Radial diffusion in the poiseuille flow of a gas mixture

In a number of experiments (see 1], in which experimental papers are listed), diffusion has been observed in the radial direction in the process of flow of a mixture along tubes at low pressures. The heavier molecules accumulate near the tube axis. The attempt made in 1] to explain this phenomenon by the influence of the Burnett contribution to the diffusion did not lead to success, and the Burnett terms in the radial diffusion velocity indicate a motion of heavy molecules away from the tube axis. In the present paper, a complete analysis is given of this phenomenon. We consider the problem of the flow of a mixture along a cylindrical tube of finite length for given pressure difference p between its ends. On the basis of the hydrodynamic equations of the Burnett and super-Burnett approximations, a consistent asymptotic (with respect to the small parameter ) solution is given; = (p/p)R/L is the relative change in the pressure along the tube at a distance of order R (R and L are the radii and length of the tube). Radial diffusion occurs in the quadratic approximation in . It is shown that the radial diffusion velocity contains new terms not present in 1]; these are due to the inhomogeneity of the temperature and the pressure over the tube section, the expansion of the gas, and the super-Burnett correction to the diffusion velocity. The most important is the thermodiffusion term, which is determined by the hydrodynamic equations of the Navier-Stokes approximation. The remaining terms have order relative to it of Kn² (Kn = 1 /R is the Knudsen number, and 1 is the mean free path of the molecules). The expression obtained for the diffusion velocity agrees in sign with the experiment.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–96, September–October, 1979.I am grateful to G. E. Skvortsov, who drew my attention to this problem, and Yu. N. Grigor'ev for discussing the results.