A Class of Planar Discrete Velocity Models for Gas Mixtures

We present a method for the construction of a simple class of physically acceptable planar discrete velocity models (DVMs) for binary gas mixtures. We want five conservation laws (no more, no less) with binary collisions. We first consider a collision with a particle at rest and different possibilities for the three other particles. We associate other particles and find semisymmetric qv _i models with q=7, 9, 11, 13 and 15, symmetric with respect to the two coordinate axes, but not to an exchange between the two axes. In order to avoid spurious mass conservation relations for the species without particle at rest, we find, for the two coordinate axes, that the tips of the momenta of the particles must be on two intervals parallel to one axis with opposite values on the other. There remain some physically acceptable q=9 (the smallest) and 11, 13, 15 models (adding multiple collisions for some others). Second, we construct the associated symmetric models qv _i^qv _i, which are superpositions of the qv _i model and another ^qv _i, rotated by /2. The possible previous defect of the spurious mass invariant for qv _i is transmitted to the symmetric one. We explain another defect coming from qv _i and ^qv _i having only one common particle, then spurious invariants exist for the momentum conservations along the two axes. We get four physically acceptable symmetric 17v _i (and three intermediate semisymmetric 13v _i models) and one 25v _i model superposition of two 11v _i and two 15v _i models (other acceptable symmetric 11v _i, 13v _i, and 25v _i models exist with multiple collisions).