Use of the Scattering Kernel Approach in Certain Gas Kinetic Problems

Abstract

The collision gain term of the Boltzmann equation may at times be transformed to a triple integral over the product of the unknown distribution and a scattering kernel in the case of a minor constituent interacting through collisions with a major constituent (scatterer). If isotropy in velocity space may be assumed, the term may be further reduced to an integral over the particle speed. The scattering kernel for hard-sphere collisions has been used in neutron slowing down and thermalization problems, but this approach does not seem to have been used widely in other gas kinetic applications. The transformation is discussed and cast in a form that is applicable to collision models other than hard-sphere scattering and to arbitrary velocity distributions of the scatterers. Kernels are obtained for isotropic constant mean-free-time scattering, and it is shown that kernels for monoenergetic velocity distributions of the scatterers are useful as well as those for Maxwellian and stationary distributions. Use of the scattering kernel approach is illustrated by finding the steady-state velocity distribution of excited atoms that are being produced at a steady rate and that are losing kinetic energy via collisions and their excitation energy by radiation. The solution for Maxwellian scatterers is well approximated at high velocities by the solution for stationary scatterers, and at low velocities by the solution for monoenergetic scatterers.