Beyond Hydrodynamics: Macroscopic Transport Equations for Rarefied Gas Flowses

Henning Struchtrup
University of Victoria

Classical hydrodynamics--the laws of Navier-Stokes and Fourier--fail in the description of processes in rarefied gases. The Boltzmann equation, on the other hand, describes a gas on the microscopic level and gives a proper description for all gas processes; its numerical solution, however, is very expensive. Macroscopic transport equations can be derived from the Boltzmann equation by averaging in velocity (moment method), and expansion in the Knudsen number (the ratio between the mean free path of a gas particle and a characteristic length of the process). Classical hydrodynamics result from expansion to first order, and higher order expansions promise to describe rarefied gases at lower cost than the Boltzmann equation. It will be shown that higher order Knudsen number expansions give meaningful equations sufficiently away from the wall, while the proper description of Knudsen boundary layers--which are dominant in slow rarefied flows--is not tied to the Knudsen number in a simple manner. Nevertheless, tests with moment systems show that a small number of moments can catch the most important Knudsen layer phenomena for Knudsen numbers below unity in sufficient accuracy. The regularized 13 moment equations are obtained by the order of magnitude method and are of third order accuracy in the bulk, but they also contain enough information to describe Knudsen layers. Analytical and numerical results for Couette, Poiseuille, and Transpiration flows and other processes will give evidence of the above statements.

Presentation (PDF File)

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