The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics

Physics of Fluids

Volumn 18, Issue 11, 2006, Pages

  Hadjiconstantinou, Nicolas G ^a  

^a USA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; KINETIC THEORY; MATHEMATICAL MODELS; MONTE CARLO METHODS; NAVIER STOKES EQUATIONS; NUMERICAL METHODS;

CONSTITUTIVE LAWS; GASEOUS HYDRODYNAMICS; NAVIER-STOKES THEORY; SECOND-ORDER SLIP;

GAS DYNAMICS;

COMPUTER SIMULATION; GAS DYNAMICS; KINETIC THEORY; MATHEMATICAL MODELS; MONTE CARLO METHODS; NAVIER STOKES EQUATIONS; NUMERICAL METHODS;

EID: 33751574666     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2393436     Document Type: Review

References (68)

1. C. M. Ho and Y. C. Tai, "Micro-electro-mechanical systems (MEMS) and fluid flows," Annu. Rev. Fluid Mech.ARVFA30066-418910.1146/ annurev.fluid.30.1.579 30, 579 (1998).
2. G. E. Karniadakis and A. Beskok, Microflows: Fundamentals and Simulation (Springer, New York, 2001).
3. M. Gad-el-Hak, "Flow physics," in Handbook of MEMS, edited by M. Gad-el-Hak (CRC, Boca Raton, FL, 2002).
4. W. G. Vincenti and C. H. Kruger, Introduction to Physical Gas Dynamics (Krieger, Malabar, FL, 1965).
5. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, England, 1970).
6. Y. Sone, Kinetic Theory and Fluid Dynamics (Birkhäuser, Boston, 2002).
7. C. Cercignani, The Boltzmann Equation and Its Applications (Springer-Verlag, New York, 1988).
8. H. Grad, "Singular and nonuniform limits of solutions of the Boltzmann equation," in Transport Theory, Proceedings of the 20th Symposium in Applied Mathematics, New York, NY, 1967, SIAM-AMS Proceedings Vol. 1, edited by R. Bellman, G. Birkhoff, and I. Abu-Shumays (American Mathematical Society, Providence, 1969).
9. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon, Oxford, 1994).
10. P. L. Bhatnagar, E. P. Gross, and M. Krook, "A model for collision processes in gases," Phys. Rev.PHRVAO0031-899X10.1103/ PhysRev.94.511 94, 511 (1954).
11. F. J. Alexander and A. L. Garcia, "The direct simulation Monte Carlo method," Comput. Phys.CPHYE20894-186610.1063/1.168619 11, 588 (1997).
12. M. N. Kogan, Rarefied Gas Dynamics (Plenum, New York, 1969).
13. E. Kennard, Kinetic Theory of Gases (McGraw-Hill, New York, 1938).
14. F. Sharipov and V. Seleznev, "Data on internal rarefied gas flows," J. Phys. Chem. Ref. DataJPCRBU0047-2689 27, 657 (1998).
15. C. Cercignani and M. Lampis, "Kinetic models for gas-surface interactions," Transp. Theory Stat. Phys.TTSPB40041-1450 1, 101 (1971).
16. C. Cercignani, P. Foresti, and F. Sernagiotto, "Dependence of the slip coefficient on the form of the collision frequency," Nuovo Cimento BNCIBAW0369-3554 57, 297 (1968).
17. T. E. Wenski, T. Olson, C. T. Rettner, and A. L. Garcia, "Simulations of air slider bearings with realistic gas-surface scattering," J. Tribol.JOTRE90742-4787 120, 639 (1998).
18. T. Ohwada, Y. Sone, and K. Aoki, "Numerical analysis of the Poiseuille and thermal transpiration flows between parallel plates on the basis of the Boltzmann equation for hard-sphere molecules," Phys. Fluids APFADEB0899-821310.1063/1.857478 1, 2042 (1989).
19. C. Cercignani, "The Kramers problem for a not completely diffusing wall," J. Math. Anal. Appl.JMANAK0022-247X 10, 568 (1965).
20. S. K. Loyalka, "Momentum and temperature-slip coefficients with arbitrary accommodation at the surface," J. Chem. Phys.JCPSA60021-960610.1063/1.1668235 48, 5432 (1968).
21. Y. Sone, "Kinetic theory analysis of linearized Rayleigh problem," J. Phys. Soc. Jpn.JUPSAU0031-901510.1143/JPSJ.19.1463 19, 1463 (1964).
22. C. Cercignani, "Higher order slip according to the linearized Boltzmann equation," Institute of Engineering Research Report No. AS-64-19, University of California, Berkeley, 1964.
23. E. B. Arkilic, K. S. Breuer, and M. A. Schmidt, "Mass flow and tangential momentum accommodation in silicon micromachined channels," J. Fluid Mech.JFLSA70022-112010.1017/S0022112001004128 437, 29 (2001).
