^{1,a)}, J. Aaron Murray

^{2}and Francisco Vega Reyes

^{1,b)}

### Abstract

The mass flux of a low-density granular binary mixture obtained previously by solving the Boltzmann equation by means of the Chapman-Enskog method is considered further. As in the elastic case, the associated transport coefficients D, D p , and D′ are given in terms of the solutions of a set of coupled linear integral equations which are approximately solved by considering the first and second Sonine approximations. The diffusion coefficients are explicitly obtained as functions of the coefficients of restitution and the parameters of the mixture (masses, diameters, and concentration) and their expressions hold for an arbitrary number of dimensions. In order to check the accuracy of the second Sonine correction for highly inelastic collisions, the Boltzmann equation is also numerically solved by means of the direct simulation Monte Carlo (DSMC) method to determine the mutual diffusion coefficient D in some special situations (self-diffusion problem and tracer limit). The comparison with DSMC results reveals that the second Sonine approximation to D improves the predictions made from the first Sonine approximation. We also study the granular segregation driven by a uni-directional thermal gradient. The segregation criterion is obtained from the so-called thermal diffusion factor Λ, which measures the amount of segregation parallel to the temperature gradient. The factor Λ is determined here by considering the second-order Sonine forms of the diffusion coefficients and its dependence on the coefficients of restitution is widely analyzed across the parameter space of the system. The results obtained in this paper extend previous works carried out in the tracer limit (vanishing mole fraction of one of the species) by some of the authors of the present paper.

V.G. and F.V.R. acknowledge the support of the Spanish Government through Grant Nos. FIS2010-16587 and MAT2009-14351-C02-02 (F.V.R.). The first grant has been partially financed by FEDER funds and by the Junta de Extremadura (Spain) through Grant No. GR10158. J.A.M. is grateful for the funding support provided by the Department of Energy (Grant No. DE-FC26-07NT43098) and the National Science Foundation (Grant No. CBET-0318999).

I. INTRODUCTION

II. BOLTZMANN KINETIC THEORY FOR GRANULAR BINARY MIXTURES: CHAPMAN-ENSKOG METHOD

III. DIFFUSION TRANSPORT COEFFICIENTS

A. Some special limits

IV. COMPARISON WITH DSMC RESULTS

V. DEPENDENCE OF THE DIFFUSION COEFFICIENTS ON THE PARAMETERS OF THE MIXTURE

VI. THERMAL DIFFUSION SEGREGATION

VII. SUMMARY AND DISCUSSION

### Key Topics

- Diffusion
- 64.0
- Thermal diffusion
- 45.0
- Boltzmann equations
- 27.0
- Mass diffusion
- 20.0
- Self diffusion
- 18.0

##### B01D17/00

## Figures

Plot of the (reduced) self-diffusion coefficient D(α)/D(1) as a function of the coefficient of restitution α as given by the first Sonine approximation (dashed line), the second Sonine approximation (solid line), and Monte Carlo simulations (symbols). Here, D(1) is the elastic value of the self-diffusion coefficient consistently obtained in each approximation. The left panel is for hard disks (d = 2) while the right panel is for hard spheres (d = 3).

Plot of the (reduced) self-diffusion coefficient D(α)/D(1) as a function of the coefficient of restitution α as given by the first Sonine approximation (dashed line), the second Sonine approximation (solid line), and Monte Carlo simulations (symbols). Here, D(1) is the elastic value of the self-diffusion coefficient consistently obtained in each approximation. The left panel is for hard disks (d = 2) while the right panel is for hard spheres (d = 3).

Plot of the (reduced) mutual diffusion coefficient D(α12)/D(1) versus the coefficient of restitution α12 in the tracer limit (x 1 → 0) for a granular gas of hard spheres with ω = 1/2, μ = 1/4 and α22 = 0.5. The dashed and solid lines are the first and second Sonine approximations, respectively, while the symbols are the Monte Carlo simulation results. Here, D(1) is the elastic value of the mutual diffusion coefficient consistently obtained in each approximation.

Plot of the (reduced) mutual diffusion coefficient D(α12)/D(1) versus the coefficient of restitution α12 in the tracer limit (x 1 → 0) for a granular gas of hard spheres with ω = 1/2, μ = 1/4 and α22 = 0.5. The dashed and solid lines are the first and second Sonine approximations, respectively, while the symbols are the Monte Carlo simulation results. Here, D(1) is the elastic value of the mutual diffusion coefficient consistently obtained in each approximation.

