Balance Laws and Constitutive Relations for Plane Flows of a Dense, Binary Mixture of Smooth, Nearly Elastic, Circular Disks

Abstract

We derive balance laws and constitutive relations for plane flows of a dense, binary mixture of smooth, nearly elastic, circular disks. The disks may differ in size and mass and in the coefficients of restitution characterizing the energy lost in collisions between like and unlike pairs. We focus attention on those parts of the fluxes and sources of momentum and energy that are due to collisions. To calculate them, we suppose that the complete pair distribution function for two colliding disks is the product of Maxwellian velocity distributions for each disk and a factor that incorporates the effects of excluded area and collisional shielding. In an Appendix we provide constitutive relations calculated in a similar way for a dense, binary mixture of smooth, nearly elastic, spheres.