Formal study of a chemical reaction by Grad expansion of the Boltzmann equation. I

The Boltzmann equations for a formal bimolecular chemical reaction in homogeneous gas phase are transformed into an infinite system of quadratic differential equations, by expanding the distribution functions of the molecules into the Grad polynomials. The properties of these expanded Boltzmann equations reflect the macroscopic laws. In particular they enable the Onsager reciprocity relations to be derived from time-reversal invariance.     

Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Our aim in this paper is to show how a probabilistic interpretation of the Boltzmann and Landau equations gives a microscopic understanding of these equations. We firstly associate stochastic jump processes with the Boltzmann equations we consider. Then we renormalize these equations following asymptotics which make prevail the grazing collisions, and prove the convergence of the associated Boltzmann jump processes to a diffusion process related to the Landau equation. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. The best feature of this approach is the microscopic understanding of the transition between the Boltzmann and the Landau equations, by an accumulation of very small jumps. We deduce from this interpretation an approximation result for a solution of the Landau equation via colliding stochastic particle systems. This result leads to a Monte-Carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from Boltzmann to Landau equations. Numerical results are given.     

Polar Coordinate Lattice Boltzmann Kinetic Modeling of Detonation Phenomena

A novel polar coordinate lattice Boltzmann kinetic model for detonation phenomena is presented and applied to investigate typical implosion and explosion processes. In this model, the change of discrete distribution function due to local chemical reaction is dynamically coupled into the modified lattice Boltzmann equation which could recover the Navier-Stokes equations, including contribution of chemical reaction, via the Chapman-Enskog expansion. For the numerical investigations, the main focuses are the nonequilibrium behaviors in these processes. The system at the disc center is always in its thermodynamic equilibrium in the highly symmetric case. The internal kinetic energies in different degrees of freedom around the detonation front do not coincide. The dependence of the reaction rate on the pressure, influences of the shock strength and reaction rate on the departure amplitude of the system from its local thermodynamic equilibrium are probed.     

On inelastic reactive collisions in kinetic theory of chemically reacting gas mixtures

A kinetic theory for a simple reversible reaction-characterized by a binary mixture of ideal gases whose constituents denoted by A and B undergo a reaction of the type A+A?B+B-is developed by considering the reactive collisions as inelastic ones. The geometry of the collision is taken into account in the line-of-centers differential cross section by allowing that a chemical reaction may occur only when the energy of the relative velocity in the direction of the line which joins the centers of the molecules at collision is larger than the activation energy. It is shown that the restitution coefficients: (i) depend explicitly on the reaction heat and on the relative translational energy in the direction of the line which joins the centers of the molecules during an inelastic collision; (ii) vanish when the reaction heat is zero; (iii) are larger or smaller than one depending on the direction of the reaction and on the sign of the reaction heat. First approximations to the distribution functions are determined from the system of Boltzmann equations for the last stage of a chemical reaction. It is shown that the deviations from the Maxwellian distribution functions and the production terms of the particle number densities: (i) vanish when the reaction heat is zero provided that the affinity is close to zero and (ii) are negative or positive depending on the sign of the reaction heat and on the direction of the reaction.     

Renormalized equilibria of a Schlögl model lattice gas

A lattice gas model for Schlögl's second chemical reaction is described and analyzed. Because the lattice gas does not obey a semi-detailed-balance condition, the equilibria are non-Gibbsian. In spite of this, a self-consistent set of equations for the exact homogeneous equilibria are described, using a generalized cluster-expansion scheme. These equations are solved in the two-particle BBGKY approximation, and the results are compared to numerical experiment. It is found that this approximation describes the equilibria far more accurately than the Boltzmann approximation. It is also found, however, that it can give rise to spurious solutions to the equilibrium equations.     

