Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T11:27:50.442Z Has data issue: false hasContentIssue false
  • English
  • Français

Rarefied gas flow around a sharp edge induced by a temperature field

Published online by Cambridge University Press:  17 January 2012

Satoshi Taguchi  and
Kazuo Aoki
Satoshi Taguchi*
Affiliation:
Department of Mechanical Engineering and Intelligent Systems, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
Kazuo Aoki
Affiliation:
Department of Mechanical Engineering and Science and Advanced Research Institute of Fluid Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan
*
†Email address for correspondence: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A rarefied gas flow thermally induced around a heated (or cooled) flat plate, contained in a vessel, is considered in two different situations: (i) both sides of the plate are simultaneously and uniformly heated (or cooled); and (ii) only one side of the plate is uniformly heated. The former is known as the thermal edge flow and the latter, typically observed in the Crookes radiometer, may be called the radiometric flow. The steady behaviour of the gas induced in the container is investigated on the basis of the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation and the diffuse reflection boundary condition by means of an accurate finite-difference method. The flow features are clarified for a wide range of the Knudsen number, with a particular emphasis placed on the structural similarity between the two flows. The limiting behaviour of the flow as the Knudsen number tends to zero (and thus the system approaches the continuum limit) is investigated for both flows. The detailed structure of the normal stress on the plate as well as the cause of the radiometric force (the force acting on the plate from the hotter to the colder side) is also clarified for the present infinitely thin plate.

JFM classification

Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Microfluidics Micro-/Nano-fluid dynamics: Non-continuum effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 694 , 10 March 2012 , pp. 191 - 224
Copyright
Copyright © The Author(s), 2012. Published by Cambridge University Press 2012 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