24. C. Aubert and S. Colin, "High-order boundary conditions for gaseous flows in rectangular microducts," Microscale Thermophys. Eng.MTENFP1089-395410.1080/108939501300005367 5, 41 (2001).
25. J. Maurer, P. Tabeling, P. Joseph, and H. Willaime, "Second-order slip laws in microchannels for helium and nitrogen," Phys. FluidsPHFLE61070-663110.1063/1.1599355 15, 2613 (2003).
26. M. Knudsen, "Die Gesetze der molecular Stromung und dieinneren Reibungstromung der Gase durch Rohren," Ann. Phys.ANPYA20003-3804 28, 75 (1909).
27. C. Cercignani and A. Daneri, "Flow of a rarefied gas between two parallel plates," J. Appl. Phys.JAPIAU0021-897910.1063/1.1729249 34, 3509 (1963).
28. A. Beskok and G. E. Karniadakis, "A model for flows in channels and ducts at micro and nano scales," Microscale Thermophys. Eng.MTENFP1089-395410.1080/108939599199864 3, 43 (1999).
29. R. G. Deissler, "An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases," Int. J. Heat Mass TransferIJHMAK0017-931010.1016/0017-9310(64)90161-9 7, 681 (1964).
30. A. Sreekanth, "Slip flow through long circular tubes," in Proceedings of the Sixth International Symposium on Rarefied Gas Dynamics, edited by L. Trilling and H. Y. Wachman (Academic, New York, 1969), Vol. 1, pp. 667-680.
31. Y. Sone, "Asymptotic theory of flow of rarefied gas over a smooth boundary I," in Proceedings of the Sixth International Symposium on Rarefied Gas Dynamics (Academic, New York, 1969), Vol. 1, pp. 243-253.
32. N. G. Hadjiconstantinou, "Comment on Cercignani's second order slip coefficient," Phys. FluidsPHFLE61070-663110.1063/1.1587155 15, 2352 (2003).
33. 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 APFADEB0899-821310.1063/1.857304 1, 1588 (1989); 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 APFADEB0899-821310.1063/1.857457 1, 363 (1989).
34. N. G. Hadjiconstantinou, "Validation of a second-order slip model for dilute gas flows," Microscale Thermophys. Eng.MTENFP1089-395410.1080/10893950590945021 9, 137 (2005).
35. N. G. Hadjiconstantinou, "Oscillatory shear-driven gas flows in the transition and free-molecular-flow regimes," Phys. FluidsPHFLE61070-663110.1063/1.1874193 17, 100611 (2005).
36. N. G. Hadjiconstantinou and H. A. Al-Mohssen, "A linearized kinetic formulation including a second-order slip model for an impulsive start problem at arbitrary Knudsen numbers," J. Fluid Mech.JFLSA70022-1120 533, 47 (2005).
37. K. S. Breuer, "Lubrication in MEMS," in Handbook of MEMS, edited by M. Gad-el-Hak (CRC, Boca Raton, FL, 2002).
38. J. H. Park, P. Bahukudumbi, and A. Beskok, "Rarefaction effects on shear driven oscillatory gas flows: A direct simulation Monte Carlo study in the entire Knudsen regime," Phys. FluidsPHFLE61070-663110.1063/1.1634563 16, 317 (2004).
39. N. G. Hadjiconstantinou, "Sound wave propagation in transition-regime micro- and nanochannels," Phys. FluidsPHFLE61070-663110.1063/1.1431243 14, 802 (2002).
40. N. G. Hadjiconstantinou and O. Simek, "Sound propagation at small scales under continuum and non-continuum transport," J. Fluid Mech.JFLSA70022-112010.1017/S0022112003005044 488, 399 (2003).
41. I. B. Crandall, Theory of Vibrating Systems and Sound (Van Nostrand, New York, 1926).
42. G. Kirchhoff, "Ueber den Einflufs der Wärmeleitung in einem Gase auf die Schallbewegung," Ann. Phys. Chem.ANPYA20003-3804 134, 177 (1868).
43. J. W. S. Rayleigh, The Theory of Sound (Macmillan, New York, 1896), Vol. 2.
44. B. J. Hamrock, Fundamentals of Fluid Film Lubrication (McGraw-Hill, New York, 1994).
45. S. Fukui and R. Kaneko, "Analysis of ultra thin gas film lubrication based on linearized Boltzmann equation: First report-derivation of a generalized lubrication equation including thermal creep flow," J. Tribol.JOTRE90742-4787 110, 253 (1988).
46. F. J. Alexander, A. L. Garcia, and B. J. Alder, "Direct simulation Monte Carlo for thin-film bearings," Phys. FluidsPHFLE61070-663110.1063/1.868377 6, 3854 (1994).
