Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T13:55:39.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic analysis of the Boltzmann–BGK equation for oscillatory flows

Published online by Cambridge University Press:  10 August 2012

Jason Nassios  and
John E. Sader
Jason Nassios
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
John E. Sader*
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia Kavli Nanoscience Institute and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: jsader@unimelb.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Kinetic theory provides a rigorous foundation for calculating the dynamics of gas flow at arbitrary degrees of rarefaction, with solutions of the Boltzmann equation requiring numerical methods in many cases of practical interest. Importantly, the near-continuum regime can be examined analytically using asymptotic techniques. These asymptotic analyses often assume steady flow, for which analytical slip models have been derived. Recently, developments in nanoscale fabrication have stimulated research into the study of oscillatory non-equilibrium flows, drawing into question the applicability of the steady flow assumption. In this article, we present a formal asymptotic analysis of the unsteady linearized Boltzmann–BGK equation, generalizing existing theory to the oscillatory (time-varying) case. We consider the near-continuum limit where the mean free path and oscillation frequency are small. The complete set of hydrodynamic equations and associated boundary conditions are derived for arbitrary Stokes number and to second order in the Knudsen number. The first-order steady boundary conditions for the velocity and temperature are found to be unaffected by oscillatory flow. In contrast, the second-order boundary conditions are modified relative to the steady case, except for the velocity component tangential to the solid wall. Application of this general asymptotic theory is explored for the oscillatory thermal creep problem, for which unsteady effects manifest themselves at leading order.

JFM classification

Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: MEMS/NEMS Micro-/Nano-fluid dynamics: Non-continuum effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 708 , 10 October 2012 , pp. 197 - 249
Copyright
Copyright © Cambridge University Press 2012

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

1. Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
2. Agarwal, R. K., Yun, K. Y. & Balakrishnan, R. 2001 Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime. Phys. Fluids 13 (10), 3061.Google Scholar
3. Ando, N. 2011 The geodesic curvatures of lines of curvature. www.sci.kumamoto-u.ac.jp/~ando/090306.pdf, pp. 1–29.Google Scholar
4. Ashurst, Wm. T., Kerstein, A. R., Kerr, R. M. & Gibson, C. H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30 (8), 2343.CrossRefGoogle Scholar
5. Baker, L. L. & Hadjiconstantinou, N. G. 2005 Variance reduction for Monte Carlo solutions of the Boltzmann equation. Phys. Fluids 17 (5), 051703.CrossRefGoogle Scholar
6. Baker, L. L. & Hadjiconstantinou, N. G. 2008 Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: a discontinuous Galerkin formulation. Intl J. Numer. Meth. Fluids 58 (4), 381–402.CrossRefGoogle Scholar
7. Bardos, C., Golse, F. & Levermore, C. D. 1993 Fluid dynamic limits of kinetic equations. Part 2. Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46 (5), 667–753.CrossRefGoogle Scholar