Plot of the (reduced) mutual diffusion coefficient D/D(1) as a function of the mass ratio μ in the tracer limit (x 1 → 0) for a granular gas of hard spheres with ω = 1/2 and a (common) coefficient of restitution α ≡ α22 = α12 = 0.5. The dashed and solid lines are the first and second Sonine approximations, respectively, while the symbols are the Monte Carlo simulation results. Here, D(1) is the elastic value of the mutual diffusion coefficient consistently obtained in each approximation.

Plot of the (reduced) mutual diffusion coefficient D/D(1) as a function of the mass ratio μ in the tracer limit (x 1 → 0) for a granular gas of hard spheres with ω = 1/2 and a (common) coefficient of restitution α ≡ α22 = α12 = 0.5. The dashed and solid lines are the first and second Sonine approximations, respectively, while the symbols are the Monte Carlo simulation results. Here, D(1) is the elastic value of the mutual diffusion coefficient consistently obtained in each approximation.

Plot of the reduced coefficient D(α)/D(1) as a function of the (common) coefficient of restitution α for hard spheres with x 1 = 0.2, σ1 = σ2 and two different values of the mass ratio μ ≡ m 1/m 2. The solid lines correspond to the results obtained from the second Sonine approximation, the dashed lines refer to the (standard) first Sonine approximation and the dotted lines correspond to the modified first Sonine approximation. Here, D(1) is the elastic value of D consistently obtained in each approximation.

Plot of the reduced coefficient D(α)/D(1) as a function of the (common) coefficient of restitution α for hard spheres with x 1 = 0.2, σ1 = σ2 and two different values of the mass ratio μ ≡ m 1/m 2. The solid lines correspond to the results obtained from the second Sonine approximation, the dashed lines refer to the (standard) first Sonine approximation and the dotted lines correspond to the modified first Sonine approximation. Here, D(1) is the elastic value of D consistently obtained in each approximation.

Plot of the reduced coefficient D p (α)/D p (1) as a function of the (common) coefficient of restitution α for hard spheres with x 1 = 0.2, σ1 = σ2 and two different values of the mass ratio μ ≡ m 1/m 2. The solid lines correspond to the results obtained from the second Sonine approximation, the dashed lines refer to the (standard) first Sonine approximation and the dotted lines correspond to the modified first Sonine approximation. Here, D p (1) is the elastic value of D p consistently obtained in each approximation.

Plot of the reduced coefficient D p (α)/D p (1) as a function of the (common) coefficient of restitution α for hard spheres with x 1 = 0.2, σ1 = σ2 and two different values of the mass ratio μ ≡ m 1/m 2. The solid lines correspond to the results obtained from the second Sonine approximation, the dashed lines refer to the (standard) first Sonine approximation and the dotted lines correspond to the modified first Sonine approximation. Here, D p (1) is the elastic value of D p consistently obtained in each approximation.

Plot of the reduced coefficient D′*(α) as a function of the (common) coefficient of restitution α for hard spheres with x 1 = 0.2, σ1 = σ2 and two different values of the mass ratio μ ≡ m 1/m 2. The solid lines correspond to the results obtained from the second Sonine approximation, the dashed lines refer to the (standard) first Sonine approximation and the dotted lines correspond to the modified first Sonine approximation.

Plot of the reduced coefficient D′*(α) as a function of the (common) coefficient of restitution α for hard spheres with x 1 = 0.2, σ1 = σ2 and two different values of the mass ratio μ ≡ m 1/m 2. The solid lines correspond to the results obtained from the second Sonine approximation, the dashed lines refer to the (standard) first Sonine approximation and the dotted lines correspond to the modified first Sonine approximation.

The ratio of the second and first Sonine approximations D[2]/D[1] to the mutual diffusion coefficient versus the mole fraction x 1 for ω = 1, α = 0.8 and two values of the mass ratio (μ = 4 and μ = 1/3).

The ratio of the second and first Sonine approximations D[2]/D[1] to the mutual diffusion coefficient versus the mole fraction x 1 for ω = 1, α = 0.8 and two values of the mass ratio (μ = 4 and μ = 1/3).

The ratio of the second and first Sonine approximations D p [2]/D p [1] to the pressure diffusion coefficient versus the mole fraction x 1 for ω = 1, α = 0.8 and two values of the mass ratio (μ = 4 and μ = 1/3).

The ratio of the second and first Sonine approximations D p [2]/D p [1] to the pressure diffusion coefficient versus the mole fraction x 1 for ω = 1, α = 0.8 and two values of the mass ratio (μ = 4 and μ = 1/3).

The ratio of the second and first Sonine approximations D′[2]/D′[1] to the thermal diffusion coefficient versus the mole fraction x 1 for ω = 1, α = 0.8 and two values of the mass ratio (μ = 4 and μ = 1/3).

The ratio of the second and first Sonine approximations D′[2]/D′[1] to the thermal diffusion coefficient versus the mole fraction x 1 for ω = 1, α = 0.8 and two values of the mass ratio (μ = 4 and μ = 1/3).

