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

Physics of Fluids

Volumn 8, Issue 2, 1996, Pages 628-638

  Sone, Yoshio ^a   Aoki, Kazuo ^a   Takata, Shigeru ^a   Sugimoto, Hiroshi ^a   Bobylev, A V ^b  

^a KYOTO UNIVERSITY   (Japan)

^b KELDYSH INSTITUTE OF APPLIED MATHEMATICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

ANALYSIS; ASYMPTOTIC; BOLTZMAN; CONDUCTION; EQUATIONS; FIELD; HEAT; TEMPERATURE;

EID: 0029671666     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.868846     Document Type: Article

References (32)

1. D. Hilbert, Grundzuge einer Allgemeinen Theorie der Linearen Integralgleichungen (Teubner, Wien, 1924), Chap. 22.
2. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, 1970), Chaps. 7, 10, and 15.
3. Y. Sone, "Asymptotic theory of flow of rarefied gas over a smooth boundary I," in Rarefied Gas Dynamics, edited by L. Trilling and H. Y. Wachman (Academic Press, New York, 1969), Vol. 1, p. 243; "Asymptotic theory of flow of rarefied gas over a smooth boundary II," in Rarefied Gas Dynamics, edited by D. Dini (Editrice Tecnico Scientifica, Pisa, 1971), Vol. 2, p. 737; " Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers," in Advances in Kinetic Theory and Continuum Mechanics, edited by R. Gatignol and Soubbaramayer (Springer-Verlag, Berlin, 1991), p. 19.
4. Y. Sone, "Asymptotic theory of flow of rarefied gas over a smooth boundary I," in Rarefied Gas Dynamics, edited by L. Trilling and H. Y. Wachman (Academic Press, New York, 1969), Vol. 1, p. 243; "Asymptotic theory of flow of rarefied gas over a smooth boundary II," in Rarefied Gas Dynamics, edited by D. Dini (Editrice Tecnico Scientifica, Pisa, 1971), Vol. 2, p. 737; " Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers," in Advances in Kinetic Theory and Continuum Mechanics, edited by R. Gatignol and Soubbaramayer (Springer-Verlag, Berlin, 1991), p. 19.
5. Y. Sone, "Asymptotic theory of flow of rarefied gas over a smooth boundary I," in Rarefied Gas Dynamics, edited by L. Trilling and H. Y. Wachman (Academic Press, New York, 1969), Vol. 1, p. 243; "Asymptotic theory of flow of rarefied gas over a smooth boundary II," in Rarefied Gas Dynamics, edited by D. Dini (Editrice Tecnico Scientifica, Pisa, 1971), Vol. 2, p. 737; " Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers," in Advances in Kinetic Theory and Continuum Mechanics, edited by R. Gatignol and Soubbaramayer (Springer-Verlag, Berlin, 1991), p. 19.
6. Y. Sone and K. Aoki, Molecular Gas Dynamics (Asakura, Tokyo, 1994), Chap. 3 (in Japanese).
7. Y. Sone and M. Wakabayashi, "Flow induced by nonlinear thermal stress in a rarefied gas," in Proceedings of Symposium on Mechanics of Space Flight (Institute of Space Sciences, Tokyo, 1988), p. 14 (in Japanese), quoted in Ref. 4.
8. A. V. Bobylev, "Quasistationary hydrodynamics for the Boltzmann equation," J. Stat. Phys. 80, 1063 (1995).
9. M. N. Kogan, "On the equations of motion of a rarefied gas," Appl. Math. Mech. 22, 597 (1958).
10. C. Cercignani, The Boltzmann Equation and Its Applications (Springer-Verlag, Berlin, 1988).
11. The k is used in carrying out analysis (mainly in Secs. II, III A, and III B), and Kn in explanation (mainly in Secs. III C and IV).
12. H. Grad, "Principles of the kinetic theory of gases," in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1958), Band XII, p. 205.
13. iH0≡0 is compatible with the Euler set of equations, which is derived from the first set of the solvability conditions [Eq. (16) with m= 1].
14. Kl is assumed to decay very rapidly with η. This introduces rearrange-ment of the ordering with respect to powers of k.
15. C. Bardos, R. E. Caflisch, and B. Nicolaenko, "The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas,"Commun. Pure Appl. Math. 39, 323 (1986).
16. C. Cercignani, "Half-space problems in the kinetic theory of gases," in Trends in Applications of Pure Mathematics to Mechanics, edited by E. Kröner and K. Kirchgässner (Springer-Verlag, Berlin, 1986), p. 35.
17. -1, which is used in the derivation of Eq. (31) (see Ref. 12).
18. Y. Sone, T. Ohwada, and K. Aoki, "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 (1989).
19. T. Ohwada, Y. Sone, and K. Aoki, "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, 1588 (1989).
20. Y. Sone and Y. Onishi, "Kinetic theory of evaporation and condensation _ Hydrodynamic equation and slip boundary condition," J. Phys. Soc. Jpn. 44, 1981 (1978).
21. E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill, New York, 1938), p. 327.
22. Y. Sone, "Thermal creep in rarefied gas," J. Phys. Soc. Jpn. 21, 1836 (1966).
23. M. N. Kogan, V. S. Galkin, and O. G. Fridlender, "Stresses produced in gases by temperature and concentration inhomogeneities. New type of free convection," Sov. Phys. Usp. 19, 420 (1976).
24. A shock layer, except a weak one, cannot correctly be described by the Navier-Stokes set. Analysis by kinetic theory is required for its correct description.
25. H. Grad, "Singular and nonuniform limits of solutions of the Boltzmann equation," in Transport Theory, edited by R. Bellman, G. Birkhoff, and I. Abu-Shumays (American Mathematical Society, Providence, RI, 1969), p. 269.
26. C. Cercignani and C. Lancellotti, "Velocity slip for curved shock waves," in Rarefied Gas Dynamics, edited by A. E. Beylich (VCH, Weinheim, 1991), p. 222.
27. Y. Sone and S. Takata, "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 (1992).
28. K. Aoki, Y. Sone, and T. Yamada, "Numerical analysis of gas flows condensing on its plane condensed phase on the basis of kinetic theory," Phys Fluids A 2, 1867 (1990).
29. H. Sugimoto and Y. Sone, "Numerical analysis of steady flows of a gas evaporating from its cylindrical condensed phase on the basis of kinetic theory," Phys. Fluids A 4, 419 (1992).
30. S. Takata, Y. Sone, and K. Aoki, "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 (1993).
31. C. L. Pekeris and Z. Alterman, "Solution of the Boltzmann-Hilbert integral equation II. The coefficients of viscosity and heat conduction," Proc. Natl. Acad. Sci. USA 43, 998 (1957).
32. T. Ohwada and Y. Sone, "Analysis of thermal stress slip flow and negative thermophoresis using the Boltzmann equation for hard-sphere molecules," Eur. J. Mech., B/Fluids 11, 389 (1992).