Collision source term integer moment

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. 

At this point, the remaining integrals in the collision source terms are with respect to the velocity density functions. We will now consider the special case in which ij/ represents the integer moments with respect to the velocity components. 

These coefficients, and the summations in Eq. (6.60), are most easily computed using a symbolic math program. Some examples of co, v, g) for selected moments up to fifth order are given in Tables 6.1-6.9. For clarity, in these tables we have denoted the velocity difference vector by g = gi,g2,g3) and g =g + gl+gy The final expressions for the collision source terms for integer moments of order j = h+h + h can now be written in the form of Eq. (6.54) as... 

We can now express the collision source term and collisional-flux term for the integer moments of order y as... 

The term in Eq. (6.167) can be rewritten in closed form using integer moments of up to third order. On the other hand, the right-hand side of Eq. (6.168) cannot be written in terms of any finite set of integer moments and must therefore be closed (e.g. using quadrature). The collision source term C2oo,i2 has four contributions ... 

The analytical expressions for will be used directly with quadrature-based moment methods to evaluate the collision source and collisional-flux terms for each integer moment. The numerical implementation of these terms in the context of quadrature is discussed in Section 6.5. 

Chapter 6 is devoted to the topic of hard-sphere collision models (and related simpler kinetic models) in the context of QBMM. In particular, the exact source terms for integer moments due to collisions are derived in the case of inelastic binary collisions between two particles with different diameters/masses, and the use of QBMM to overcome the closure problem is illustrated. 

---