47. L. Graetz, "On the thermal conduction of liquids," Ann. Phys. Chem.ANPYA20003-3804 25, 337 (1885).
48. A. F. Mills, Heat Transfer (Irwin, Homewood, IL, 1992).
49. E. M. Sparrow and S. H. Lin, "Laminar heat transfer in tubes under slip flow conditions," J. Heat TransferJHTRAO0022-1481 84, 363 (1962).
50. N. G. Hadjiconstantinou and O. Simek, "Constant-wall-temperature Nusselt number in micro and nano channels," J. Heat TransferJHTRAO0022-148110.1115/1.1447931 124, 356 (2002).
51. N. G. Hadjiconstantinou, "Dissipation in small scale gaseous flows," J. Heat TransferJHTRAO0022-148110.1115/1.1571088 125, 944 (2003).
52. M. A. Gallis, D. J. Rader, and J. R. Torczynski, "Calculations of the near-wall thermophoretic force in rarefied gas flow," Phys. FluidsPHFLE61070-663110.1063/1.1518692 14, 4290 (2002).
53. J. Tyndall, "On dust and disease," Proc. R. Inst. G. B. 6, 1 (1870).
54. M. A. Gallis, J. R. Torczynski, and D. J. Rader, "An approach for simulating the transport of spherical particles in a rarefied gas flow via the direct simulation Monte Carlo," Phys. FluidsPHFLE61070-663110.1063/1.1409367 13, 3482 (2001).
55. H. S. Wijesinghe and N. G. Hadjiconstantinou, "A discussion of hybrid atomistic-continuum methods for multiscale hydrodynamics," Int. J. Multiscale Comp. Eng.ZZZZZZ1543-1649 2, 189 (2004).
56. N. G. Hadjiconstantinou, "Discussion of recent developments in hybrid atomistic-continuum methods for multiscale hydrodynamics," Bull. Pol. Acad. Sci.: Tech. Sci.BASSEP0239-7528 53, 335 (2005).
57. W. Wagner, "A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation," J. Stat. Phys.JSTPBS0022-471510.1007/BF01055714 66, 1011 (1992).
58. F. J. Alexander, A. L. Garcia, and B. J. Alder, "A consistent Boltzmann algorithm," Phys. Rev. Lett.PRLTAO0031-900710.1103/ PhysRevLett.74.5212 74, 5212 (1995).
59. F. J. Alexander, A. L. Garcia, and B. J. Alder, "Cell size dependence of transport coefficients in stochastic particle algorithms," Phys. FluidsPHFLE61070-663110.1063/1.869674 10, 1540 (1998); F. J. Alexander, A. L. Garcia, and B. J. Alder,erratum, Phys. FluidsPHFLE61070-663110.1063/1.870278 12, 731 (2000).
60. N. G. Hadjiconstantinou, "Analysis of discretization in the direct simulation Monte Carlo," Phys. FluidsPHFLE61070-663110.1063/ 1.1289393 12, 2634 (2000).
61. D. J. Rader, M. A. Gallis, J. R. Torczynski, and W. Wagner, "DSMC convergence behavior of the hard-sphere-gas thermal conductivity for Fourier heat flow," Phys. FluidsPHFLE61070-663110.1063/1.2213640 18, 077102 (2006).
62. A. L. Garcia and W. Wagner, "Time step truncation error in direct simulation Monte Carlo," Phys. FluidsPHFLE61070-663110.1063/ 1.1289691 12, 2621 (2000).
63. N. G. Hadjiconstantinou, A. L. Garcia, M. Z. Bazant, and G. He, "Statistical error in particle simulations of hydrodynamic phenomena," J. Comput. Phys.JCTPAH0021-999110.1016/ S0021-9991(03)00099-8 187, 274 (2003).
64. L. L. Baker and N. G. Hadjiconstantinou, "Variance reduction for Monte Carlo solutions of the Boltzmann equation," Phys. FluidsPHFLE61070-663110.1063/1.1899210 17, 051703 (2005).
65. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, New York, 1994).
66. L. L. Baker and N. G. Hadjiconstantinou, "Variance reduction in particle methods for solving the Boltzmann equation," in Proceedings of the Fourth International Conference on Nanochannels, Microchannels and Minichannels, Limerick, Ireland, 2006 (ASME, New York, 2006), paper ICNMM2006-96089.
67. R. G. Lord, "Some further extensions of the Cercignani-Lampis gas-surface interaction model," Phys. FluidsPHFLE61070-663110.1063/ 1.868557 7, 1159 (1995).
68. F. Sharipov, "Application of the Cercignani-Lampis scattering kernel to calculations of rarefied gas flows. I. Plane flow between two parallel plates," Eur. J. Mech. B/FluidsEJBFEV0997-754610.1016/ S0997-7546(01)01160-8 21, 113 (2002).