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Kramers’ problem and the Knudsen minimum: a theoretical analysis using a linearized 26-moment approach

Research paper by Xiao-Jun Gu, David R. Emerson, Gui-Hua Tang

Indexed on: 13 Nov '09Published on: 13 Nov '09Published in: Continuum Mechanics and Thermodynamics



Abstract

A set of linearized 26 moment equations, along with their wall boundary conditions, are derived and used to study low-speed gas flows dominated by Knudsen layers. Analytical solutions are obtained for Kramers’ defect velocity and the velocity-slip coefficient. These results are compared to the numerical solution of the BGK kinetic equation. From the analysis, a new effective viscosity model for the Navier–Stokes equations is proposed. In addition, an analytical expression for the velocity field in planar pressure-driven Poiseuille flow is derived. The mass flow rate obtained from integrating the velocity profile shows good agreement with the results from the numerical solution of the linearized Boltzmann equation. These results are good for Knudsen numbers up to 3 and for a wide range of accommodation coefficients. The Knudsen minimum phenomenon is also well captured by the present linearized 26-moment equations.