14 research outputs found
Mass transport of an impurity in a strongly sheared granular gas
Transport coefficients associated with the mass flux of an impurity immersed
in a granular gas under simple shear flow are determined from the inelastic
Boltzmann equation. A normal solution is obtained via a Chapman-Enskog-like
expansion around a local shear flow distribution that retains all the
hydrodynamic orders in the shear rate. Due to the anisotropy induced by the
shear flow, tensorial quantities are required to describe the diffusion process
instead of the conventional scalar coefficients. The mass flux is determined to
first order in the deviations of the hydrodynamic fields from their values in
the reference state. The corresponding transport coefficients are given in
terms of the solutions of a set of coupled linear integral equations, which are
approximately solved by considering the leading terms in a Sonine polynomial
expansion. The results show that the deviation of these generalized
coefficients from their elastic forms is in general quite important, even for
moderate dissipation.Comment: 6 figure
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
Inherent Rheology of a Granular Fluid in Uniform Shear Flow
In contrast to normal fluids, a granular fluid under shear supports a steady
state with uniform temperature and density since the collisional cooling can
compensate locally for viscous heating. It is shown that the hydrodynamic
description of this steady state is inherently non-Newtonian. As a consequence,
the Newtonian shear viscosity cannot be determined from experiments or
simulation of uniform shear flow. For a given degree of inelasticity, the
complete nonlinear dependence of the shear viscosity on the shear rate requires
the analysis of the unsteady hydrodynamic behavior. The relationship to the
Chapman-Enskog method to derive hydrodynamics is clarified using an approximate
Grad's solution of the Boltzmann kinetic equationComment: 10 pages, 4 figures; substantially enlarged version; to be published
in PR
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.
Nonlinear viscosity and velocity distribution function in a simple longitudinal flow
A compressible flow characterized by a velocity field is
analyzed by means of the Boltzmann equation and the Bhatnagar-Gross-Krook
kinetic model. The sign of the control parameter (the longitudinal deformation
rate ) distinguishes between an expansion () and a condensation ()
phenomenon. The temperature is a decreasing function of time in the former
case, while it is an increasing function in the latter. The non-Newtonian
behavior of the gas is described by a dimensionless nonlinear viscosity
, that depends on the dimensionless longitudinal rate . The
Chapman-Enskog expansion of in powers of is seen to be only
asymptotic (except in the case of Maxwell molecules). The velocity distribution
function is also studied. At any value of , it exhibits an algebraic
high-velocity tail that is responsible for the divergence of velocity moments.
For sufficiently negative , moments of degree four and higher may diverge,
while for positive the divergence occurs in moments of degree equal to or
larger than eight.Comment: 18 pages (Revtex), including 5 figures (eps). Analysis of the heat
flux plus other minor changes added. Revised version accepted for publication
in PR
Steady state properties of a mean field model of driven inelastic mixtures
We investigate a Maxwell model of inelastic granular mixture under the
influence of a stochastic driving and obtain its steady state properties in the
context of classical kinetic theory. The model is studied analytically by
computing the moments up to the eighth order and approximating the
distributions by means of a Sonine polynomial expansion method. The main
findings concern the existence of two different granular temperatures, one for
each species, and the characterization of the distribution functions, whose
tails are in general more populated than those of an elastic system. These
analytical results are tested against Monte Carlo numerical simulations of the
model and are in general in good agreement. The simulations, however, reveal
the presence of pronounced non-gaussian tails in the case of an infinite
temperature bath, which are not well reproduced by the Sonine method.Comment: 23 pages, 10 figures, submitted for publicatio
Non-Newtonian Couette-Poiseuille flow of a dilute gas
The steady state of a dilute gas enclosed between two infinite parallel
plates in relative motion and under the action of a uniform body force parallel
to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation
is analytically solved for this Couette-Poiseuille flow to first order in the
force and for arbitrary values of the Knudsen number associated with the shear
rate. This allows us to investigate the influence of the external force on the
non-Newtonian properties of the Couette flow. Moreover, the Couette-Poiseuille
flow is analyzed when the shear-rate Knudsen number and the scaled force are of
the same order and terms up to second order are retained. In this way, the
transition from the bimodal temperature profile characteristic of the pure
force-driven Poiseuille flow to the parabolic profile characteristic of the
pure Couette flow through several intermediate stages in the Couette-Poiseuille
flow are described. A critical comparison with the Navier-Stokes solution of
the problem is carried out.Comment: 24 pages, 5 figures; v2: discussion on boundary conditions added; 10
additional references. Published in a special issue of the journal "Kinetic
and Related Models" dedicated to the memory of Carlo Cercignan