Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T16:56:08.290Z Has data issue: false hasContentIssue false
  • English
  • Français

Poiseuille and thermal transpiration flows of a highly rarefied gas: over-concentration in the velocity distribution function

Published online by Cambridge University Press:  16 February 2011

SHIGERU TAKATA  and
HITOSHI FUNAGANE
SHIGERU TAKATA*
Affiliation:
Department of Mechanical Engineering and Science, Kyoto University, Kyoto 606-8501, Japan Advanced Research Institute of Fluid Science and Engineering, Kyoto University, Kyoto 606-8501, Japan
HITOSHI FUNAGANE
Affiliation:
Department of Mechanical Engineering and Science, Kyoto University, Kyoto 606-8501, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Poiseuille and thermal transpiration flows of a highly rarefied gas are investigated on the basis of the linearized Boltzmann equation, with a special interest in the over-concentration of molecules on velocities parallel to the walls. An iterative approximation procedure with an explicit error estimate is presented, by which the structure of the over-concentration is clarified. A numerical computation on the basis of the procedure is performed for a hard-sphere molecular gas to construct a database that promptly gives the induced net mass flow for an arbitrary value of large Knudsen numbers. An asymptotic formula of the net mass flow is also presented for molecular models belonging to Grad's hard potential. Finally, the resemblance of the profiles between the heat flow of the Poiseuille flow and the flow velocity of the thermal transpiration is pointed out. The reason is also given.

JFM classification

Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Non-continuum effects Rarefied Gas Flow: Rarefied Gas Flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 669 , 25 February 2011 , pp. 242 - 259
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andries, P., LeTallec, P., Perlat, J. P. & Perthame, B. 2000 The Gaussian–BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluids 19, 813–830.CrossRefGoogle Scholar
Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. Part I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525.CrossRefGoogle Scholar
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press.CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd (rev.) edn. Dover.Google Scholar
Cercignani, C. 1963 Plane Poiseuille flow and Knudsen minimum effect. In Rarefied Gas Dynamics (ed. Laurmann, J. A.), vol. II, pp. 92–101. Academic.Google Scholar
Cercignani, C. 1988 The Boltzmann Equation and its Applications. Springer.CrossRefGoogle Scholar
Cercignani, C. 2006 Slow Rarefied Flows. Birkhäuser.CrossRefGoogle Scholar
Cercignani, C. & Sernagiotto, F. 1966 Cylindrical Poiseuille flow of a rarefied gas. Phys. Fluids 9, 40–44.CrossRefGoogle Scholar
Chen, C.-C., Chen, I.-K., Liu, T.-P. & Sone, Y. 2007 Thermal transpiration for the linearized Boltzmann equation. Commun. Pure Appl. Math. 60, 0147–0163.CrossRefGoogle Scholar
Grad, H. 1963 Asymptotic theory of the Boltzmann equation. Part II. In Rarefied Gas Dynamics (ed. Laurmann, J. A.), vol. I, pp. 26–59. Academic.Google Scholar
Holway, L. H. Jr 1963 Approximation procedures for kinetic theory. PhD thesis, Harvard University, Cambridge, MA.Google Scholar
Holway, L. H. Jr 1966 New statistical models for kinetic theory: methods of construction. Phys. Fluids 9, 1658–1673.CrossRefGoogle Scholar
Kennard, E. H. 1938 Kinetic Theory of Gases. McGraw-Hill.Google Scholar
Kosuge, S., Sato, K., Takata, S. & Aoki, K. 2005 Flows of a binary mixture of rarefied gases between two parallel plates. In Rarefied Gas Dynamics (ed. Capitelli, M.), pp. 150–155. AIP.Google Scholar
Kosuge, S. & Takata, S. 2008 Database for flows of binary gas mixtures through a plane microchannel. Eur. J. Mech. B Fluids 27, 444–465.CrossRefGoogle Scholar
Koura, K. & Matsumoto, H. 1991 Variable soft sphere molecular model for inverse-power-law or Lennard–Jones potential. Phys. Fluids A 3, 2459–2465.CrossRefGoogle Scholar
Loyalka, S. K. 1971 Kinetic theory of thermal transpiration and mechanocaloric effect. Part I. J. Chem. Phys. 55, 4497–4503.CrossRefGoogle Scholar
McCormack, F. J. 1973 Construction of linearized kinetic models for gaseous mixtures and molecular gases. Phys. Fluids 16, 2095–2105.CrossRefGoogle Scholar
Mori, M. 2005 Discovery of the double exponential transformation and its developments. Publ. RIMS Kyoto Univ. 41, 897–935.CrossRefGoogle Scholar
Mori, M. & Sugihara, M. 2001 The double-exponential transformation in numerical analysis. J. Comput. Appl. Math. 127, 287–296.CrossRefGoogle Scholar
Niimi, H. 1968 Thermal creep flow of rarefied gas through a cylindrical tube. J. Phys. Soc. Japan 24, 225.CrossRefGoogle Scholar
Niimi, H. 1971 Thermal creep flow of rarefied gas between two parallel plates. J. Phys. Soc. Japan 30, 572–574.CrossRefGoogle Scholar
Ohwada, T., Sone, Y. & Aoki, K. 1989 Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1, 2042–2049.CrossRefGoogle Scholar
Sone, Y. 1969 Asymptotic theory of flow of rarefied gas over a smooth boundary. Part I. In Rarefied Gas Dynamics (ed. Trilling, L. & Wachman, H. Y.), vol. I, pp. 243–253. Academic.Google Scholar
Sone, Y. 1991 Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers. In Advances in Kinetic Theory and Continuum Mechanics (ed. Gatignol, R. & Soubbaramayer, ), pp. 19–31. Springer.CrossRefGoogle Scholar
Sone, Y. 2007 Molecular Gas Dynamics. Birkhäuser. Supplemental notes and errata are available at: http://hdl.handle.net/2433/66098.CrossRefGoogle Scholar
Sone, Y., Ohwada, T. & Aoki, K. 1989 Temperature jump and Knudsen layer in a rarefied gas over a plane wall: numerical analysis of the linearized Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1, 363–370.CrossRefGoogle Scholar
Takahasi, H. & Mori, M. 1974 Double exponential formulas for numerical integration. Publ. RIMS Kyoto Univ. 9, 721–741.CrossRefGoogle Scholar
Takata, S. 2009 Symmetry of the linearized Boltzmann equation and its application. J. Stat. Phys. 136, 751–784. ((5/2)γ2 on line 18, p. 762, and line 24, p. 763, in this reference is a misprint of (5/4)γ2.)CrossRefGoogle Scholar
Welander, P. 1954 On the temperature jump in a rarefied gas. Ark. Fys. 7, 507–553.Google Scholar