8. Bardos, C., Golse, F. & Levermore, C. D. 1998 Acoustic equation and Stokes limits for the Boltzmann equation. C. R. Acad. Sci. I 327, 323–328.CrossRefGoogle Scholar
9. Bargatin, I., Kozinsky, I. & Roukes, M. L. 2007 Efficient electrothermal actuation of multiple modes of high-frequency nanoelectromechanical resonators. Appl. Phys. Lett. 90 (9), 093116.CrossRefGoogle Scholar
10. Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
11. Berger, R., Gerber, C., Lang, H. P. & Gimzewski, J. K. 1997 Micromechanics: a toolbox for femtoscale science: ‘towards a laboratory on a tip’. Microelectron. Engng 35 (1–4), 373–379.CrossRefGoogle Scholar
12. Bhatnagar, P. L., Gross, E. P. & Krook, M. 1954 A model for collision processes in gases. Part 1. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 (3), 511.CrossRefGoogle Scholar
13. Binnig, G., Quate, C. F. & Gerber, C. 1986 Atomic force microscope. Phys. Rev. Lett. 56 (9), 930–933.CrossRefGoogle ScholarPubMed
14. Bird, G. A. 1963 Approach to translational equilibrium in a rigid sphere gas. Phys. Fluids 6, 1518–1519.CrossRefGoogle Scholar
15. Bird, G. A. 1978 Monte Carlo simulation of gas flows. Annu. Rev. Fluid Mech. 10 (1), 11–31.CrossRefGoogle Scholar
16. Bird, G. A. 1998 Recent advances and current challenges for DSMC. Comput. Math. Appl. 35 (1-2), 1–14.CrossRefGoogle Scholar
17. Bobylev, A. V. 1982 The Chapman–Enskog and Grad methods for solving the Boltzmann equation. Akad. Nauk SSSR Dokl. 262, 71–75.Google Scholar
18. Boltzmann, L. 1872 Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Wiener Berichte 66, 275–370.Google Scholar
19. Boskovic, S., Chon, J. W. M., Mulvaney, P. & Sader, J. E. 2002 Rheological measurements using microcantilevers. J. Rheol. 46 (4), 891.CrossRefGoogle Scholar
20. Burg, T. P., Godin, M., Knudsen, S. M., Shen, W., Carlson, G., Foster, J. S., Babcock, K. & Manalis, S. R. 2007 Weighing of biomolecules, single cells and single nanoparticles in fluid. Nature 446 (7139), 1066–1069.CrossRefGoogle ScholarPubMed
21. Burnett, D. 1935 The distribution of velocities in a slightly non-uniform gas. Proc. Lond. Math. Soc. 2 (1), 385.CrossRefGoogle Scholar
22. Cao, B.-Y., Sun, J., Chen, M. & Guo, Z.-Y. 2009 Molecular momentum transport at fluid–solid interfaces in MEMS/NEMS: a review. Intl J. Molecular Sci. 10 (11), 4638–4706.Google ScholarPubMed
23. Cartan, H. 1977 Course de Calcul Différentiel. Hermann.Google Scholar
24. Cercignani, C. 1962 Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem. Ann. Phys. 20 (2), 219–233.CrossRefGoogle Scholar
25. Cercignani, C. 1964 Higher order slip according to the linearized Boltzmann equation. Tech. Rep., California University Berkeley Institute of Engineering Research.Google Scholar
26. Cercignani, C. 1988 The Boltzmann Equation and its Applications. Springer.CrossRefGoogle Scholar
27. Cercignani, C. 2000 Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge University Press.Google Scholar
28. Chapman, S. 1916 On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas. Phil. Trans. R. Soc. Lond. Ser. A, Containing Papers of a Mathematical or Physical Character 216, 279–348.Google Scholar
29. Chapman, S. & Cowling, T. G. 1960 The Mathematical Theory of Non-uniform Gases. Cambridge University Press.Google Scholar
30. Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S. & Yakhot, V. 2003 Extended Boltzmann kinetic equation for turbulent flows. Science 301 (5633), 633–636.CrossRefGoogle ScholarPubMed