基于格子Boltzmann方法的超声速非平衡流模拟

基于D1Q4可压缩格子Boltzmann模型,按照流通矢量分裂方法的思路,采用坐标旋转技术构造求解三维带化学反应Navier-Stokes方程对流通量求解器.结合有限体积法求解三维化学非平衡流Navier-Stokes方程,采用时间算子分裂算法解决化学反应刚性问题,数值模拟超声速化学非平衡流的三个经典算例.数值结果表明:在高马赫数下,采用D1Q4可压缩格子Boltzmann模型构造的三维对流通量求解器数值模拟中没有出现非物理解,同时在超声速化学非平衡流场中正确分辨激波、燃烧波等物理现象,精度和分辨率均较高,验证了本文构造的三维对流通量求解器的可靠性,拓宽了D1Q4可压缩格子Boltzmann模型的应用范围,为计算超声速化学非平衡流提供一种新方法.     

Unified System of Differential Equations for Plasmas and Boundary Layers

Starting from the Boltzmann equation the moment equations up to the third order for a non-quasineutral plasma are derived. It is shown that for a non-quasineutral plasma, in the distinction from the second order set of moment equations, the systems of the first order and of the third order can be closed in a physically reasonable manner by a relatively simple method when near the walls the ion gas has a high drift velocity caused by an electric field. From this follows that in non-quasineutral plasmas the ion pressure must be neglected in the first order set of equations consisting of the equations of continuity and of momentum transfer. Terms of the first order in the ion pressure related to the electron pressure are taken into consideration for the third order system. In this way one obtains a unified set of differential equations to treat both the quasineutral plasma core and the space charge sheath at the walls jointly taking into account a non-vanishing ion temperature. This set of equations is applied to the positive column in the free-fall regime.     

Transport equations for chiral fermions to order ? and electroweak baryogenesis: Part I

This is the first in a series of two papers. In this first part, we use the Schwinger-Keldysh formalism to derive semiclassical Boltzmann transport equations, accurate to order ?, for massive chiral fermions, scalar particles, and for the corresponding CP-conjugate states. Our considerations include complex mass terms and mixing fermion and scalar fields, such that CP-violation is naturally included, rendering the equations particularly suitable for studies of baryogenesis at a first order electroweak phase transition. We provide a quantitative criterion in which case the reduction to the diagonal kinetic equations in the mass eigenbasis is justified, leading to a quasiparticle picture even in the case of mixing scalar or fermionic particles. Within the approximations we make, it is possible to first study the Boltzmann equations without the collision term. In a second paper [Ann. Phys. xxx (2004) xxx] we discuss the collision terms and reduce the Boltzmann equations to fluid equations.     

A reliable treatment of the homotopy analysis method for viscous flow over a non-linearly stretching sheet in presence of a chemical reaction and under influence of a magnetic field

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].      

Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows

A lattice Boltzmann flux solver (LBFS) is presented in this work for simulation of incompressible viscous and inviscid flows. The new solver is based on Chapman-Enskog expansion analysis, which is the bridge to link Navier-Stokes (N-S) equations and lattice Boltzmann equation (LBE). The macroscopic differential equations are discretized by the finite volume method, where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers. The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh, tie-up of mesh spacing and time interval, limitation to viscous flows. LBFS is validated by its application to simulate the viscous decaying vortex flow, the driven cavity flow, the viscous flow past a circular cylinder, and the inviscid flow past a circular cylinder. The obtained numerical results compare very well with available data in the literature, which show that LBFS has the second order of accuracy in space, and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.     

Numerical method based on the lattice Boltzmann model for the Fisher equation

In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.     

Casson fluid flow: Free convective heat and mass transfer over an unsteady permeable stretching surface considering viscous dissipation

The present study investigates a Casson fluid flow in the presence of free convection of combined heat and mass transfer toward an unsteady permeable stretching sheet with thermal radiation, viscous dissipation and chemical reaction. The governing partial differential equations are reduced to a system of nonlinear ordinary differential equations and then solved by an efficient Runge–Kutta–Fehlberg method. The dimensionless velocity is decreased by increasing values of the chemical reaction and magnetic parameter while fluid temperature is significantly reduced by increasing values of the Prandtl number. The heat transfer rate is reduced with increasing values of thermal radiation and magnetic parameters.     

