Analysis of the slip coefficient and defect velocity in the Knudsen layer of a rarefied gas using the linearized moment equations

The linearized R13 and R26 moment equations are used to study Kramers’ problem. Analytical solutions for the defect velocity and slip coefficient are derived and compared with numerical results from the kinetic theory. It is found that the linearized R26 equations can capture the Knudsen layer fairly accurately in terms of the defect velocity and slip coefficient, while the linearized R13 equations underpredict the kinetic data. At the wall, however, the kinetic models predict a slightly higher value for the defect velocity than the linearized R26 equations. In general, the linearized R26 equations perform well for both specular and diffusive walls.

Figure 1

Defect velocity profile for Kramers’ problem: comparison between the moment equation solutions (lines) and kinetic theory (symbols) [27]. (Note: the original data in Ref. [27] are presented in terms of the mean free path defined by $l=\left(2/\sqrt{\pi }\right)\lambda$. The data were converted to be consistent with the present definition of the mean free path $\lambda$.)

Figure 2

Analysis of the slip coefficient: comparison between the moment equation solutions (lines) and kinetic theory (symbols) [27]. (Note: the original data in Ref. [27] are presented in terms of the mean free path defined by $l=\left(2/\sqrt{\pi }\right)\lambda$. The data were converted to be consistent with the present definition of the mean free path $\lambda$.)