31. Chen, S., Chen, H., Martnez, D. & Matthaeus, W. 1991 Lattice Boltzmann model for simulation of magnetohydrodynamics. Phys. Rev. Lett. 67 (27), 3776–3779.Google ScholarPubMed
32. Chun, J. & Koch, D. L. 2005 A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number. Phys. Fluids 17 (10), 107107.CrossRefGoogle Scholar
33. Cleland, A. N. 2002 Foundations of Nanomechanics. Springer.Google Scholar
34. Clercx, H. J. H. & van Heijst, G. J. F. 2009 Two-dimensional Navier–Stokes turbulence in bounded domains. Appl. Mech. Rev. 62 (2), 020802.CrossRefGoogle Scholar
35. Colosqui, C. E., Karabacak, D. M., Ekinci, K. L. & Yakhot, V. 2010 Lattice Boltzmann simulation of electromechanical resonators in gaseous media. J. Fluid Mech. 652, 241–257.CrossRefGoogle Scholar
36. Craighead, H. G. 2000 Nanoelectromechanical systems. Science 290 (5496), 1532–1535.Google ScholarPubMed
37. Enskog, D. 1917 Kinetische Theorie der Vorgänge in mässig verdünnten Gasen. PhD thesis, Uppsala.Google Scholar
38. Fan, J. & Shen, C. 2001 Statistical simulation of low-speed rarefied gas flows. J. Comput. Phys. 167 (2), 393–412.CrossRefGoogle Scholar
39. Frisch, U., Hasslacher, B. & Pomeau, Y. 1986 Lattice-gas automata for the Navier–Stokes equation. Phys. Rev. Lett. 56 (14), 1505–1508.CrossRefGoogle ScholarPubMed
40. Giessibl, F. J. 2003 Advances in atomic force microscopy. Rev. Mod. Phys. 75 (3), 949–983.CrossRefGoogle Scholar
41. Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407.CrossRefGoogle Scholar
42. Grad, H. 1958 Principles of the Kinetic Theory of Gases. Springer.CrossRefGoogle Scholar
43. Groth, C. P. T. & McDonald, J. G. 2009 Towards physically realizable and hyperbolic moment closures for kinetic theory. Contin. Mech. Thermodyn. 21 (6), 467–493.Google Scholar
44. Gu, X.-J. & Emerson, D. R. 2007 A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions. J. Comput. Phys. 225 (1), 263–283.CrossRefGoogle Scholar
45. Hadjiconstantinou, N. G. 2003 Comment on Cercignani’s second-order slip coefficient. Phys. Fluids 15 (8), 2352.CrossRefGoogle Scholar
46. Hadjiconstantinou, N. G. 2005a Oscillatory shear-driven gas flows in the transition and free-molecular-flow regimes. Phys. Fluids 17 (10), 100611.CrossRefGoogle Scholar
47. Hadjiconstantinou, N. G. 2005b Validation of a second-order slip model for dilute gas flows. Microscale Therm. Engng 9 (2), 137–153.CrossRefGoogle Scholar
48. Hadjiconstantinou, N. G., Garcia, A. L., Bazant, M. Z. & He, G. 2003 Statistical error in particle simulations of hydrodynamic phenomena. J. Comput. Phys. 187 (1), 274–297.CrossRefGoogle Scholar
49. Higuera, F. J & Jiménez, J. 1989 Boltzmann approach to lattice gas simulations. Europhys. Lett. 9 (7), 663–668.CrossRefGoogle Scholar
50. Hilbert, D. 1900 Mathematische Probleme. In Vortrag, gehalten auf dem internationalen Mathematiker, pp. 253–297. Vadenhöck and Ruprecht.Google Scholar
51. Hilbert, D. 1912 Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. Teubner.Google Scholar
52. de Izarra, L., Rouet, J.-L. & Izrar, B. 2011 High-order lattice Boltzmann models for gas flow for a wide range of Knudsen numbers. Phys. Rev. E 84 (6), 1–7.CrossRefGoogle ScholarPubMed
53. Jin, S. & Slemrod, M. 2001 Regularization of the Burnett equations via relaxation. J. Stat. Phys. 103 (5), 1009–1033.CrossRefGoogle Scholar
54. Kennard, E. H. 1938 Kinetic Theory of Gases. McGraw-Hill.Google Scholar
55. Knudsen, M. 1909a Die Gesetze der molekular Strömung und der inneren Reibungströmung der Gase durch Röhren. Ann. Phys. 28, 75–130.CrossRefGoogle Scholar
