Poiseuille Flow and Thermal Transpiration of a Rarefied Polyatomic Gas Through a Circular Tube with Applications to Microflows

Funagane, Hitoshi ; Takata, Shigeru ; Aoki, Kazuo ; Kugimoto, Ko

Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 1, pp. 19-46

Résumé

As the first step, a rarefied polyatomic gas in a long and straight circular tube is considered, and the flow caused by a small uniform pressure gradient (Poiseuille flow) and the flow induced by a small uniform temperature gradient along the tube (thermal transpiration) are investigated, using the ellipsoidal statistical (ES) model of the Boltzmann equation for a polyatomic gas. It is shown that the solutions to these problems can be reduced to those based on the Bhatnagar-Gross-Krook (BGK) model for a monatomic gas. Numerical results of the velocity profiles, mass-flow rates, etc. for the Nitrogen gas, obtained by exploiting the existing database based on the BGK model, are shown. As the second step, a rarefied polyatomic gas in a long circular pipe is considered in the following situation: (i) the pressure and temperature variations along the pipe can be arbitrary and large; (ii) the length scale of variations is much longer than the radius of the pipe; (iii) the pipe may consist of circular tubes with different radii connected one after another. It is shown that, in this situation, the pressure distribution along the pipe is described by a macroscopic equation of diffusion type, with the diffusion coefficients consisting of the mass-flow rates of the Poiseuille flow and thermal transpiration, and an appropriate condition at the junction where the cross section changes suddenly. Then, the system is applied to a polyatomic gas flow through a single long pipe caused by a large pressure difference imposed at both ends and to a Knudsen compressor consisting of many alternately arranged thinner and thicker circular tubes.