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

Hitoshi Funagane; Shigeru Takata; Kazuo Aoki; Ko Kugimoto

Bollettino dell'Unione Matematica Italiana (2011)

  • Volume: 4, Issue: 1, page 19-46
  • ISSN: 0392-4041

Abstract

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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.

How to cite

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Funagane, Hitoshi, et al. "Poiseuille Flow and Thermal Transpiration of a Rarefied Polyatomic Gas Through a Circular Tube with Applications to Microflows." Bollettino dell'Unione Matematica Italiana 4.1 (2011): 19-46. <http://eudml.org/doc/290739>.

@article{Funagane2011,
abstract = {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.},
author = {Funagane, Hitoshi, Takata, Shigeru, Aoki, Kazuo, Kugimoto, Ko},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {19-46},
publisher = {Unione Matematica Italiana},
title = {Poiseuille Flow and Thermal Transpiration of a Rarefied Polyatomic Gas Through a Circular Tube with Applications to Microflows},
url = {http://eudml.org/doc/290739},
volume = {4},
year = {2011},
}

TY - JOUR
AU - Funagane, Hitoshi
AU - Takata, Shigeru
AU - Aoki, Kazuo
AU - Kugimoto, Ko
TI - Poiseuille Flow and Thermal Transpiration of a Rarefied Polyatomic Gas Through a Circular Tube with Applications to Microflows
JO - Bollettino dell'Unione Matematica Italiana
DA - 2011/2//
PB - Unione Matematica Italiana
VL - 4
IS - 1
SP - 19
EP - 46
AB - 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.
LA - eng
UR - http://eudml.org/doc/290739
ER -

References

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