Non-stationary two-dimensional potential flows by the Broadwell model equations

The two-dimensional Broadwell model of discrete kinetic theory is studied i n order to clarify the physical relevance of its solutions in comparison to the solutions of the continuous Boltzmann equation. This is achieved by de termining completely, in closed form, all non-stationary potential hows wit h steady limiting conditions and isotropic pressure tensor at infinity. Sev eral classes of exact solutions are also constructed when some of the above hypotheses are dropped. Most results are made possible by suitable transfo rmations, which reduce essentially a complicated overdetermined system of p artial differential equations to solving explicitly a Liouville equation. T he structure of the obtained solutions, and especially the unphysical featu res that they exhibit, are finally commented on. It is remarkable that, for the problem considered here, there is no solution showing the typical qual itative features which characterize the continuous Boltzmann equation. (C) 2000 Editions scientifiques et medicales Elsevier SAS.