Simulation of Combustion Field with Lattice Boltzmann Method

Turbulent combustion is ubiquitously used in practical combustion devices. However, even chemically non-reacting turbulent flows are complex phenomena, and chemical reactions make the problem even more complicated. Due to the limitation of the computational costs, conventional numerical methods are impractical in carrying out direct 3D numerical simulations at high Reynolds numbers with detailed chemistry. Recently, the lattice Boltzmann method has emerged as an efficient alternative for numerical simulation of complex flows. Compared with conventional methods, the lattice Boltzmann scheme is simple and easy for parallel computing. In this study, we present a lattice Boltzmann model for simulation of combustion, which includes reaction, diffusion, and convection. We assume the chemical reaction does not affect the flow field. Flow, temperature, and concentration fields are decoupled and solved separately. As a preliminary simulation, we study the so-called counter-flow laminar flame. The particular flow geometry has two opposed uniform combustible jets which form a stagnation flow. The results are compared with those obtained from solving Navier–Stokes equations.     

Investigation of Stability and Hydrodynamics of Different Lattice Boltzmann Models

Stability and hydrodynamic behaviors of different lattice Boltzmann models including the lattice Boltzmann equation (LBE), the differential lattice Boltzmann equation (DLBE), the interpolation-supplemented lattice Boltzmann method (ISLBM) and the Taylor series expansion- and least square-based lattice Boltzmann method (TLLBM) are studied in detail. Our work is based on the von Neumann linearized stability analysis under a uniform flow condition. The local stability and hydrodynamic (dissipation) behaviors are studied by solving the evolution operator of the linearized lattice Boltzmann equations numerically. Our investigation shows that the LBE schemes with interpolations, such as DLBE, ISLBM and TLLBM, improve the numerical stability by increasing hyper-viscosities at large wave numbers (small scales). It was found that these interpolated LBE schemes with the upwind interpolations are more stable than those with central interpolations because of much larger hyper-viscosities.     

Application of Sobolev gradient method to Poisson–Boltzmann system

The idea of a weighted Sobolev gradient, introduced and applied to singular differential equations in [1], is extended to a Poisson–Boltzmann system with discontinuous coefficients. The technique is demonstrated on fully nonlinear and linear forms of the Poisson– Boltzmann equation in one, two, and three dimensions in a finite difference setting. A comparison between the weighted gradient and FAS multigrid is given for large jump size in the coefficient function.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

The Boltzmann equation from quantum field theory

We show from first principles the emergence of classical Boltzmann equations from relativistic nonequilibrium quantum field theory as described by the Kadanoff–Baym equations. Our method applies to a generic quantum field, coupled to a collection of background fields and sources, in a homogeneous and isotropic spacetime. The analysis is based on analytical solutions to the full Kadanoff–Baym equations, using the WKB approximation. This is in contrast to previous derivations of kinetic equations that rely on similar physical assumptions, but obtain approximate equations of motion from a gradient expansion in momentum space. We show that the system follows a generalized Boltzmann equation whenever the WKB approximation holds. The generalized Boltzmann equation, which includes off-shell transport, is valid far from equilibrium and in a time dependent background, such as the expanding universe.     

Enhancement of the LABSWE for shallow water flows

In the lattice Boltzmann method for the shallow water equations (LABSWE), the force term involves the first order derivative related to a bed slope, which has to be determined using the centred scheme for an accurate solution. However, such calculation contradicts the spirit of only simple arithmetic calculations required in the lattice Boltzmann method. This paper shows how to remove this drawback from the LABSWE by incorporating the bed level into the lattice Boltzmann equation in a consistent manner with the lattice Boltzmann approach. Three numerical tests have been presented to verify the proposed method.     

On physical assumptions for the validity of master equations in nuclear reaction theory

The analysis is given of physical assumptions used in different approaches to the derivation of master equations in nuclear reaction theory. The connection of these assumptions is traced with the necessary physical conditions for the derivation of the Boltzmann equations in classical approaches.     

On a class of solutions of nonlinear Boltzmann equations

In a recent paper by Krook and Wu, the nonlinear Boltzmann equation for an infinite, spatially homogeneous, isotropic monoatomic gas of constant density and kinetic energy and with an elastic differential cross section that varies inversely as relative speed has been reduced to an infinite sequence of moment equations. The present note observes that the moment equations are successively integrable and shows that as time goes to infinity, the distribution tends to be Maxwellian.