Plot of the thermal diffusion factor Λ[2] obtained from the second Sonine approximation as a function of the diameter ratio σ1/σ2 for an ordinary binary mixture (α ij = 1) of hard spheres when both species have the same mass density (m 1/m 2 = (σ1/σ2)3). Three different values of the mole fraction are considered: (a) x 1 = 0.2, (b) x 1 = 0.5, and (c) x 1 = 0.8.

Plot of the thermal diffusion factor Λ[2] obtained from the second Sonine approximation as a function of the diameter ratio σ1/σ2 for an ordinary binary mixture (α ij = 1) of hard spheres when both species have the same mass density (m 1/m 2 = (σ1/σ2)3). Three different values of the mole fraction are considered: (a) x 1 = 0.2, (b) x 1 = 0.5, and (c) x 1 = 0.8.

Plot of the thermal diffusion factors Λ[2] and Λ[1] as a function of the mole fraction x 1 for m 1 = m 2, σ1 = σ2 and different values of the coefficients of restitution: (a) α11 = α22 = 0.5, α12 = 0.9 and (b) α11 = 0.8, α22 = 0.9, α12 = 0.7. The solid lines correspond to the second Sonine approximation Λ[2] while the dashed lines refer to the first Sonine approximation Λ[1].

Plot of the thermal diffusion factors Λ[2] and Λ[1] as a function of the mole fraction x 1 for m 1 = m 2, σ1 = σ2 and different values of the coefficients of restitution: (a) α11 = α22 = 0.5, α12 = 0.9 and (b) α11 = 0.8, α22 = 0.9, α12 = 0.7. The solid lines correspond to the second Sonine approximation Λ[2] while the dashed lines refer to the first Sonine approximation Λ[1].

Plot of the second Sonine approximation to the ratio Λ(α)/Λ(1) as a function of the (common) coefficient of restitution α for x 1 = 0.5, σ1/σ2 = 2 and three different values of the mass ratio: (a) m 1/m 2 = 4, (b) m 1/m 2 = 8, and (c) m 1/m 2 = 1/4. Here, Λ(1) refers to the elastic value of the thermal diffusion factor.

Plot of the second Sonine approximation to the ratio Λ(α)/Λ(1) as a function of the (common) coefficient of restitution α for x 1 = 0.5, σ1/σ2 = 2 and three different values of the mass ratio: (a) m 1/m 2 = 4, (b) m 1/m 2 = 8, and (c) m 1/m 2 = 1/4. Here, Λ(1) refers to the elastic value of the thermal diffusion factor.

Plot of the second Sonine approximation to the ratio Λ(α)/Λ(1) as a function of the (common) coefficient of restitution α for x 1 = 0.5, m 1/m 2 = 2 and three different values of the size ratio: (a) σ1/σ2 = 1, (b) σ1/σ2 = 3, and (c) σ1/σ2 = 5. Here, Λ(1) refers to the elastic value of the thermal diffusion factor.

Plot of the second Sonine approximation to the ratio Λ(α)/Λ(1) as a function of the (common) coefficient of restitution α for x 1 = 0.5, m 1/m 2 = 2 and three different values of the size ratio: (a) σ1/σ2 = 1, (b) σ1/σ2 = 3, and (c) σ1/σ2 = 5. Here, Λ(1) refers to the elastic value of the thermal diffusion factor.

BNE/RBNE phase diagram for inelastic hard spheres at α = 0.8 and two different values of the mole fraction x 1: x 1 = 0.1 (a) and x 1 = 0.5 (b). Points above the curves correspond to Λ > 0 (BNE) while points below the curves correspond to Λ < 0 (RBNE). The dashed and solid lines are the results obtained from the first and second Sonine approximations, respectively.

BNE/RBNE phase diagram for inelastic hard spheres at α = 0.8 and two different values of the mole fraction x 1: x 1 = 0.1 (a) and x 1 = 0.5 (b). Points above the curves correspond to Λ > 0 (BNE) while points below the curves correspond to Λ < 0 (RBNE). The dashed and solid lines are the results obtained from the first and second Sonine approximations, respectively.

BNE/RBNE phase diagram for inelastic hard spheres with x 1 = 0.7 and three different values of the (common) coefficient of restitution α. Points above the curves correspond to Λ > 0 (BNE) while points below the curves correspond to Λ < 0 (RBNE).

BNE/RBNE phase diagram for inelastic hard spheres with x 1 = 0.7 and three different values of the (common) coefficient of restitution α. Points above the curves correspond to Λ > 0 (BNE) while points below the curves correspond to Λ < 0 (RBNE).

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