7,412 research outputs found
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
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
Anomalous transport of impurities in inelastic Maxwell gases
A mixture of dissipative hard grains generically exhibits a breakdown of
kinetic energy equipartition. The undriven and thus freely cooling binary
problem, in the tracer limit where the density of one species becomes minute,
may exhibit an extreme form of this breakdown, with the minority species
carrying a finite fraction of the total kinetic energy of the system. We
investigate the fingerprint of this non-equilibrium phase transition, akin to
an ordering process, on transport properties. The analysis, performed by
solving the Boltzmann kinetic equation from a combination of analytical and
Monte Carlo techniques, hints at the possible failure of hydrodynamics in the
ordered region. As a relevant byproduct of the study, the behaviour of the
second and fourth-degree velocity moments is also worked out.Comment: The title has been changed. The paper has been enlarged with respect
to our first version. 13 pages, 9 figures. To be published in EPJ
Diffusion of impurities in a granular gas
Diffusion of impurities in a granular gas undergoing homogeneous cooling
state is studied. The results are obtained by solving the Boltzmann--Lorentz
equation by means of the Chapman--Enskog method. In the first order in the
density gradient of impurities, the diffusion coefficient is determined as
the solution of a linear integral equation which is approximately solved by
making an expansion in Sonine polynomials. In this paper, we evaluate up to
the second order in the Sonine expansion and get explicit expressions for
in terms of the restitution coefficients for the impurity--gas and gas--gas
collisions as well as the ratios of mass and particle sizes. To check the
reliability of the Sonine polynomial solution, analytical results are compared
with those obtained from numerical solutions of the Boltzmann equation by means
of the direct simulation Monte Carlo (DSMC) method. In the simulations, the
diffusion coefficient is measured via the mean square displacement of
impurities. The comparison between theory and simulation shows in general an
excellent agreement, except for the cases in which the gas particles are much
heavier and/or much larger than impurities. In theses cases, the second Sonine
approximation to improves significantly the qualitative predictions made
from the first Sonine approximation. A discussion on the convergence of the
Sonine polynomial expansion is also carried out.Comment: 9 figures. to appear in Phys. Rev.
Navier-Stokes transport coefficients of -dimensional granular binary mixtures at low density
The Navier-Stokes transport coefficients for binary mixtures of smooth
inelastic hard disks or spheres under gravity are determined from the Boltzmann
kinetic theory by application of the Chapman-Enskog method for states near the
local homogeneous cooling state. It is shown that the Navier-Stokes transport
coefficients are not affected by the presence of gravity. As in the elastic
case, the transport coefficients of the mixture verify a set of coupled linear
integral equations that are approximately solved by using the leading terms in
a Sonine polynomial expansion. The results reported here extend previous
calculations [V. Garz\'o and J. W. Dufty, Phys. Fluids {\bf 14}, 1476 (2002)]
to an arbitrary number of dimensions. To check the accuracy of the
Chapman-Enskog results, the inelastic Boltzmann equation is also numerically
solved by means of the direct simulation Monte Carlo method to evaluate the
diffusion and shear viscosity coefficients for hard disks. The comparison shows
a good agreement over a wide range of values of the coefficients of restitution
and the parameters of the mixture (masses and sizes).Comment: 6 figures, to be published in J. Stat. Phy
Mass transport in a strongly sheared binary mixture of Maxwell molecules
Transport coefficients associated with the mass flux of a binary mixture of
Maxwell molecules under uniform shear flow are exactly determined from the
Boltzmann kinetic equation. A normal solution is obtained via a
Chapman--Enskog-like expansion around a local shear flow distribution that
retains all the hydrodynamics orders in the shear rate. In the first order of
the expansion the mass flux is proportional to the gradients of mole fraction,
pressure, and temperature but, due to the anisotropy induced in the system by
the shear flow, mutual diffusion, pressure diffusion and thermal diffusion
tensors are identified instead of the conventional scalar coefficients. These
tensors are obtained in terms of the shear rate and the parameters of the
mixture (particle masses, concentrations, and force constants). The description
is made both in the absence and in the presence of an external thermostat
introduced in computer simulations to compensate for the viscous heating. As
expected, the analysis shows that there is not a simple relationship between
the results with and without the thermostat. The dependence of the three
diffusion tensors on the shear rate is illustrated in the tracer limit case,
the results showing that the deviation of the generalized transport
coefficients from their equilibrium forms is in general quite important.
Finally, the generalized transport coefficients associated with the momentum
and heat transport are evaluated from a model kinetic equation of the Boltzmann
equation.Comment: 6 figure
Tracer diffusion in granular shear flows
Tracer diffusion in a granular gas in simple shear flow is analyzed. The
analysis is made from a perturbation solution of the Boltzmann kinetic equation
through first order in the gradient of the mole fraction of tracer particles.
The reference state (zeroth-order approximation) corresponds to a Sonine
solution of the Boltzmann equation, which holds for arbitrary values of the
restitution coefficients. Due to the anisotropy induced in the system by the
shear flow, the mass flux defines a diffusion tensor instead of a
scalar diffusion coefficient. The elements of this tensor are given in terms of
the restitution coefficients and mass and size ratios. The dependence of the
diffusion tensor on the parameters of the problem is illustrated in the
three-dimensional case. The results show that the influence of dissipation on
the elements is in general quite important, even for moderate values
of the restitution coefficients. In the case of self-diffusion (mechanically
equivalent particles), the trends observed in recent molecular dynamics
simulations are similar to those obtained here from the Boltzmann kinetic
theory.Comment: 5 figure
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