The polynomial expansion method for boundary value problems of transport in rarefied gases

The principles of rarefied gas dynamics are finding application in many are as of practical interest ranging from aerosol science to micro-fabrication to vacuum and space applications. Many methods have become available for so lving the Boltzmann equation which is the complicated integro-differential equation that describes the distribution of the gas molecules and forms the basis for the field of rarefied gas dynamics. Moments methods are among th e most popular of these solution techniques due to their relative simplicit y and overall versatility. In the half-range moment method of solving bound ary value problems based on the Boltzmann equation, difficult bracket integ rals are encountered. We discuss here a simplification of such integrals sp ecific to the spherical geometry and applicable for arbitrary values of the Knudsen number. A polynomial expansion in velocity space is used to repres ent the discontinuous factor in these integrals. The accuracy of the method is verified by a comparison of the analytical results with available numer ical results.