A unified implicit scheme for kinetic model equations. Part I. Memory reduction technique

A unified gas-kinetic scheme for axisymmetric flow in all Knudsen number regimes

We have developed and, to some extent, solved a number of kinetic equations for displaced correlation functions in a classical fluid. These functions, of which the Van Hove neutron scattering function is a special example, are one-particle distribution functions obtained from a Gibbs ensemble which is initially, at $t=0$, in equilibrium except for one labeled particle whose distribution $W(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ at $t=0$ differs from its equilibrium value $\ensuremath{\rho}{h}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$, where $\ensuremath{\rho}$ is the average fluid density, and ${h}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ is the Maxwellian velocity distribution function. We investigate the time evolution of the (self-) distribution function of this labeled particle, ${f}_{s}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}},t)$, as well as the deviation from equilibrium, $\ensuremath{\eta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}},t)$, of the total one-particle distribution function. The latter represents the density of fluid particles, labeled and unlabeled, at position $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ and velocity $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$. Since both ${f}_{s}$ and $\ensuremath{\eta}$ are linear functionals of $W$, they will satisfy exactly a linear non-Markovian kinetic equation of the form $f=\mathrm{B}f+\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}\mathrm{M}({t}^{\ensuremath{'}})f(t\ensuremath{-}{t}^{\ensuremath{'}})$. B is a time-in-dependent and M a time-dependent (memory) operator (nonsingular in $t$). Our kinetic equations (first- and higher-order) are based on neglecting or approximating M in such a way that the short-time behavior of ${f}_{s}$ and $\ensuremath{\eta}$ is described exactly. The rationale behind this scheme is that our choice of initial ensemble is precisely of the type generally assumed in the "derivation" of kinetic equations. The calculation of B is straightforward and depends in a very important way on whether the interparticle potential in the fluid is smooth or contains a hard core. In the former case, the first-order kinetic equation is of the Vlasov type with an effective potential given by the equilibrium direct correlation function, while in the latter case, B contains, in addition, a linear Enskog-type collision term. We show that this Vlasov equation (also derived previously by many authors) gives a damping linear in the wave number $k$ for small $k$ instead of the hydrodynamic ${k}^{2}$ dependence. The kinetic equation for systems with hard cores does not give correct hydrodynamic behavior. (For a one-dimensional system of hard rods, the first-order kinetic equation is exact.) We also obtain and solve a second-order kinetic equation, which is a generalized Vlasov-Fokker-Planck-type equation, for systems with continuous potentials.

A Monte Carlo simulation method is used to study a dilute gas of hard spheres under uniform shear flow in the hydrodynamic regime over a wide range of shear rates. The results for the components of the pressure tensor and also for the distribution function are presented and compared with those obtained by using the Bhatnagar-Gross-Krook model kinetic equation. The agreement is fairly good for the pressure tensor and also for the velocity distribution function in the small-velocity region, while discrepancies appear for large velocities. The possibility of using a maximum-entropy method to construct an approximate distribution function is discussed. It is shown that the inclusion of the pressure tensor leads to a distribution function significantly different from the one obtained in the simulation.

Comparison of the Shakhov and ellipsoidal models for the Boltzmann equation and DSMC for ab initio-based particle interactions

The electron kinetic model for investigating the transport and ionization rate coefficients of argon glow discharge dusty plasma is developed from the Boltzmann equation. Both of the electron–neutral and electron–dust collisions are considered as collision terms in the kinetic equation. The kinetic equation is simplified by employing the local approximation and nonlocal approach under the same discharge conditions, and the corresponding simplified kinetic equations are known as local and nonlocal kinetic equations respectively. Then the electron energy distribution function (EEDF) is obtained by numerically solving the local and nonlocal kinetic equations and the dust charging equations simultaneously. Based on the obtained EEDFs, the effective electron temperature, electron mobility, electron diffusion coefficient and ionization rate coefficient are calculated for different discharge conditions. It is shown that the EEDFs calculated from the local kinetic model clearly differ from the nonlocal EEDFs and both the local and nonlocal EEDFs are also clearly different with Maxwellian distributions. The appearance of dust particles results in an obvious decrease of high energy electrons and increase of low energy electrons when axial electric field is low. With the increase of axial electric field, the influence of dust particles on the EEDFs becomes smaller. The electron mobility and diffusion coefficients calculated on the basis of local and nonlocal EEDFs do not differ greatly to the dust-free ones. While, when dust density nd = 106 cm−3, the electron mobility increases obviously compared with the dust-free results at low axial electric field and decreases with the increasing axial electric field until they are close to the dust-free ones. Meanwhile, electron diffusion coefficients for dusty case become smaller and decrease with the increasing axial electric field. The ionization rate coefficients decrease when dust particles are introduced and they approach the dust-free results gradually with the increasing axial electric field.

