Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite-difference analysis of the nonlinear Boltzmann equation for hard-sphere molecules

Physics of Fluids

Volumn 8, Issue 8, 1996, Pages 2153-2160

  Ohwada, Taku ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

HEAT TRANSFER; RAREFIED GAS; TEMPERATURE DISTRIBUTIONS;

EID: 0029831262     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.868989     Document Type: Article

References (43)

1. C. S. Wang Chang and G. E. Uhlenbeck, "The heal transport between two parallel plates as functions of the Knudsen number," University of Michigan Report M999, 1953.
2. E. P. Gross and S. Ziering, "Heat flow between parallel plates," Phys. Fluids 2, 701 (1959).
3. S. Ziering, "Shear and heat flow for Maxwellian molecules," Phys. Fluids 3, 503 (1960).
4. C. Y. Liu and L. Lees, "Kinetic theory description of plane compressible Couette flow," in Rarefied Gas Dynamics, edited by L. Talbot (Academic Press, New York, 1961), p. 391.
5. D. R. Willis, "Heat transfer in a rarefied gas between parallel plates at large temperature ratios," in Rarefied Gas Dynamics, edited by J. L. Laurmann (Academic Press, New York, 1963), p. 209.
6. F. Frankowski, Z. Alterman, and C. L. Pekeris, "Heat transport between parallel plates in a gas of rigid sphere molecules," Phys. Fluids 8, 245 (1965).
7. P. Bassanini, C. Cereignani, and C. D. Pagani, "Comparison of kinetic theory analyses of linearized heat transfer between parallel plates," Int. J. Heat Mass Transfer 10, 447 (1967).
8. P. Bassanini, C. Cercignani, and C. D. Pagani, "Influence of the accommodation caefficient on the heat transfer in a rarefeid gas," Int. J. Heat Mass Transfer 11, 1359 (1968).
9. W. P. Teagan and G. S. Springer, "Heat-transfer and density-distribution mesurments between parellel plates in the transition regime," Phys. Fluids 11, 497 (1968).
10. D. A. Alofs, R. C. Flagan, and G. S. Springer, "Density distribution measurements in rarefied gases contained between parellel plates at high temperature differences," Phys. Fluids 14, 529 (1971).
11. J. W. Cipolla and C. Cercignani, "Effect of molecular and boundary conditions on linearized heat transfer," in Rarefied Gas Dynamics, edited by D. Dini (Editrice Tecnico Scientifica, Pisa, 1971), p. 767.
12. J. R. Thomas, Jr., T. S. Chang, and C. E. Siewert, "Heat transfer between parallel plates with arbitrary surface accommodation," Phys. Fluids 16, 2116 (1973).
13. Y. Sone and Y. Onishi, "Heat transfer through a rarefied gas between coaxial circular cylinders with large temperature difference," in Rarefied Gas Dynamics, edited by J. L. Potter (AIAA, New York, 1977), p. 295.
14. T. Ohwada, K. Aoki, and Y. Sone, "Heat transfer and temperature distribution in a rarefied gas between two parallel plates with different temperatures: Numerical analysis of the Boltzmann equation for a hard sphere molecule," in Rarefied Gas Dynamics: Theoretical and Computational Techniques, edited by E. P. Muntz, D. P. Weaver, and D. H. Campbell (AIAA, Washington, DC, 1989), p. 70.
15. T. Inamuro and B. Strutevant, "Numerical study of discrete-velocity gases," Phys. Fluids A 2, 2196 (1990).
16. D. C. Wadsworth, "Slip effects in a confined rarefied gas. I: Temperature slip," Phys. Fluids A 5, 1831 (1993).
17. 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).
18. Y. Sone, T. Ohwada, and K. Aoki, "Evaporation and condensation on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules," Phys. Fluids A 1, 1398 (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. T. Ohwada, Y. Sone, and K. Aoki, "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 (1989).
21. 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).
22. 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).
23. M. Wakabayashi, T. Ohwada, and F. Golse, "Numerical analysis of the shear and thermal creep flows of a rarefied gas over the plane wall of a Maxwell-type boundary condition on the basis of the linearized Boltzmann equation for hard-sphere molecules," Eur. J. Mech. B/Fluids 15, 175 (1996).
24. T. Ohwada, "Srtucture of normal shock waves: Direct numerical analysis of the Baltzmann equation for hard-sphere molecules," Phys. Fluids A 5, 217 (1993): T. Ohwada, "Numerical analysis of normal shock waves on the basis of the Baltzmann equation for hard-sphere molecules," in Rarefeild Gas Dynamics: Theory and Simulations, edited by B. D. Shizgal and D. P. Weaver (AIAA, Washington, DC, 1994), p. 482.
25. T. Ohwada, "Srtucture of normal shock waves: Direct numerical analysis of the Baltzmann equation for hard-sphere molecules," Phys. Fluids A 5, 217 (1993): T. Ohwada, "Numerical analysis of normal shock waves on the basis of the Baltzmann equation for hard-sphere molecules," in Rarefeild Gas Dynamics: Theory and Simulations, edited by B. D. Shizgal and D. P. Weaver (AIAA, Washington, DC, 1994), p. 482.
26. H. Grad, "Asymptotic theory of solutions of the Boltzmann equation II, in Ref. 5, p. 26.
27. C. Cercignani, The Boltzmann Equation and Its Applications (Springer-Verlag, Berlin, 1988), p. 174.
28. E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill, New York, 1938), p. 315.
29. 2)].
30. The value of ΔT is not necessary in the analysis of the linearized problem.
31. r). Thus, strictly speaking, the nonlinear collision integral at the wall is not accurate, although it is confirmed in the numerical test using the BGK equation that the error caused by this approximation is very small.
32. The results for the BGK equation shown in the present paper are obtained by the finite-difference computation. The accurate results for the linearized BGK equation and the Maxwell-type boundary condition have also been given in Refs. 8 and 12.
33. The curve shown as the results of 8-moment method (hard-sphere) in Fig. 6 of Ref. 9 is not for α=0.826 but is very close to that for α=1.
34. G. A. Bird, "Monte-Carlo simulation in an engineering context," in Rarefeild Gas Dynamics, edited by S. S. Fisher (AIAA, New York, 1981), Part 1, p. 239.
35. S. Chapman and T. G. Cowling, The Mathematical Theory of NonUniform Gases (Cambridge University Press, London, 1970), P. 172.
36. J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1967), p. 547.
37. In Ref. 27, p. 327.
38. Y. Sone, "Thermal creep in rarefied gas," J. Phys. Soc. Jpn. 21, 1836 (1966).
39. Y. Sone, K. Sawada, and H. Hirano, "A simple experiment on the strenth of thermal creep flow of a rarefeid gas over a flat wall," Eur. J. Mech. B Fluids 13, 299 (1994).
40. Y. Sone, "Flow induced by thermal stress in rarefied gas," Phys. Fluids 15, 1418 (1972).
41. M. N. Kogan, V. S. Galkin, and O. G. Fridlender, "Stresses produced in canvection," Sov. Phys. Usp. 19, 420 (1976).
42. K. Aoki, Y. Sone, and N. Masukawa, "A rarefied gas flow induced by a temperature field," in Rarefeild Gas Dynamics, edited by J. Harvey and G. Lord (Oxford University Press, Oxford, 1996), p. 35.
43. Y. Sone, K. Aoki, S. Takata, H. Sugimoto, and A. V. Bobylev, "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 (1996).