A Class of Planar Discrete Velocity Models for Gas Mixtures
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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 qvi 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 qvi∪^qvi, which are superpositions of the qvi model and another ^qvi, rotated by π/2. The possible previous defect of the spurious mass invariant for qvi is transmitted to the symmetric one. We explain another defect coming from qvi and ^qvi having only one common particle, then “spurious” invariants exist for the momentum conservations along the two axes. We get four physically acceptable symmetric 17vi (and three intermediate semisymmetric 13vi models) and one 25vi model superposition of two 11vi and two 15vi models (other acceptable symmetric 11vi, 13vi, and 25vi models exist with multiple collisions).
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