56. Knudsen, M. 1909b Eine Revision der Gleichgewichtsbedingung der Gase: Thermische Molekularströmung. Ann. Phys. 336 (1), 205–229.CrossRefGoogle Scholar
57. Lagubeau, G., Le Merrer, M., Clanet, C. & Quéré, D. 2011 Leidenfrost on a ratchet. Nat. Phys. 7 (5), 395–398.CrossRefGoogle Scholar
58. Lavrik, N. V., Sepaniak, M. J. & Datskos, P. G. 2004 Cantilever transducers as a platform for chemical and biological sensors. Rev. Sci. Instrum. 75 (7), 2229.CrossRefGoogle Scholar
59. Lee, C. J. 1994 Unique determination of solutions to the Burnett equations. AIAA J. 32, 985–990.CrossRefGoogle Scholar
60. Loyalka, S. K. 1971 Kinetic theory of thermal transpiration and mechanocaloric effect. Part 1. J. Chem. Phys. 55 (9), 4497.CrossRefGoogle Scholar
61. Loyalka, S. K., Petrellis, N. & Storvick, T. S. 1979 Some exact numerical results for the BGK model: Couette, Poiseuille and thermal creep flow between parallel plates. Z. Angew. Math. Phys. 30 (3), 514–521.CrossRefGoogle Scholar
62. Manela, A. & Hadjiconstantinou, N. G. 2010 Gas-flow animation by unsteady heating in a microchannel. Phys. Fluids 22 (6), 062001.CrossRefGoogle Scholar
63. Maurer, J., Tabeling, P., Joseph, P. & Willaime, H. 2003 Second-order slip laws in microchannels for helium and nitrogen. Phys. Fluids 15 (9), 2613.CrossRefGoogle Scholar
64. Maxwell, J. C. 1878 On stresses in rarefied gases arising from inequalities of temperature. Proc. R. Soc. Lond. 27 (185–189), 304–308.Google Scholar
65. Maxwell, J. C. 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
66. McNamara, G. & Zanetti, G. 1988 Use of the Boltzmann equation to simulate lattice gas automata. Phys. Rev. Lett. 61 (20), 2332–2335.Google ScholarPubMed
67. Motamedi, R. & Wood-Adams, P. M. 2010 Measurement of fluid properties using an acoustically excited atomic force microscope micro-cantilever. J. Rheol. 54 (5), 959.CrossRefGoogle Scholar
68. Ohwada, T. & Sone, Y. 1992 Analysis of thermal stress slip flow and negative thermophoresis using the Boltzmann equation for hard-sphere molecules. Eur. J. Mech. B 11 (4), 389–414.Google Scholar
69. Ohwada, T., Sone, Y. & Aoki, K. 1989a 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.CrossRefGoogle Scholar
70. Ohwada, T., Sone, Y. & Aoki, K. 1989b 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
71. Onishi, Y. & Sone, Y. 1979 Kinetic theory of slightly strong evaporation and condensation: hydrodynamic equation and slip boundary condition for finite Reynolds number. J. Phys. Soc. Japan 47 (5), 1676–1685.Google Scholar
72. Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159–205.CrossRefGoogle Scholar
73. Park, J. H., Bahukudumbi, P. & Beskok, A. 2004 Rarefaction effects on shear driven oscillatory gas flows: a direct simulation Monte Carlo study in the entire Knudsen regime. Phys. Fluids 16 (2), 317.Google Scholar
74. Pitakarnnop, J., Varoutis, S., Valougeorgis, D., Geoffroy, S., Baldas, L. & Colin, S. 2009 A novel experimental setup for gas microflows. Microfluid. Nanofluid. 8 (1), 57–72.Google Scholar
75. Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
76. Prandtl, L. 1904 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In Proceedings of 3rd International Mathematics Congress, Heidelberg (ed. Krazer, A. ), pp. 484–491. Teubner.Google Scholar
77. Qian, Y-H., D’Humières, D. & Lallemand, P. 1992 Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17 (6), 479.CrossRefGoogle Scholar