1. Aoki, K., Bardos, C., Dogbe, C. & Golse, F. 2001a A note on the propagation of boundary induced discontinuities in kinetic theory. Math. Models Meth. Appl. Sci. 11 (9), 1581–1595.CrossRefGoogle Scholar
2. Aoki, K., Degond, P. & Mieussens, L. 2009 Numerical simulations of rarefied gases in curved channels: Thermal creep, circulating flow, and pumping effect. Commun. Comput. Phys. 6 (5), 919–954.CrossRefGoogle Scholar
3. Aoki, K., Kanba, K. & Takata, S. 1997 Numerical analysis of a supersonic rarefied gas flow past a flat plate. Phys. Fluids 9 (4), 1144–1161.CrossRefGoogle Scholar
4. Aoki, K., Sone, Y. & Masukawa, N. 1995 A rarefied gas flow induced by a temperature field. In Rarefied Gas Dynamics (ed. Harvey, J. & Lord, G. ), pp. 35–41. Oxford University Press.Google Scholar
5. Aoki, K., Sone, Y., Nishino, K. & Sugimoto, H. 1991 Numerical analysis of unsteady motion of a rarefied gas caused by sudden changes of wall temperature with special interest in the propagation of a discontinuity in the velocity distribution function. In Rarefied Gas Dynamics (ed. Beylich, A. E. ), p. 222. VCH.Google Scholar
6. Aoki, K., Sone, Y. & Waniguchi, Y. 1998 A rarefied gas flow induced by a temperature field: Numerical analysis of the flow between two coaxial elliptic cylinders with different uniform temperatures. Comput. Maths Applics. 35 (1/2), 15–28.CrossRefGoogle Scholar
7. Aoki, K., Takata, S., Aikawa, H. & Golse, F. 2001b A rarefied gas flow caused by a discontinuous wall temperature. Phys. Fluids 13 (9), 2645–2661 and erratum: 13, 3843 (2001).CrossRefGoogle Scholar
8. Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525.CrossRefGoogle Scholar
9. Bird, G. A. 1976 Molecular Gas Dynamics. Oxford University Press.Google Scholar
10. Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press.CrossRefGoogle Scholar
11. Cercignani, C. 2000 Propagation phenomena in classical and relativistic rarefied gases. Transp. Theory Stat. Phys. 29, 607–614.CrossRefGoogle Scholar
12. Chu, C. K. 1965 Kinetic-theoretic description of the formation of a shock wave. Phys. Fluids 8 (1), 12–22.CrossRefGoogle Scholar
13. Cornella, B. M., Ketsdever, A. D., Gimelshein, N. E. & Gimelshein, S. F. 2011 Impact of separation distance on multi-vane radiometer configurations. In Rarefied Gas Dynamics (ed. Levin, D. A., Wysong, I. J. & Garcia, A. L. ), pp. 712–717. AIP.Google Scholar
14. Einstein, A. 1924 Zur Theorie der Radiometerkräfte. Z. Phys. 27, 1–6.CrossRefGoogle Scholar
15. Galkin, V. S., Kogan, M. N. & Fridlender, O. G. 1971 Free convection in a gas in the absence of external forces. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (3), 98–107.Google Scholar
16. Garcia, R. D. M. & Siewert, C. E. 2009 The linearized Boltzmann equation with Cercignani-Lampis boundary conditions: basic flow problems in a plane channel. Eur. J. Mech. B/Fluids 28, 387–396.CrossRefGoogle Scholar
17. Gimelshein, N. E., Gimelshein, S. F., Ketsdever, A. D. & Selden, N. P. 2011a Shear force in radiometric flows. In Rarefied Gas Dynamics (ed. Levin, D. A., Wysong, I. J. & Garcia, A. L. ), pp. 661–666. AIP.Google Scholar
18. Gimelshein, S. F., Gimelshein, N. E., Ketsdever, A. D. & Selden, N. P. 2011b Analysis and applications of radiometeric forces in rarefied gas flows. In Rarefied Gas Dynamics (ed. Levin, D. A., Wysong, I. J. & Garcia, A. L. ), pp. 693–700. AIP.Google Scholar
19. Kennard, E. H. 1938 Kinetic Theory of Gases. McGraw-Hill.Google Scholar
20. Kim, C. 2011 Formation and propagation of discontinuity for Boltzmann equation in non-convex domains. Commun. Math. Phys. 308 (3), 641–701.CrossRefGoogle Scholar
21. Kitamura, M., Tsutahara, M., Taguchi, S., Mitani, R., Enomura, M. & Zhang, X. 2009 A study of promotion of sublimation phenomenon of freeze drying by using thermal edge flow. Trans. Japan Soc. Mech. Engrs B 75 (756), 1642–1648 (in Japanese).CrossRefGoogle Scholar
22. Kogan, M. N., Galkin, V. S. & Fridlender, O. G. 1976 Stresses produced in gases by temperature and concentration inhomogeneities. New type of free convection. Sov. Phys. Usp. 19, 420–428.CrossRefGoogle Scholar
23. Loeb, L. B. 1961 Kinetic Theory of Gases, 3rd edn. Dover.Google Scholar
24. Maxwell, J. C. 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
25. Ohwada, T., Sone, Y. & Aoki, K. 1989a Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1 (9), 1588–1599.CrossRefGoogle Scholar
26. Ohwada, T., Sone, Y. & Aoki, K. 1989b 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 (12), 2042–2049 and erratum: 2, 639 (1990).CrossRefGoogle Scholar
27. Ota, M., Nakao, T. & Sakamoto, M. 2001 Numerical simulation of molecular motion around laser microengine blades. Math. Comput. Simul. 55, 223–230.CrossRefGoogle Scholar
28. Reynolds, O. 1876 On the forces caused by the communication of heat between a surface and a gas; and on a new photometer. Phil. Trans. R. Soc. Lond. 166, 725–735.Google Scholar
29. Scandurra, M., Iacopetti, F. & Colona, P. 2007 Gas kinetic forces on thin plates in the presence of thermal gradients. Phys. Rev. E 75, 026308–5.CrossRefGoogle ScholarPubMed
