A method has been developed for treating the Boltzmann equation for a binary gas mixture with inverse fifth power molecular interaction. This method involves replacing the complete collision integral in the Boltzmann equation with a kinetic model. This kinetic model reproduces the collisional transfer of momentum and energy for Maxwell molecules, as well as conserving mass, momentum and energy. Several interesting linearizations of the model equations are presented, among which is a simplified form particularly well suited to the case of disparate masses.

A systematic approximation has been constructed for the moment equations of kinetic theory which describe source flow expansion. For small source Knudsen number, the moments are expanded and solutions are obtained near the source. These solutions are nonuniformly valid far from the source, breaking down when transition flow is encountered. To analyze the rarefied regime, the equations are rescaled taking account that the flow is hypersonic in the transition regime. This allows us to apply a hypersonic approximation to the moment equations in the rarefied regime and subsequently match this to the inviscid solution. For spherical expansion we resolve the problem to a relaxation process with two translation temperatures, one along streamlines T∥, and the other transverse to streamlines T⊥. We obtain expressions for the terminal Mach number in terms of source Knudsen number and intermolecular force law, and a simple rarefaction criteria is found which states that transition flow is encountered when T∥ − T⊥ ≅ Tisentropic.

In gas mixtures with roughly equal masses, self-collisions and cross-collisions between species are of equal importance in the approach of the mixture to a Maxwellian distribution about a single temperature and velocity and the usual Chapman-Enskog transport theory obtains. In a disparate-mass mixture, however, the effects of self- and cross-collisions are not equal and one has an epochal relaxation, with the temperature relaxation occurring on the longest time scale. The clear distinction between a single molecular relaxation time and a hydrodynamic time scale which makes the usual Chapman-Enskog theory possible is absent for the disparate-mass case, and a more complex ordering of time scales is necessary to obtain hydrodynamic solutions. In the present work, for neutral, disparate-mass mixtures, the Boltzmann collision integrals are replaced by relaxation-type kinetic models and terms in the model equation are ordered according to mass ratio. A set of two-fluid transport equations is then derived by obtaining normal solutions of these model equations. The derived transport equations and coefficients are discussed, and their correspondence to the usual Chapman-Enskog results is shown.