Collisional pair distribution function

In order to evaluate the collisional integrals Pc, qc, and y, explicitly, it is important to know the specific form of the pair distribution function /(2)(vi, ri V2, ry. t). The pair distribution function /(2) may be related to the single-particle velocity distribution function f l by introducing a configurational pair-correlation function g(ri, r2). In the following, we first introduce the distribution functions and then derive the expression of /(2) in terms of f l by assuming /(1) is Maxwellian and particles are nearly elastic (i.e., 1 — e 1). 

Define an L-particle configurational distribution function n(L)(ri, r2. /t) such that (L)(rj, r2. rifSri Sty is the probability of finding a particle in each of the volume elements 8r, Sr2. centered on rj, r2. rt- The one-particle distribution function n(X (r) is just the number density n of particles at r. For a homogeneous bulk phase, we have 

For an amorphous mass of particles there is no correlation between particles that are far apart. The joint probability of finding particles at r and ri is simply the product of the individual probabilities. Let us define a configurational pair-correlation function g(ri, r2) as 

For a gas at equilibrium, i.e., no mean deformation, there is a spatial homogeneity and thus g(ri,r2) depends only on the separation distance r = ri — r2. Then g = go(r) is termed the radial distribution function, which may be interpreted as the ratio of the local number density at a distance r from the central particle to the bulk number density. For a system of identical spheres, the radial distribution function go O ) at contact (i.e., r = dp) can be expressed in terms of the volume fraction of solids ap as 

Assume that a complete pair distribution function can be expressed as the product of the spatial pair distribution function and the two single-particle velocity distribution functions. Thus, we have 

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Now, we proceed to the evaluation of the collisional integrals Pc, qc, and y by using Eq. (5.293) as the form of a collisional pair distribution function. Use the coordinates in Fig. 5.12, in which ez is chosen to be parallel to the relative velocity vn. 9 and 0 are the polar angles of k with respect to ez and the plane of ez and ex, respectively. ex, ey, and ez are the three mutually perpendicular unit vectors corresponding to each coordinate in Fig. 5.12. k, as mentioned in the previous section, is the unit normal on the collision point directed from the center of particle 1 to the center of particle 2. Thus, we have... 

To generalize the formulation of the collision frequency a collisional pair distribution function ri,ci,r2,C2) is sometimes introduced, following... 

For completeness it is noted that these relations rely on the molecular chaos assumption (see Sect. 2.4.2), anticipating that the collisional pair distribution function can be approximated by [53, 90] ... 

By the assumption of particulate chaos, the collisional pair distribution function was expanded in Taylor series following a similar approach as for mono-particle mixtures (4.25) ... 

Pair distribution function in collisional theory Terms defined in granular flow constitutive equations... 

The collisional pressure tensor naturally divides into two separate integrals when expanding the pair distribution function by truncated Taylor series. From Eq. (4.349), the integral could thus be separated into two parts. The pressure... 

For moderately dense systems, say v<0.49, at which a first-order phase transition from a random to order collisional gas is first possible (Adler and Wainwright 1957), the complete pair distribution function for a colliding pair is assumed to be the product of the single particle distribution function of each sphere, evaluated at its center, and a factor gQ that incorporates the influence of the volume occupied by the spheres on their collision frequency. This factor is the equilibrium radial distribution function, evaluated at the point of contact. It is given as a function of v by Carnahan and Starling (1970) as... 

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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