2,512 research outputs found
Collisional invariants for the phonon Boltzmann equation
For the phonon Boltzmann equation with only pair collisions we characterize
the set of all collisional invariants under some mild conditions on the
dispersion relation
The relativistic kinetic dispersion relation: Comparison of the relativistic Bhatnagar-Gross-Krook model and Grad's 14-moment expansion
In this paper, we study the Cauchy problem of the linearized kinetic
equations for the models of Marle and Anderson-Witting, and compare these
dispersion relations with the 14-moment theory. First, we propose a
modification of the Marle model to improve the resultant transport coefficients
in accord with those obtained by the full Boltzmann equation. Using the
modified Marle model and Anderson-Witting model, we calculate dispersion
relations that are kinetically correct within the validity of the BGK
approximation. The 14-moment theory that includes the time derivative of
dissipation currents has causal structure, in contrast to the acausal
first-order Chapman-Enskog approximation. However, the dispersion relation of
the 14-moment theory does not accurately describe the result of the kinetic
equation. Thus, our calculation indicates that keeping these second-order terms
does not simply correspond to improving the physical description of the
relativistic hydrodynamics.Comment: 20 pages, 22 figures, accepted for publication in Physica
Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic Matlab program
The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic
model for many applications in thermodynamics, econophysics and sociodynamics.
Despite recent hardware improvements, the solution of the Boltzmann equation
remains extremely challenging from the computational point of view, in
particular by deterministic methods (free of stochastic noise). This work aims
to improve a deterministic direct method recently proposed [V.V. Aristov,
Kluwer Academic Publishers, 2001] for solving the HIBE with a generic
collisional kernel and, in particular, for taking care of the late dynamics of
the relaxation towards the equilibrium. Essentially (a) the original problem is
reformulated in terms of particle kinetic energy (exact particle number and
energy conservation during microscopic collisions) and (b) the computation of
the relaxation rates is improved by the DVM-like correction, where DVM stands
for Discrete Velocity Model (ensuring that the macroscopic conservation laws
are exactly satisfied). Both these corrections make possible to derive very
accurate reference solutions for this test case. Moreover this work aims to
distribute an open-source program (called HOMISBOLTZ), which can be
redistributed and/or modified for dealing with different applications, under
the terms of the GNU General Public License. The program has been purposely
designed in order to be minimal, not only with regards to the reduced number of
lines (less than 1,000), but also with regards to the coding style (as simple
as possible).Comment: 35 pages, 4 figures, it describes the code HOMISBOLTZ to be
distributed with the pape
A volume-based hydrodynamic approach to sound wave propagation in a monatomic gas
We investigate sound wave propagation in a monatomic gas using a volume-based
hydrodynamic model. In Physica A vol 387(24) (2008) pp6079-6094, a microscopic
volume-based kinetic approach was proposed by analyzing molecular spatial
distributions; this led to a set of hydrodynamic equations incorporating a
mass-density diffusion component. Here we find that these new mass-density
diffusive flux and volume terms mean that our hydrodynamic model, uniquely,
reproduces sound wave phase speed and damping measurements with excellent
agreement over the full range of Knudsen number. In the high Knudsen number
(high frequency) regime, our volume-based model predictions agree with the
plane standing waves observed in the experiments, which existing kinetic and
continuum models have great difficulty in capturing. In that regime, our
results indicate that the "sound waves" presumed in the experiments may be
better thought of as "mass-density waves", rather than the pressure waves of
the continuum regime.Comment: Revised to aid clarification (no changes to presented model); typos
corrected, figures added, paper title change
A continuum model of gas flows with localized density variations
We discuss the kinetic representation of gases and the derivation of macroscopic equations governing the thermomechanical behavior of a dilute gas viewed at the macroscopic level as a continuous medium. We introduce an approach to kinetic theory where spatial distributions of the molecules are incorporated through a mean-free-volume argument. The new kinetic equation derived contains an extra term involving the evolution of this volume, which we attribute to changes in the thermodynamic properties of the medium. Our kinetic equation leads to a macroscopic set of continuum equations in which the gradients of thermodynamic properties, in particular density gradients, impact on diffusive fluxes. New transport terms bearing both convective and diffusive natures arise and are interpreted as purely macroscopic expansion or compression. Our new model is useful for describing gas flows that display non-local-thermodynamic-equilibrium (rarefied gas flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluid flows
From the Boltzmann equation to fluid mechanics on a manifold
We apply the Chapman-Enskog procedure to derive hydrodynamic equations on an
arbitrary surface from the Boltzmann equation on the surface
Call for Papers
Studia austriaca XX (2012) - Call for Paper
Call for Papers
Studia theodisca – Hölderliniana I (2014): Call for Papers
Call for Papers
Studia theodisca XIX (2012) - Call for Paper
- …