78. Radtke, G. a., Hadjiconstantinou, N. G. & Wagner, W. 2011 Low-noise Monte Carlo simulation of the variable hard sphere gas. Phys. Fluids 23 (3), 030606.CrossRefGoogle Scholar
79. Ramanathan, S. & Koch, D. L. 2009 An efficient direct simulation Monte Carlo method for low Mach number noncontinuum gas flows based on the Bhatnagar–Gross–Krook model. Phys. Fluids 21 (3), 033103.Google Scholar
80. Ramanathan, S., Koch, D. L. & Bhiladvala, R. B. 2010 Noncontinuum drag force on a nanowire vibrating normal to a wall: simulations and theory. Phys. Fluids 22 (10), 103101.Google Scholar
81. Reynolds, O. 1879 On certain dimensional properties of matter in the gaseous state. Part 1. Experimental researches on thermal transpiration of gases through porous plates and on the laws of transpiration and impulsion, including an experimental proof that gas is not a continuous plenum. Phil. Trans. R. Soc. Lond. 170, 727–845.Google Scholar
82. Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Proc. R. Soc. Lond. 35 (224–226), 84.Google Scholar
83. Reynolds, O. 1895 On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. R. Soc. Lond. A 186, 123–164.Google Scholar
84. Sader, J. E. 1998 Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J. Appl. Phys. 84 (1), 64.CrossRefGoogle Scholar
85. Schlichting, H. 1960 Boundary-Layer Theory. McGraw-Hill.Google Scholar
86. Sharipov, F. & Kalempa, D. 2007 Gas flow near a plate oscillating longitudinally with an arbitrary frequency. Phys. Fluids 19 (1), 017110.CrossRefGoogle Scholar
87. Sharipov, F. & Kalempa, D. 2008 Oscillatory Couette flow at arbitrary oscillation frequency over the whole range of the Knudsen number. Microfluid. Nanofluid. 4 (5), 363–374.CrossRefGoogle Scholar
88. Shavaliyev, M. S. 1993 Super-Burnett corrections to the stress tensor and the heat flux in a gas of Maxwellian molecules. J. Appl. Math. Mech. 57 (3), 573–576.CrossRefGoogle Scholar
89. Shen, S., Chen, G., Crone, R. M. & Anaya-Dufresne, M. 2007 A kinetic-theory based first order slip boundary condition for gas flow. Phys. Fluids 19 (8), 086101.CrossRefGoogle Scholar
90. Shi, Y., Brookes, P., Yap, Y. & Sader, J. E. 2011 Accuracy of the lattice Boltzmann method for low-speed noncontinuum flows. Phys. Rev. E 83 (4), 2–5.CrossRefGoogle ScholarPubMed
91. Shi, Y. & Sader, J. E. 2010 Lattice Boltzmann method for oscillatory Stokes flow with applications to micro- and nanodevices. Phys. Rev. E 81 (3), 1–14.CrossRefGoogle ScholarPubMed
92. Soderholm, L. H. 2007 Hybrid Burnett equations: a new method of stabilizing. Transp. Theory Stat. Phys. 36 (4), 495–512.CrossRefGoogle Scholar
93. Sone, Y. 1964 Kinetic theory analysis of linearized Rayleigh problem. J. Phys. Soc. Japan 19 (8), 1463–1473.CrossRefGoogle Scholar
94. Sone, Y. 1965 Effect of sudden change of wall temperature in rarefied gas. J. Phys. Soc. Japan 20 (2), 222–229.CrossRefGoogle Scholar
95. Sone, Y. 1966 Thermal creep in rarefied gas. J. Phys. Soc. Japan 21, 1836–1837.CrossRefGoogle Scholar
96. Sone, Y. 1968a Asymptotic behaviour of diffusion of tangential velocity discontinuity in rarefied gas. Phys. Fluids 11, 1935.CrossRefGoogle Scholar
97. Sone, Y. 1968b Flow of rarefied gas through a circular pipe. Phys. Fluids 11, 1672.CrossRefGoogle Scholar
98. Sone, Y. 1969 Asymptotic theory of flow of rarefied gas over a smooth boundary. Part 1. In Rarefied Gas Dynamics (ed. Trilling, L. & Wachman, H. Y. ), pp. 243–253. Academic.Google Scholar
99. Sone, Y. 1974 Asymptotic theory of flow of rarefied gas over a smooth boundary. Part 2. Trans. Japan Soc. Aeronaut. Space Sci. 17, 113–122.Google Scholar
