Collision model Boltzmann

We shall start with the Boltzmann collision model, which belongs to the last group. The Boltzmann-induced distribution FB. established at the instant t = 0 of the strong collision, in the case of an isotropic medium, is given by... 

The elastic Boltzmann collision model has a unique equilibrium solution defined by C = 0. This Maxwellian distribution is given by... 

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. 

Starting from the Boltzmann collision model, we write down the formula for the Boltzmann orientational distribution FB pertaining to weak electromagnetic... 

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. 

In this section we will briefly review the collision model for binary hard-sphere collisions using the notation in Fox Vedula (2010). The change in the number-density function due to elastic hard-sphere collisions (Boltzmann, 1872 Cercignani, 1988 Chapman Cowling, 1961 Enksog, 1921) obeys an (unclosed) integral expression of the form ... 

Owing to the presence of the pair correlation function, the collision model in Eq. (6.2) is unclosed. Thus, in order to close the kinetic equation (Eq. 6.1), we must provide a closure for written in terms of /. The simplest closure is the Boltzmann Stofizahlansatz (Boltzmann, 1872) ... 

Second, the SF (254) is used in the case of the Gross collision model as a constitutive element of the formula for the complex susceptibility Xg I 1 this case the orientational distribution Fq, differing from FB, changes radically the calculated low-frequency spectrum, while the far-IR spectrum is very close to that given by the Boltzmann susceptibility Xb (252). We shall return to this point in Section IX.D. 

Now we consider an important self-consistent Gross collision model. In this model just as in the Boltzmann one (with F = FB) the velocity distribution function is assumed to be unperturbed by a.c. field. We represent the Gross induced distribution with F = Fq as... 

Keywords Binary drop colUsions Bouncing Coalescence Collision model Crossing separation Gaseous environment Immiscible liquids Lattice-Boltzmann simulation Miscible liquids Navier-Stokes simulation Reflexive separation Satellite droplets Spray flow simulation SPH simulation Stretching separation... 

Any solution about Boltzmann equation needs an expression for collision term Q(f). The complexity of it, carry the search of simple models of collision processes, it will permit to make easy the mathematical analysis. Perhaps collision model more known was suggested simultaneously by Bhatnagar, Gross and Krook (Bhatnagar et al., 1954) and it is known like BGK ... 

The basic ingredients in the formulation of kinetic models are the matrix elements of the memory function calculated using an appropriate set of momentum basis functions. In the analysis of the Boltzmann collision operator it has been found convenient to use the Sonine polynomials. However, the k r dependence in (124) destroys the rotational symmetry present in the Boltzmann collision operator for a spherically symmetric potential therefore it is equally appropriate to choose the function... 

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 model of electron scattering in high-mobility systems applied in the simulations is rather simplified. Especially, the assumption that the electron velocity is randomized at each scattering to restore the Maxwell-Boltzmann distribution may be an oversimplification. If the dissipation of energy by electron collisions in a real system is less efficient than that assumed in the simulation, the escape probability is expected to further increase. 

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