The Vlasov equation is not applicable for those turbulent plasmas in which certain higher-order correlation functions may play just as important a part as the single-particle distribution function. We construct the many-particle distribution functions by summing certain selected products of the single-particle distribution functions and the pair correlation functions, and show that, in the Rostoker-Rosenbluth fluid limit, those distribution functions satisfy the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy and enable us to truncate it at the second equation. The resulting equations contain an asymmetric feature in that the single-particle distribution function develops in time with inclusion of the correlation effects in the collision term, while the time evolution of the pair correlation function is governed by a solution of the linearized Vlasov equation. It is shown explicitly that those kinetic equations become identical to the quasilinear equations without mode coupling if the contribution of the growing plasma oscillation is singled out in the pair correlation function.

A comparative study of an asymptotic preserving scheme and unified gas-kinetic scheme in continuum flow limit

General synthetic iterative scheme for non-equilibrium dense gas flows

Leaching kinetics of scheelite in hydrochloric acid solution containing hydrogen peroxide as complexing agent

In [1] R. G. Mukhometov published a proof of the following result which remains the strongest among those known in integral geometry: a simple Riemannian metric on a compact manifold with boundary is uniquely determined in a prescribed conformal class by the distances between boundary points (a metric is called simple, if a unique geodesic joints each pair of points). The proof is based on some differential identity for the kinetic equation on a Riemannian manifold. Using the identity, R. G. Mukhometov “in passing” obtained the following two strong results: he proved uniqueness for a solution to the linear problem of integral geometry and found a formula that expresses the volume of a Riemannian manifold in terms of the distances between boundary points. Unfortunately, the formula is given in [1] in regretfully noninvariant form. Closed results were obtained by I. M. Bernstĕin and M. L. Gerver [2]. Mukhometov’s identity has a cumbersome formulation and very difficult proof. The present article was originally intended as purely methodological: to find an exposition of Mukhometov’s results appropriate for their inclusion into the author’s lecture course on integral geometry. As it often happens in such matters, the research goes beyond the preliminary framework, leading to the results of interest in their own right. First, the stability estimate that we obtained in the nonlinear problem of determining a metric differs from the similar estimate by Mukhometov; the differences can occur essential for some questions. Second, our formula for the volume of a Riemannian manifold has an invariant form and depends linearly on the boundary distances; this circumstance turned out to be unexpected for the author. In the present article, it is Mukhometov’s results on the linear problem of integral geometry that undergo the broadest generalization. To clarify the nature of these generalizations, we recall that the integral geometry problem is equivalent to the inverse problem of determining a source in the stationary kinetic equation. The latter equation has a simple physical meaning: it describes the distribution of particles (or a radiation) moving along certain trajectories (in our case along the geodesics of a Riemannian metric) and not interacting with each other and with a medium. If we wish to take account of interaction of particles with the medium, then we have to insert extra summands into the equation. The simplest of such summands describes attenuation of particles by the medium. From the standpoint of integral geometry, the taking of attenuation into account leads to the fact that, in the main integral operator, there appears a weight factor depending exponentially

A Kinetic Model for Vehicular Traffic: Existence of Stationary Solutions

Numerical simulation of heat transfer flows by a direct solution of generalized kinetic models

The acceleration of energetic particles and their propagation in magnetic fields of the solar wind and the Galaxy is a topical astrophysical problem. Cosmic rays (CR) affect communications and the operation of on-board spacecraft electronics and disturb the magnetosphere and the ionosphere of the Earth. The scattering of particles by magnetic-field irregularities is the primary mechanism governing the CR propagation in the interplanetary medium. If the scattering of energetic particles in the interplanetary medium is relatively inefficient (i.e., the mean free path is comparable to the heliocentric distance), a kinetic equation should be used to characterize the CR propagation. The Fokker–Planck kinetic equation is used to analyze the propagation of charged energetic particles in a magnetic field represented as a superposition of a mean uniform field and magnetic inhomogeneities of various scales. This kinetic equation corresponds to multiple small-angle scattering, and its collision integral characterizes particle diffusion in the momentum space. A system of differential equations for the spherical harmonics of the CR distribution function is obtained based on the kinetic equation. The CR transport equations are derived and solved. The evolution of the CR distribution function under anisotropic scattering of particles by magnetic-field fluctuations is examined. It is demonstrated that the angular distribution function of particles depends to a considerable extent on the degree of their scattering anisotropy. The temporal dependence of the CR distribution function is analyzed, and the parameter characterizing the scattering anisotropy is estimated.

Titanium dioxide: A heterogeneous catalyst for dark peroxidation superior to iron oxide

Numerical modeling of rarefied gas flow through a slit into vacuum based on the kinetic equation