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Diffusion transport coefficients for granular binary mixtures at low density: Thermal diffusion segregation
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10.1063/1.4800775
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2 Department of Chemical and Biological Engineering, University of Colorado, Boulder, Colorado 80309, USA
Phys. Fluids 25, 043302 (2013)
/content/aip/journal/pof2/25/4/10.1063/1.4800775
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/4/10.1063/1.4800775
View: Figures

## Figures

FIG. 1.

Plot of the (reduced) self-diffusion coefficient (α)/(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, (1) is the elastic value of the self-diffusion coefficient consistently obtained in each approximation. The left panel is for hard disks ( = 2) while the right panel is for hard spheres ( = 3).

FIG. 2.

Plot of the (reduced) mutual diffusion coefficient )/(1) versus the coefficient of restitution α in the tracer limit ( → 0) for a granular gas of hard spheres with ω = 1/2, μ = 1/4 and α = 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, (1) is the elastic value of the mutual diffusion coefficient consistently obtained in each approximation.

FIG. 3.

Plot of the (reduced) mutual diffusion coefficient /(1) as a function of the mass ratio μ in the tracer limit ( → 0) for a granular gas of hard spheres with ω = 1/2 and a (common) coefficient of restitution α ≡ α = α = 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, (1) is the elastic value of the mutual diffusion coefficient consistently obtained in each approximation.

FIG. 4.

Plot of the reduced coefficient (α)/(1) as a function of the (common) coefficient of restitution α for hard spheres with = 0.2, σ = σ and two different values of the mass ratio μ ≡ / . 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, (1) is the elastic value of consistently obtained in each approximation.

FIG. 5.

Plot of the reduced coefficient (α)/ (1) as a function of the (common) coefficient of restitution α for hard spheres with = 0.2, σ = σ and two different values of the mass ratio μ ≡ / . 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, (1) is the elastic value of consistently obtained in each approximation.

FIG. 6.

Plot of the reduced coefficient ′*(α) as a function of the (common) coefficient of restitution α for hard spheres with = 0.2, σ = σ and two different values of the mass ratio μ ≡ / . 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.

FIG. 7.

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

FIG. 8.

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

FIG. 9.

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

FIG. 10.

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

FIG. 11.

Plot of the thermal diffusion factors Λ[2] and Λ[1] as a function of the mole fraction for = , σ = σ and different values of the coefficients of restitution: (a) α = α = 0.5, α = 0.9 and (b) α = 0.8, α = 0.9, α = 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].

FIG. 12.

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

FIG. 13.

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

FIG. 14.

BNE/RBNE phase diagram for inelastic hard spheres at α = 0.8 and two different values of the mole fraction : = 0.1 (a) and = 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.

FIG. 15.

BNE/RBNE phase diagram for inelastic hard spheres with = 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).

/content/aip/journal/pof2/25/4/10.1063/1.4800775
2013-04-16
2014-04-24

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