Comment on "Theory and computer simulation for the equation of state of additive hard-disk fluid mixtures"

A flaw in the comparison between two different theoretical equations of state for a binary mixture of additive hard disks and Monte Carlo results, as recently reported in C. Barrio and J. R. Solana, Phys. Rev. E 63, 011201 (2001), is pointed out. It is found that both proposals, which require the equation of state of the single component system as input, lead to comparable accuracy but the one advocated by us [A. Santos, S. B. Yuste, and M. L\'{o}pez de Haro, Mol. Phys. 96, 1 (1999)] is simpler and complies with the exact limit in which the small disks are point particles.Comment: 4 pages, including 1 figur

The semiclassical--Sobolev orthogonal polynomials: a general approach

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$_S= +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr D}r}>,$$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time.Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theor

The second and third Sonine coefficients of a freely cooling granular gas revisited

In its simplest statistical-mechanical description, a granular fluid can be modeled as composed of smooth inelastic hard spheres (with a constant coefficient of normal restitution $\alpha$) whose velocity distribution function obeys the Enskog-Boltzmann equation. The basic state of a granular fluid is the homogeneous cooling state, characterized by a homogeneous, isotropic, and stationary distribution of scaled velocities, $F(\mathbf{c})$. The behavior of $F(\mathbf{c})$ in the domain of thermal velocities ($c\sim 1$) can be characterized by the two first non-trivial coefficients ($a_2$ and $a_3$) of an expansion in Sonine polynomials. The main goals of this paper are to review some of the previous efforts made to estimate (and measure in computer simulations) the $\alpha$-dependence of $a_2$ and $a_3$, to report new computer simulations results of $a_2$ and $a_3$ for two-dimensional systems, and to investigate the possibility of proposing theoretical estimates of $a_2$ and $a_3$ with an optimal compromise between simplicity and accuracy.Comment: 12 pages, 5 figures; v2: minor change

Navier-Stokes transport coefficients of dd-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

Characterization and control of phase fluctuations in elongated Bose-Einstein condensates

Quasi one dimensional Bose-Einstein condensates (BECs) in elongated traps exhibit significant phase fluctuations even at very low temperatures. We present recent experimental results on the dynamic transformation of phase fluctuations into density modulations during time-of-flight and show the excellent quantitative agreement with the theoretical prediction. In addition we confirm that under our experimental conditions, in the magnetic trap density modulations are strongly suppressed even when the phase fluctuates. The paper also discusses our theoretical results on control of the condensate phase by employing a time-dependent perturbation. Our results set important limitations on future applications of BEC in precision atom interferometry and atom optics, but at the same time suggest pathways to overcome these limitations.Comment: 9 pages, 7 figure

Cocoa Butter Saturated with Supercritical Carbon Dioxide: Measurements and Modelling of Solubility, Volumetric Expansion, Density and Viscosity

International audienceThe use of supercritical carbon dioxide technology for lipid processing has recently developed at the expense of traditional processes. For designing new processes the knowledge of thermophysical properties is a prerequisite. This work is focused on the characterization of physical and thermodynamic properties of CO2-cocoa butter (CB) saturated mixture. Measurements of density, volumetric expansion, viscosity and CO2 solubility were carried out on CB-rich phase at 313 and 353 K and pressures up to 40 MPa. The experimental techniques have previously been compared and validated. Density measurements of CB and CB saturated with CO2, were performed using the vibrating tube technology at pressures ranging from 0.1 to 25 MPa. Experimental values correlated well with the modified Tait equation. CO2 solubility measurements were coupled to those of density in the same pressures ranges. At 25 MPa, the solubility of CO2 is 31.4 % and 28.7 % at 313 and 353 K. Phase behaviour was investigated using a view cell in order to follow the expansion of the CB-rich phase with the rise in pressure. Volumetric expansion up to 47 % was measured and correlated to the CO2 solubility. Phase inversion was observed at 313 K and 40 MPa. Lastly, we developed an innovative falling ball viscometer for high pressure measurements. Viscosity measurements were carried out up to 25 MPa showing a viscosity reduction up to 90 % upon CO2 dissolution. These results were correlated with two empirical models. A new model here presented, was favourably compared with the Grunberg and Nissan model. All the experimental results are consistent with the available literature data for the CB-CO2 mixture and other fat systems. This work is a new contribution to the characterization of physical and thermodynamic behaviour of fats in contact with CO2 which is necessary to design supercritical fluid processes for fats processing

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 $D$ 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 $D$ up to the second order in the Sonine expansion and get explicit expressions for $D$ 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 $D$ 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.

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 $\alpha$, so that the only free parameter is the (reduced) thermal gradient $\epsilon$. It turns out that the reduced moments of order $k$ are polynomials of degree $k-2$ in $\epsilon$, with coefficients that are nonlinear functions of $\alpha$. In particular, the rheological properties ($k=2$) are independent of $\epsilon$ and coincide exactly with those of the simple shear flow. The heat flux ($k=3$) 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

Estudia comparativo de los cariatipos de Acrotylus insubricus Scop. y A. fischeri Azam. (Orthoptera: Acrididae)

Mediante el estudio de1 cariotipo de Acrotylus insubricus y A. fischeri se confirma que se trata de dos especies distintas