8 research outputs found

### A note on the violation of the Einstein relation in a driven moderately dense granular gas

The Einstein relation for a driven moderately dense granular gas in
$d$-dimensions is analyzed in the context of the Enskog kinetic equation. The
Enskog equation neglects velocity correlations but retains spatial correlations
arising from volume exclusion effects. As expected, there is a breakdown of the
Einstein relation $\epsilon=D/(T_0\mu)\neq 1$ relating diffusion $D$ and
mobility $\mu$, $T_0$ being the temperature of the impurity. The kinetic theory
results also show that the violation of the Einstein relation is only due to
the strong non-Maxwellian behavior of the reference state of the impurity
particles. The deviation of $\epsilon$ from unity becomes more significant as
the solid volume fraction and the inelasticity increase, especially when the
system is driven by the action of a Gaussian thermostat. This conclusion
qualitatively agrees with some recent simulations of dense gases [Puglisi {\em
et al.}, 2007 {\em J. Stat. Mech.} P08016], although the deviations observed in
computer simulations are more important than those obtained here from the
Enskog kinetic theory. Possible reasons for the quantitative discrepancies
between theory and simulations are discussed.Comment: 6 figure

### Non-equilibrium phase transition in a sheared granular mixture

The dynamics of an impurity (or tracer particle) immersed in a dilute
granular gas under uniform shear flow is investigated. A non-equilibrium phase
transition is identified from an exact solution of the inelastic Boltzmann
equation for a granular binary mixture in the tracer limit, where the impurity
carries either a vanishing (disordered phase) or a finite (ordered phase)
fraction of the total kinetic energy of the system. In the disordered phase,
the granular temperature ratio (impurity "temperature" over that of the host
fluid) is finite, while it diverges in the ordered phase. To correctly capture
this extreme violation of energy equipartition, we show that the picture of an
impurity enslaved to the host fluid is insufficient

### Third and fourth degree collisional moments for inelastic Maxwell models

The third and fourth degree collisional moments for $d$-dimensional inelastic
Maxwell models are exactly evaluated in terms of the velocity moments, with
explicit expressions for the associated eigenvalues and cross coefficients as
functions of the coefficient of normal restitution. The results are applied to
the analysis of the time evolution of the moments (scaled with the thermal
speed) in the free cooling problem. It is observed that the characteristic
relaxation time toward the homogeneous cooling state decreases as the
anisotropy of the corresponding moment increases. In particular, in contrast to
what happens in the one-dimensional case, all the anisotropic moments of degree
equal to or less than four vanish in the homogeneous cooling state for $d\geq
2$.Comment: 15 pages, 3 figures; v2: addition of two new reference

### Hydrodynamics of inelastic Maxwell models

An overview of recent results pertaining to the hydrodynamic description
(both Newtonian and non-Newtonian) of granular gases described by the Boltzmann
equation for inelastic Maxwell models is presented. The use of this
mathematical model allows us to get exact results for different problems.
First, the Navier--Stokes constitutive equations with explicit expressions for
the corresponding transport coefficients are derived by applying the
Chapman--Enskog method to inelastic gases. Second, the non-Newtonian
rheological properties in the uniform shear flow (USF) are obtained in the
steady state as well as in the transient unsteady regime. Next, an exact
solution for a special class of Couette flows characterized by a uniform heat
flux is worked out. This solution shares the same rheological properties as the
USF and, additionally, two generalized transport coefficients associated with
the heat flux vector can be identified. Finally, the problem of small spatial
perturbations of the USF is analyzed with a Chapman--Enskog-like method and
generalized (tensorial) transport coefficients are obtained.Comment: 40 pages, 10 figures; v2: final version published in a special issue
devoted to "Granular hydrodynamics

### Transport coefficients for inelastic Maxwell mixtures

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 {\em 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.Comment: 9 figures, to be published in J. Stat. Phy