Planar (x,z) mixtures discrete Boltzmann equations with species either in odd or even ∣x∣+∣z∣ values

Abstract

We present large size “physical” discrete Boltzmann equations or discrete velocity models results for binary mixtures of gas (light mass 1 and heavy mass M) in a half-plane (coordinates x,z, interface at z=0 without velocities except the rest-particle) where the sums ∣x∣+∣z∣ are either odd or even for heavy or light species. The models fill all integer coordinates z≠0 of the plane with only a z spatial dependence [densities with (±x, z) are equal]. Here, to previous results, we generalize with new mathematical tools, to any binary mixture with M ratio of any even to odd values, M=2p∕(2q+1) and ratio of odd to odd values M=(2p+1)∕(2q+1), p,q arbitrary integers. With only binary collisions, we construct large size “physical” discrete models (only mass, energy, and momentum along the z-axis invariants, without other invariants which are called spurious). We prove the existence of a “physical” square (diagonals along the x and z axes) grid where for the first component all ∣x∣+∣z∣⩽ a sufficient value (function of p and q) while for the other “physical” components we add 2, 4, etc., to the sufficient value. The numerical “physical” applications, with only binary collisions, could be done with this grid. The heavy species Hausdorff’ dimension has the limit 2 when the number of velocities →∞.