The collision term for arbitrary moments

The goal of the next series of steps is to obtain a form of Eq. (6.13) wherein the pair correlation function depends on only one spatial location (i.e. the point where the particles touch during a collision). We start by adding and subtracting the pair distribution function / (x - r/pXi2, vi X, V2) to / (x, vi x + ipXi2, V2) in Eq. (6.13)  

Using Taylor-series expansions about the collision contact point. 

Note that the sums on the right-hand sides can be rewritten as multinomial expansions. We will make use of this fact at several points in the derivation. 

In most applications, the summation terms for n = 1,. oo are neglected in the definitions of C and G, and henceforth we will do the same. This approximation amounts to assuming that dp is much smaller than the characteristic length scale of the spatial gradients of (i.e. dp a ) and thus that only terms up to order dp are important. 

The next step is to provide a closure for the pair correlation function appearing in the collision source and collisional-flux terms. For moderately dense flows, the collision frequency for finite-size particles is known to be larger than that found using the Boltzmann Stofizahlansatz (Carnahan Starling, 1969 Enksog, 1921). In order to account for this effect, the pair correlation function can be modeled as the product of two single-particle velocity distribution functions and a radial distribution function  

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