30. Selden, N., Ngalande, C., Gimelshein, N., Gimelshein, S. & Ketsdever, A. 2009a Origins of radiometric forces on a circular vane with a temperature gradient. J. Fluid Mech. 634, 419–431.CrossRefGoogle Scholar
31. Selden, N., Ngalande, C., Gimelshein, S., Muntz, E. P., Alexeenko, A. & Ketsdever, A. 2009b Area and edge effects in radiometric forces. Phys. Rev. E 79, 041201.CrossRefGoogle ScholarPubMed
32. Sharipov, F. & Seleznev, V. 1998 Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data 27 (3), 657–706.CrossRefGoogle Scholar
33. Sone, Y. 1966 Thermal creep in rarefied gas. J. Phys. Soc. Japan 21, 1836–1837.CrossRefGoogle Scholar
34. Sone, Y. 1969 Asymptotic theory of flow of rarefied gas over a smooth boundary I. In Rarefied Gas Dynamics (ed. Trilling, L. & Wachman, H. Y. ), vol. 1. pp. 243–253. Academic Press.Google Scholar
35. Sone, Y. 1971 Asymptotic theory of flow of rarefied gas over a smooth boundary II. In Rarefied Gas Dynamics (ed. Dini, D. ), vol. 2. pp. 737–749. Editrice Tecnico Scientfica.Google Scholar
36. Sone, Y. 1973 New kind of boundary layer over a convex solid boundary in a rarefied gas. Phys. Fluids 16 (9), 1422–1424.CrossRefGoogle Scholar
37. Sone, Y. 1984 Highly rarefied gas around a group of bodies with various temperature distributions i small temperature variation. J. Méc. Théor. Appl. 3, 315–328.Google Scholar
38. Sone, Y. 1985 Highly rarefied gas around a group of bodies with various temperature distributions ii arbitrary temperature variation. J. Méc. Théor. Appl. 4, 1–14.Google Scholar
39. 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
40. Sone, Y. 2000 Flows induced by temperature fields in a rarefied gas and their ghost effect on the behaviour of a gas in the continuum limit. Annu. Rev. Fluid Mech. 32, 779–811.CrossRefGoogle Scholar
41. Sone, Y. 2002 Kinetic Theory and Fluid Dynamics. Birkhäuser, Supplementary Notes and Errata: Kyoto University Research Information Repository (http://hdl.handle.net/2433/66099).CrossRefGoogle Scholar
42. Sone, Y. 2007 Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkhäuser, Supplementary Notes and Errata: Kyoto University Research Information Repository (http://hdl.handle.net/2433/66098).CrossRefGoogle Scholar
43. Sone, Y., Aoki, K., Takata, S., Sugimoto, H. & Bobylev, A. V. 1996 Inappropriateness of the heat-conduction equation for description of a temperature field of a stationary gas in the continuum limit: examination by asymptotic analysis and numerical computation of the Boltzmann equation. Phys. Fluids 8, 628–638 and erratum: 8, 841.CrossRefGoogle Scholar
44. Sone, Y & Sugimoto, H 1990 Strong evapouration from a plane condensed phase. In Adiabatic Waves in Liquid-Vapor Systems (ed. Meier, G. E. A. & Thompson, P. A. ), pp. 293–304. Springer.CrossRefGoogle Scholar
45. Sone, Y. & Sugimoto, H. 1993 Kinetic theory analysis of steady evapourating flows from a spherical condensed phase into a vacuum. Phys. Fluids A 5, 1491–1511.CrossRefGoogle Scholar
46. Sone, Y. & Takata, S. 1992 Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the S layer at the bottom of the Knudsen layer. Transp. Theory Stat. Phys. 21, 501–530.CrossRefGoogle Scholar
47. Sone, Y. & Tanaka, S. 1980 Thermal stress slip flow induced in rarefied gas between noncoaxial circular cylinders. In Theoretical and Applied Mechanics (ed. Rimrott, F. P. J. & Tabarrok, B. ), pp. 405–416. North-Holland.Google Scholar
48. Sone, Y. & Yoshimoto, M. 1997 Demonstration of a rarefied gas flow induced near the edge of a uniformly heated plate. Phys. Fluids 9, 3530–3534.CrossRefGoogle Scholar
49. Sugimoto, H. 2009 Experiment on the gas separation effect of the pump driven by the thermal edge flow. In Rarefied Gas Dynamics (ed. Abe, T. ), pp. 1123–1128. AIP.Google Scholar
50. Sugimoto, H. & Sone, Y. 1992 Numerical analysis of steady flows of a gas evapourating from its cylindrical condensed phase on the basis of kinetic theory. Phys. Fluids A 4, 419–440.CrossRefGoogle Scholar
51. Sugimoto, H. & Sone, Y. 2005 Vacuum pump without a moving part by thermal edge flow. In Rarefied Gas Dynamics (ed. Capitelli, M. ), pp. 168–173. AIP.Google Scholar
52. Taguchi, S. & Aoki, K. 2011 Numerical analysis of rarefied gas flow induced around a flat plate with a single heated side. In Rarefied Gas Dynamics (ed. Levin, D. A., Wysong, I. J. & Garcia, A. L. ), pp. 790–795. AIP.Google Scholar
53. Takata, S., Funagane, H. & Aoki, K. 2010 Fluid modelling for the Knudsen compressor: case of polyatomic gases. Kinet. Relat. Mod. 3 (2), 353–372.CrossRefGoogle Scholar
54. Takata, S., Sone, Y. & Aoki, K. 1993 Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 5, 716–737.CrossRefGoogle Scholar
55. Wadsworth, D. C. & Muntz, E. P. 1996 A computational study of radiometric phenomena for powering microactuators with unlimited displacements and large available forces. J. Microelectromech. Syst. 5 (1), 59–65.CrossRefGoogle Scholar
56. Welander, P. 1954 On the temperature jump in a rarefied gas. Ark. Fys. 7, 507.Google Scholar