES Ifl Influence of roughness on the properties of the system. Rotational degrees of freedom are also affected by the collisional dissipation.very complex problem Some previous results for monodisperse gases: Golshtein&Shapiro, JFM (1993); Huthmann&Zippelius, PRE (1997); Huthmann et al. PRE (1999); Zippelius, Physica A (2006).. No equipartition of energy : T tr 6= T rot

45 To the best of my knowledge, no results for multicomponent systems. Objective: Analyze the homogeneous state (HCS) ( bσ g 0 ij )= α ij( bσ g ij ), 0 < α ij 1 ( bσ g 0 ij )= β ij( bσ g ij ), 1 β ij 1 Coefficent of tangential restitution β ij = 1, α ij < 1 Smooth spheres

46 Boltzmann collision operator d 1 Z Z J ij [f i, f j ] = σ ij dv 2 Z dw 2 dσbσ Θ(σ g bσ g bσ g )(σ g 12 ) h i α ij β ij f i f j f i f j Collisional transfer of momentum and energy in terms of averages involving f and f averages involving f i and f j Simplifications: Isotropic distributions NO correlation between rotational and translational degrees of freedom

47 f = tr i)f rot i (v i, w i ) f i (v i i (w i ) Maxwell distributions Ã fi tr (v i)=n i mi 2πT tr i! 3/2 exp Ã m iv 2 i 2T tr i! Partial temperatures : T tr i = m i 3 hv2 i i T rot i = I i 3 hw2 i i

48 Partial cooling rates : ζ tr J ij [v 2 ij e i ] n i ihv 2 i i ζij rot ej ij [w 2 i ] n 2 i hw i i Parameter space: n o mi,n i, σ i, I i, α ij, β ij Study of HCS Homogenous cooling state f i (v i, t) = f i (v i, T (t))

49 t T tr i = X j X ζ tr T tr ζij tr T i tr rot i t T rot = X j X ζ rot ij T i rot Normal solution Temperature ratios are constant X j ζij tr = X j ζ rot ij T tr /T tr T rot /T rot T tr /T rot Particular case: binary mixture T 1 /T 2, T 1 /T 2, T /T

50 β = 1 α =1 Smooth spheres Elastic collisions

51 CONCLUSIONS Hydrodynamic y y description (derived from kinetic theory) appears to be a powerful tool for analysis and predictions of rapid flow gas dynamics of granular mixtures. New and interesting result: partial temperatures (which measure the mean kinetic i energy of each species) are different (breakdown of energy equipartition theorem). Not surprising feature since granular matter is inherently in non-equilibrium. Energy nonequipartition has important and new quantitiative effects on macroscopic properties (transport coefficients, Einstein and Onsager relations, segregation criterion, ) of the system

52 PERSPECTIVES Extension to inelastic rough spheres Influence of interstitial fluid on grains (suspensions) Granular hydrodynamics d for far from equilibrium i steady states

UEx (Spain) José émaría Montanero Francisco Vega Reyes

Dufty (University of Florida, USA) Christine Hrenya, y,

(Un. Federal Paraná, Brazil) Ministerio de Educación y

53 UEx (Spain) José émaría Montanero Francisco Vega Reyes Andrés Santos External collaborations James W. Dufty (University of Florida, USA) Christine Hrenya, y, Steven Dahl (University of Colorado, USA) Gilberto Kremer (Un. Federal Paraná, Brazil) Ministerio de Educación y Ciencia (Spain) Grant no. FIS Junta de Extremadura Grant No. GRU09038

54 Thanks for your attention and. HASTA LA PRÓXIMA, CUATES

BOLTZMANN KINETIC THEORY FOR INELASTIC MAXWELL MIXTURES

BOLTZMANN KINETIC THEORY FOR INELASTIC MAXWELL MIXTURES Vicente Garzó Departamento de Física, Universidad de Extremadura Badajoz, SPAIN Collaborations Antonio Astillero, Universidad de Extremadura José