Impurity in a Maxwellian unforced granular fluid

Ben‐Naim, E.; Krapivsky, P. L.

doi:10.1140/epje/i2002-10034-0

Cited by 25 publications

(37 citation statements)

References 27 publications

(53 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The form of ω i j is closer to the original model of Maxwell molecules for ordinary gas mixtures [7]. The plain vanilla Maxwell model has been previously considered by several authors [8][9][10][11] in homogeneous problems pertaining to granular mixtures.…”

mentioning

confidence: 70%

See 1 more Smart Citation

Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit

Garzó

Trizac

2011

Granular Matter

View full text Add to dashboard Cite

The homogeneous Boltzmann equation for inelastic Maxwell mixtures is considered to study the dynamics of tracer particles or impurities (solvent) immersed in a uniform granular gas (solute). The analysis is based on exact results derived for a granular binary mixture in the homogeneous cooling state (HCS) that apply for arbitrary values of the parameters of the mixture (particle masses m i , mole fractions c i , and coefficients of restitution α i j ). In the tracer limit (c 1 → 0), it is shown that the HCS supports two distinct phases that are evidenced by the corresponding value of E 1 /E, the relative contribution of the tracer species to the total energy. Defining the mass ratio μ ≡ m 1 /m 2 , there indeed exist two critical values μ (−) HCS and μ (+) HCS (which depend on the coefficients of restitution), such that E 1 /E = 0 for μ (−) HCS < μ < μ (+) HCS (disordered or normal phase), while E 1 /E = 0 for μ < μ (−) HCS and/or μ > μ (+) HCS (ordered phase).Keywords Inelastic Maxwell mixtures · Tracer limit · Non-equilibrium phase transition Granular assemblies depart from molecular systems not only from the difference of the length scales involved, but more importantly in that the interactions among constituents are dissipative [1]. In conjunction with the use of powerful experimental and numerical techniques, the application of non-equilibrium statistical mechanics to the field has yielded much progress in the last 20 years, whereas the questions

show abstract

mentioning

confidence: 70%

“…The coefficients A 22 and A 21 can be easily obtained from Eqs. (8) and (9) by change of indices 1 ↔ 2. In addition, as pointed out earlier, the temperature T (t) behaves for long times as T (t) = T (0)e λτ where λ is a nonlinear function of α i j and the parameters of the mixture.…”

mentioning

confidence: 99%

Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit

Garzó

Trizac

2011

Granular Matter

View full text Add to dashboard Cite

show abstract

“…(27). The distribution f (0) r is given by the scaling form (28) except that n r → n r (r, t) and T → T (r, t) are local quantities and v → V = v − u(r, t).…”

Section: Chapman-enskog Solution Of the Boltzmann Equation For Immmentioning

confidence: 99%

“…The functional dependence of γ 1 on x 1 can be obtained from the HCS condition (29) by using the expressions (27) for the partial cooling rates ζ…”

Section: Appendix B: Derivation Of the Transport Coefficientsmentioning

confidence: 99%

Transport Coefficients for Inelastic Maxwell Mixtures

Garzó

Astillero

2005

J Stat Phys

View full text Add to dashboard Cite

The Boltzmann equation for inelastic Maxwell models is used to determine the Navier-Stokes transport coefficients of a granular binary mixture in d dimensions. The Chapman-Enskog method is applied to solve the Boltzmann equation for states near the (local) homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the corresponding transport coefficients are identified. There are seven relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the pressure energy coefficient, and the thermal conductivity. All these coefficients are exactly obtained in terms of the coefficients of restitution and the ratios of mass, concentration, and particle sizes. The results are compared with known transport coefficients of inelastic hard spheres obtained analytically in the leading Sonine approximation and by means of Monte Carlo simulations. The comparison shows a reasonably good agreement between both interaction models for not too strong dissipation, especially in the case of the transport coefficients associated with the mass flux.

show abstract

“…Historically, it played an important role in the development of kinetic theory [32][33][34][35][36], and it remains the subject of current research [37][38][39]. Recently, it has been realized that the Maxwell model is analytically tractable even for inelastic collisions [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. In this review, we detail dynamics of uniform inelastic gases, isolated impurities in uniform gases, and binary mixtures.…”

Section: Introductionmentioning

confidence: 99%