43 research outputs found
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
An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux
In the steady Couette flow of a granular gas the sign of the heat flux
gradient is governed by the competition between viscous heating and inelastic
cooling. We show from the Boltzmann equation for inelastic Maxwell particles
that a special class of states exists where the viscous heating and the
inelastic cooling exactly compensate each other at every point, resulting in a
uniform heat flux. In this state the (reduced) shear rate is enslaved to the
coefficient of restitution , so that the only free parameter is the
(reduced) thermal gradient . It turns out that the reduced moments of
order are polynomials of degree in , with coefficients that
are nonlinear functions of . In particular, the rheological properties
() are independent of and coincide exactly with those of the
simple shear flow. The heat flux () is linear in the thermal gradient
(generalized Fourier's law), but with an effective thermal conductivity
differing from the Navier--Stokes one. In addition, a heat flux component
parallel to the flow velocity and normal to the thermal gradient exists. The
theoretical predictions are validated by comparison with direct Monte Carlo
simulations for the same model.Comment: 16 pages, 4 figures,1 table; v2: minor change
Computer simulations of an impurity in a granular gas under planar Couette flow
We present in this work results from numerical solutions, obtained by means
of the direct simulation Monte Carlo (DSMC) method, of the Boltzmann and
Boltzmann--Lorentz equations for an impurity immersed in a granular gas under
planar Couette flow. The DSMC results are compared with the exact solution of a
recent kinetic model for the same problem. The results confirm that, in steady
states and over a wide range of parameter values, the state of the impurity is
enslaved to that of the host gas: it follows the same flow velocity profile,
its concentration (relative to that of the granular gas) is constant in the
bulk region, and the impurity/gas temperature ratio is also constant. We
determine also the rheological properties and nonlinear hydrodynamic transport
coefficients for the impurity, finding a good semi-quantitative agreement
between the DSMC results and the theoretical predictions.Comment: 23 pages, 11 figures; v2: minor change
System of elastic hard spheres which mimics the transport properties of a granular gas
The prototype model of a fluidized granular system is a gas of inelastic hard
spheres (IHS) with a constant coefficient of normal restitution . Using
a kinetic theory description we investigate the two basic ingredients that a
model of elastic hard spheres (EHS) must have in order to mimic the most
relevant transport properties of the underlying IHS gas. First, the EHS gas is
assumed to be subject to the action of an effective drag force with a friction
constant equal to half the cooling rate of the IHS gas, the latter being
evaluated in the local equilibrium approximation for simplicity. Second, the
collision rate of the EHS gas is reduced by a factor , relative
to that of the IHS gas. Comparison between the respective Navier-Stokes
transport coefficients shows that the EHS model reproduces almost perfectly the
self-diffusion coefficient and reasonably well the two transport coefficients
defining the heat flux, the shear viscosity being reproduced within a deviation
less than 14% (for ). Moreover, the EHS model is seen to agree
with the fundamental collision integrals of inelastic mixtures and dense gases.
The approximate equivalence between IHS and EHS is used to propose kinetic
models for inelastic collisions as simple extensions of known kinetic models
for elastic collisionsComment: 20 pages; 6 figures; change of title; few minor changes; accepted for
publication in PR
Impurity in a granular gas under nonlinear Couette flow
We study in this work the transport properties of an impurity immersed in a
granular gas under stationary nonlinear Couette flow. The starting point is a
kinetic model for low-density granular mixtures recently proposed by the
authors [Vega Reyes F et al. 2007 Phys. Rev. E 75 061306]. Two routes have been
considered. First, a hydrodynamic or normal solution is found by exploiting a
formal mapping between the kinetic equations for the gas particles and for the
impurity. We show that the transport properties of the impurity are
characterized by the ratio between the temperatures of the impurity and gas
particles and by five generalized transport coefficients: three related to the
momentum flux (a nonlinear shear viscosity and two normal stress differences)
and two related to the heat flux (a nonlinear thermal conductivity and a cross
coefficient measuring a component of the heat flux orthogonal to the thermal
gradient). Second, by means of a Monte Carlo simulation method we numerically
solve the kinetic equations and show that our hydrodynamic solution is valid in
the bulk of the fluid when realistic boundary conditions are used. Furthermore,
the hydrodynamic solution applies to arbitrarily (inside the continuum regime)
large values of the shear rate, of the inelasticity, and of the rest of
parameters of the system. Preliminary simulation results of the true Boltzmann
description show the reliability of the nonlinear hydrodynamic solution of the
kinetic model. This shows again the validity of a hydrodynamic description for
granular flows, even under extreme conditions, beyond the Navier-Stokes domain.Comment: 23 pages, 11 figures; v2: Preliminary DSMC results from the Boltzmann
equation included, Fig. 11 is ne
Third and fourth degree collisional moments for inelastic Maxwell models
The third and fourth degree collisional moments for -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 .Comment: 15 pages, 3 figures; v2: addition of two new reference
Segregation by thermal diffusion of an intruder in a moderately dense granular fluid
A solution of the inelastic Enskog equation that goes beyond the weak
dissipation limit and applies for moderate densities is used to determine the
thermal diffusion factor of an intruder immersed in a dense granular gas under
gravity. This factor provides a segregation criterion that shows the transition
between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by
varying the parameters of the system (masses, sizes, density and coefficients
of restitution). The form of the phase-diagrams for the BNE/RBNE transition
depends sensitively on the value of gravity relative to the thermal gradient,
so that it is possible to switch between both states for given values of the
parameters of the system. Two specific limits are considered with detail: (i)
absence of gravity, and (ii) homogeneous temperature. In the latter case, after
some approximations, our results are consistent with previous theoretical
results derived from the Enskog equation. Our results also indicate that the
influence of dissipation on thermal diffusion is more important in the absence
of gravity than in the opposite limit. The present analysis extends previous
theoretical results derived in the dilute limit case [V. Garz\'o, Europhys.
Lett. {\bf 75}, 521 (2006)] and is consistent with the findings of some recent
experimental results.Comment: 10 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
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