Solving the Boltzmann Equation

Various attempts have been made to obtain approximate solutions to the Boltzmann equation. Two of these methods were suggested independently by Chapman [10] [11] and by Enskog [24] giving identical results. In this book emphasis is placed on the Enskog method, rather than the Chapman one, as most modern work follows the Enskog approach since it is less intuitive and more systematic, although still very demanding mathematically. 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

The zero order approximation to / is valid when the system is at equilibrium and the gas properties contain no or very small macroscopic gradients. In particular, when the system is at equilibrium the heat fluxes and the viscous stresses vanish. 

The Navier-Stokes equations are valid whenever the relative changes in p, T and v in the distance of the mean free path are small compared to unity. Inasmuch as the Enskog theory is rather long and involved, we will only provide a brief outline of the problem and the method of attack, and then rather discuss the important results. 

When the second order approximations to the pressure tensor and the heat flux vector are inserted into the general conservation equation, one obtains the set of PDEs for the density, velocity and temperature which are called the Burnett equations. In principle, these equations are regarded as valid for non-equilibrium flows. However, the use of these equations never led to any noticeable success (e.g., [28], pp. 150-151) [39], p. 464), merely due to the severe problem of providing additional boundary conditions for the higher order derivatives of the gas properties. Thus the second order approximation will not be considered in further details in this book. 

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Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

In a fluid model the correct calculation of the source terms of electron impact collisions (e.g. ionization) is important. These source terms depend on the EEDF. In the 2D model described here, the source terms as well as the electron transport coefficients are related to the average electron energy and the composition of the gas by first calculating the EEDF for a number of values of the electric field (by solving the Boltzmann equation in the two-term approximation) and constructing a lookup table. 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

When solving the Boltzmann equation, it is common to solve for the distribution function as a function of velocity rather than as a function of momentum, that is, for /(r, v, t) instead of /(r, p, t). In this case Eq. 12.74 is converted to... 

Most of the work in solving the Boltzmann equation for electrons has been for the relatively simple conditions of electron swarm experiments. In these experiments, electrons are released from a cathode in low concentrations and drift under the influence of a uniform applied electric field in a low-pressure gas towards an anode at which the electrons are collected. If... 

An alternative to solving the Boltzmann equation direcdy is the use of particle simulation techniques, sometimes referred to as Monte Carlo methods (58, 59). Major difficulties with the Monte Carlo approach include self-consistency, inclusion of ions, and extension to two spatial dimensions. However, these difficulties are probably not insurmountable, and the Monte Carlo approach may well turn out to be a very powerful tool for discharge analysis. 

Higher-order approximations have not been proven to be useful when the Navier-Stokes equations are invalid, the most satisfactory procedure is to solve the Boltzmann equation by methods that do not rely on the assumption of near-equilibrium flow. 

The distribution of the positions and velocities of electrons at a given instant can be found by solving the Boltzmann equation... 

Note that in 1949 Harold Grad (1923-1986) published an alternative method of solving the Boltzmann equation systematically by expanding the solution into a series of orthogonal polynomials. [36]... 

Aristov, V. V. 2001 Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Dordrecht Kluwer Academic Publishers. 

As noted above, solving the Boltzmann equation is problematic because of the multidimensionality of the problem. A promising approach to calculating the electron distribution function in low pressure plasmas is the so-called non-local approach to electron kinetics. This was proposed by Bernstein and Holstein [56] and popularized by Tsendin [57], who initially suggested this approach for the positive column of a DC discharge. Since then, the non-local approach has been applied to a variety of low pressure gas discharge systems [58]. 

Sitarski-Nowakowski Approach None of the approaches given above describes the dependence of the transition regime mass flux on the molecular mass ratio z of the condensing/evaporating species and the surrounding gas. Sitarski and Nowakowski (1979) applied the 13-moment method of Grad (Hirschfelder et al. 1954) to solve the Boltzmann equation to obtain... 

To solve the Boltzmann equation Enskog s method of successive approximations is generally used This consists of writing firstly,... 

In the transfer regime, which is important for CCN formation and activation, several approaches are known to solve the Boltzmann equation Fuchs theory, Fuchs and Sutugin approach, Dahneke approach, Sitarski and Nowakowski approach see Seinfeld and Pandis (1998), Pruppacher and Klett (1997)). In the following, we will consider only the case uptake by cloud droplets, i. e. the continuum regime. For more details, see Poschl et al. (2005), Davidovits et al. (2006) and Morita and Garrett (2008). It should be noted that termination and symbols are different to those used in the scientific literature (Poschl et al. 2005). The mean-free path is... 

Sitarski and Nowakowski [19] relaxed the assumption of tv = 1.0, and used the 13-moment method of Grad [20] to solve the Boltzmann equation for isothermal condensation. This method is less accurate than the methods applied by Loyatka and his colleagues, but it yields good results in the near-continuum regime. They assumed hard sphere molecules in their treatment of molecular interactions. Their result is... 

Physically, gas flows in the transition regime, where Kn is 0(1), are characterized by the formation of narrow, highly nonequilibrium zones (Knudsen layers) of thickness of the order of the molecular mean free path X the flow structure is then determined by the fast kinetic processes. Moreover, in the case of unsteady flows, an initial Knudsen time interval is of the order Tq = X/v, where v is the molecular velocity. Thus, the Knudsen layer can be computed accurately only by directly solving the Boltzmann equation. 

The principal difficulty in solving the Boltzmann equation lies in the analytically intractable collision term. For small disturbances from equilibrium, the collision term may be linearized. Another approach is the calculation of transfer processes about a particle using a relaxation model for the collision term. It would be expected that such models would be most successful in near-free-molecular conditions where the "free-streaming" terms are much more important than collisions between host-gas molecules. The so-called BGK model is perhaps the most widely applied of these models [2.5,6]. 

To incorporate these ideas into a method of solving the Boltzmann equation, we assume that we are interested in a time after some initial state, and that /(r, v, t) has the form... 

A more accurate and rigorous treatment must consider the intermolecular forces of attraction and repulsion between molecules and also the different sizes of molecules A and B. Chapman and Enskog (H3) solved the Boltzmann equation, which does not utilize the mean free path X but uses a distribution function. To solve the equation, a relation between the attractive and repulsive forces between a given pair of molecules must be used. For a pair of nonpolar molecules a reasonable approximation to the forces is the Lennard-Jones function. 

In the present analysis, the EEDF is determined by solving the Boltzmann equation as a fimction of the reduced electric field E/N so that the electron mean energy equals 3/2 times the electron temperature experimentally measured by the probe. The Boltzmann equation is simultaneously solved with the master equations for the vibrational distribution function (VDF) of the N2 X iZg+ state, since the EEDF of N2-based plasma is strongly affected by the VDF of N2 molecules owing to superelastic collisions with vibrationally excited N2 molecules. A more detailed account of obtaining the EEDF is given in the next section. 

Only when a complete description including all seven variables is required is it necessary to solve the Boltzmann equation in all its generalities, Frequently, simplifying assumptions and limiting conditions can be imposed which reduce the integrodifTerential equation to more tractable form. Thus much of the subject of reactor analysis is devoted to the development and the application of simplified analytical models which define, within the limits of engineering needs, the nuclear characteristics of the reactor complex. 

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