100. Sone, Y. 2000 Kinetic Theory and Fluid Dynamics. Birkhäuser.Google Scholar
101. Sone, Y. 2007 Molecular Gas Dynamics: Theory, Techniques, and Applications. Springer.CrossRefGoogle Scholar
102. 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, Fluid Dyn. 1 (2), 363–370.CrossRefGoogle Scholar
103. Sone, Y. & Onishi, Y. 1973 Kinetic theory of evaporation and condensation. J. Phys. Soc. Japan 35 (6), 1773–1776.CrossRefGoogle Scholar
104. Sone, Y. & Onishi, Y. 1978 Kinetic theory of evaporation and condensation: hydrodynamic equation and slip boundary condition. J. Phys. Soc. Japan 44 (6), 1981–1994.CrossRefGoogle Scholar
105. Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–106.Google Scholar
106. Struchtrup, H. 2005 Failures of the Burnett and super-Burnett equations in steady state processes. Contin. Mech. Thermodyn. 17 (1), 43–50.CrossRefGoogle Scholar
107. Struchtrup, H. & Torrilhon, M. 2003 Regularization of Grad’s 13 moment equations: derivation and linear analysis. Phys. Fluids 15 (9), 2668.CrossRefGoogle Scholar
108. Takata, S., Aoki, K., Hattori, M. & Hadjiconstantinou, N. G. 2012 Parabolic temperature profile and second-order temperature jump of a slightly rarefied gas in an unsteady two-surface problem. Phys. Fluids 24 (3), 032002.CrossRefGoogle Scholar
109. Tang, G., Gu, X.-J., Barber, R., Emerson, D. R. & Zhang, Y. 2008 Lattice Boltzmann simulation of nonequilibrium effects in oscillatory gas flow. Phys. Rev. E 78 (2), 1–8.CrossRefGoogle ScholarPubMed
110. Tibbs, K. W., Baras, F. & Garcia, A. L. 1997 Anomalous flow profile due to the curvature effect on slip length. Phys. Rev. E 56 (2), 2282–2283.CrossRefGoogle Scholar
111. Torrilhon, M. & Struchtrup, H. 2008 Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227 (3), 1982–2011.CrossRefGoogle Scholar
112. Vargo, S. E., Muntz, E. P., Shiflett, G. R. & Tang, W. C. 1999 Knudsen compressor as a micro- and macroscale vacuum pump without moving parts or fluids. J. Vac. Sci. Technol. A 17 (4), 2308.CrossRefGoogle Scholar
113. Vincenti, W. G. & Kruger, C. H. Jr. 1965 Introduction to Physical Gas Dynamics, 8th edition. Krieger.Google Scholar
114. Welander, P. 1954 On the temperature jump in a rarefied gas. Arkiv Fysik 7, 507–553.Google Scholar
115. Weng, H. C. & Chen, C.-K. 2008 A challenge in Navier–Stokes-based continuum modelling: Maxwell–Burnett slip law. Phys. Fluids 20 (10), 106101.CrossRefGoogle Scholar
116. Williams, M. M. R. 1971 Boundary-value problems in the kinetic theory of gases. Part 2. Thermal creep. J. Fluid Mech. 45 (04), 759.CrossRefGoogle Scholar
117. Würger, A. 2011 Leidenfrost gas ratchets driven by thermal creep. Phys. Rev. Lett. 107 (16), 1–4.CrossRefGoogle ScholarPubMed
118. Yang, Y. T., Callegari, C., Feng, X. L., Ekinci, K. L. & Roukes, M. L. 2006 Zeptogram-scale nanomechanical mass sensing. Nano Lett. 6 (4), 583–586.CrossRefGoogle ScholarPubMed
119. Yap, Y. W. & Sader, J. E. 2012 High accuracy numerical solutions of the Boltzmann Bhatnagar–Gross–Krook equation for steady and oscillatory Couette flows. Phys. Fluids 24 (3), 032004.CrossRefGoogle Scholar
120. Yu, H., Girimaji, S. & Luo, L.-S. 2005 Lattice Boltzmann simulations of decaying homogeneous isotropic turbulence. Phys. Rev. E 71 (1), 1–5.CrossRefGoogle ScholarPubMed
121. Zhong, X., MacCormack, R. W. & Chapman, D. R. 1993 Stabilization of the Burnett equations and application to hypersonic flows. AIAA J. 31 (6), 1036–1043.CrossRefGoogle Scholar