Boltzmann equation collision term

Discussion of the Equation.—The Boltzmann equation describes the manner in which the distribution function for a system of particles, /x = /(r,vx,f), varies in terms of its independent variables r, the position of observation vx, the velocity of the particles considered and the time, t. The variation of the distribution function due to the external forces acting on the particles and the action of collisions are both considered. In the integral expression on the right of Eq. (1-39), the Eqs. (1-21) are used to express the velocities after collision in terms of the velocities before collision the dynamics of the collision process are taken into account in the expression for x(6,e), from Eqs. (1-11) and (1-12), which enters into the k of Eqs. (1-21). Alternatively, as will be shown to be useful later, the velocities before and after collision may be expressed, by Eq. (1-20), in terms of G,g, and g the dynamics of the collision comes into the relation between g and g of Eq. (1-19). 

Since /S Tj0) = , its integral over the collision term is zero (conservation of momentum in a collision). Thus the result of multiplying the Boltzmann equation by and integrating is ... 

Consider electrons of mass m and velocity v, and atoms of mass M and velocity V we have mjM 1. The distribution function for the electrons will be denoted by /(v,<) (we assume no space dependence) that for the atoms, F( V), assumed Maxwellian as usual, in the collision integral, unprimed quantities refer to values before collision, while primed quantities are the values after collision. In general, we would have three Boltzmann equations (one each for the electrons, ions, and neutrals), each containing three collision terms (one for self-collisions, and one each for collisions with the other two species). We are interested only in the equation for the electron distribution function by the assumption of slight ionization, we neglect the electron-electron... 

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. 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

The most recent effort in this direction is the work of Cohen,8 who established a systematic generalization of the Boltzmann equation. This author obtained the explicit forms of the two-, three-, and four-particle collision terms. His approach is formally very similar to the cluster expansion of Mayer in the equilibrium case. 

Choh and Uhlenbeck6 developed Bogolubov s ideas and extended his formal results. They established a generalized Boltzmann equation which takes account of three-particle collisions. The extension of their results to higher orders in the concentration poses no problem in principle, but it appears difficult, in this formalism, to write a priori the collision term with an arbitrary number of particles. 

Hie singlet distribution ff changes when a collision occurs between the particle ot and any one of the remaining 3 = (N — 1) particles (labelled 2 during the collision process). The probability of particles ot and /3 being able to collide depends upon their mutual position in space and their velocity, which is described by the doublet density f"0. Now, in the Boltzmann equation analysis, this leads to a factorisation of the doublet density in terms of the singlet density of both particles. Because any correlation in the position and motion of these particles is lost by this procedure, an alternative approach must be tried to estimate the doublet density. 

In terms of the collision integrals, the Boltzmann equation can be rewritten as... 

Here a is the differential cross-section, and depends only on Pi Pi = l/>3 Pa and on (/U - p2) p2 Pa)-The precise number of molecules in the cell fluctuates around the value given by the Boltzmann equation, because the collisions occur at random, and only their probability is given by the Stosszahlansatz. Our aim is to compute these fluctuations. If / differs little from the equilibrium distribution one may replace the Boltzmann equation by its linearized version. It is then possible to include the fluctuations by adding a Langevin term, whose strength is determined by means of the fluctuation-dissipation theorem.510 As demonstrated in IX.4, however, the Langevin approach is unreliable outside the linear domain. We shall therefore start from the master equation and use the -expansion. The whole procedure consists of four steps. 

On the other hand, the first term in the collision part of the Boltzmann equation may be written... 

The dominant term in the Boltzmann Equation (84) is assumed to be the collision term, i.e., the second term on the right-hand side of (84). This then implies that, as t —> 00, solutions to Equation (82), denoted 11l0 (r. v p. u, e) represent a good approximation to the asymptotic solution to (84). Consequently, we choose... 

The osmotic pressure between two flat surfaces can be derived within the Poisson-Boltzmann approximation. The PB equation was originally developed to describe ion distributions outside a large charged surface. However, there are extended PB equations where polymers have been included [32]. The expression for the osmotic pressure given below is valid in the absence of polymers. In the PB equation the correlations between ions are neglected, which means that Pei is identically zero. Furthermore, the ions are normally treated as point particles which means that the collision term disappears. Thus for symmetric systems only two terms remain, the kinetic pressure and the bulk pressure. The net pressure can be written as... 

However, the intermolecular force laws play a central role in the model determining the molecular interaction terms (i.e., related to the collision term on the RHS of the Boltzmann equation). Classical kinetic theory proceeds on the assumption that this law has been separately established, either empirically or from quantum theory. The force of interaction between two molecules is related to the potential energy as expressed by... 

Preliminarily, deriving the terms on the LHS of the Boltzmann equation, we assume that the effects of collisions are negligible. The molecular motion is thus purely translational. We further assume that in an average sense each molecule of mass m is subjected to an external forces per unit mass, F(r,t), which doesn t depend on the molecular velocity c. This restriction excludes magnetic forces, while the gravity and electric fields which are more common in chemical and metallurgical reactors are retained. 

Therefore, in the limit of no molecular interactions, for which the collision term ( )coiiision vanishes, the Boltzmann equation yields... 

This relation enable us to simplify the formulation of the general equation of change considerably. Fortunately, the fundamental fluid dynamic conservation equations of continuity, momentum, and energy are thus derived from the Boltzmann equation without actually determining the form of either the collision term or the distribution function /. 

The conservation equations (2.202), (2.207) and (2.213) are rigorous (i.e., for mono-atomic gases) consequences of the Boltzmann equation (2.185). It is important to note that we have derived the governing conservation equations without knowing the exact form of the collision term, the only requirement is that we are considering summation invariant properties of mono-atomic gases. That is, we are considering properties that are conserved in molecular collisions. 

The flrst step in the Enskog expansion is to introduce a perturbation parameter e into the Boltzmann equation to enforce a state of equilibrium flow as the gas is dominated by a large collision term ... 

The starting point for the kinetic theory of dilute mono-atomic gases is the Boltzmann equation determining the evolution of the distribution function in time and space. The formulation of the collision term is restricted to gases that are sufficiently dilute so that only binary collisions need to be taken into account. It is also required that the molecular dimensions are small in comparison with the mean distance between the molecules, hence the transfer of molecular properties is solely regarded as a consequence of the free motion of molecules between collisions. 

Multiplying the RHS of the Boltzmann equation (4.1) with tpdC and thereafter integrating the resulting term over the velocity space, one obtains the rate of change by collisions in the property -0 summed over all the particles in a unit volume ([11], sect 3.11) ... 

T source term in generalized Boltzmann type of equation representing the effects of particle coalescence, breakage and collisions J c)) collision term in the Boltzmann equation... 

Hydrodynamic models are derived from the mesoscale model (e.g. the Boltzmann equation) using a Chapman-Enskog expansion in powers of the Knudsen number (Bardos et al., 1991 Cercignani et al, 1994 Chapman Cowling, 1961 Ferziger Kaper, 1972 Jenkins Mancini, 1989). The basic idea is that the collision term will drive the velocity distribution n towards an equilibrium function eq (i-e. the solution to C( eq) = 0), and thus the deviation from equilibrium can be approximated by n -i- Knui. From the... 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

Note that the transport term on the left-hand side of Eq. (6.1) can be larger or smaller in magnitude than the collision term. For cases in which the collision term is much more important than the transport term, the solution to Eq. (6.1) with the Boltzmann collision model is a local Maxwellian wherein ap. Up, and p depend on space and time but / is well approximated by Eq. (6.10). In this limit, the particles behave as an ideal gas and the mean velocity obeys the Euler equation. 

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