This article discusses the development of continuum models to describe processes in gases in which the particle collisions cannot maintain thermal equilibrium. Such a situation typically is present in rarefied or diluted gases, for flows in microscopic settings, or in general whenever the Knudsen number—the ratio between the mean free path of the particles and a macroscopic length scale—becomes significant. The continuum models are based on the stochastic description of the gas by Boltzmann's equation in kinetic gas theory. With moment approximations, extended fluid dynamic equations can be derived, such as the regularized 13-moment equations. Moment equations are introduced in detail, and typical results are reviewed for channel flow, cavity flow, and flow past a sphere in the low–Mach number setting for which both evolution equations and boundary conditions are well established. Conversely, nonlinear, high-speed processes require special closures that are still under development. Current approaches are examined, along with the challenge of computing shock wave profiles based on continuum equations.


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Literature Cited

  1. Agarwal RK, Yun KY, Balakrishnan R. 2001. Beyond Navier-Stokes: Burnett equations for flows in the continuum-transition regime. Phys. Fluids 13:3061–85 Erratum. 2002 Phys. Fluids 14:1818 [Google Scholar]
  2. Alaoui M, Santos A. 1992. Poiseuille flow driven by an external force. Phys. Fluids A 4:1273–82 [Google Scholar]
  3. Alsmeyer H. 1976. Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech. 74:497–513 [Google Scholar]
  4. Aoki K, Sone Y. 1987. Temperature field induced around a sphere in a uniform flow of a rarefied gas. Phys. Fluids 30:2286–88 [Google Scholar]
  5. Aoki K, Takata S, Nakanishi T. 2002. Poiseuille type flow of a rarefied gas between two parallel plates driven by a uniform external force. Phys. Rev. E 65:026315 [Google Scholar]
  6. Au JD. 2001. Lösung nichtlinearer Probleme in der Erweiterten Thermodynamik PhD Thesis, Tech. Univ. Berlin [Google Scholar]
  7. Au JD, Torrilhon M, Weiss W. 2001. The shock tube study in extended thermodynamics. Phys. Fluids 13:2423–32 [Google Scholar]
  8. Bailey CL, Barber RW, Emerson DR, Lockerby DA, Reese JM. 2005. A critical review of the drag force on a sphere in the transition flow regime. AIP Conf. Proc. 762:743–48 [Google Scholar]
  9. Bao F, Lin J. 2008. Burnett simulation of gas flow and heat transfer in micro Poiseuille flow. Int. J. Heat Mass Transf. 51:7593–600 [Google Scholar]
  10. Baranyai A, Evans DJ, Daivis PJ. 1992. Isothermal shear-induced heat flow. Phys. Rev A 46:7593–600 [Google Scholar]
  11. Barber RW, Emerson DR. 2003. Numerical simulation of low Reynolds number slip flow past a confined microsphere. AIP Conf. Proc. 663:808–15 [Google Scholar]
  12. Barbera E, Brini F. 2011. Heat transfer in gas mixtures: advantages of an extended thermodynamics approach. Phys. Lett. A 375:827–31 [Google Scholar]
  13. Barbera E, Müller I, Reitebuch D, Zhao NR. 2004. Determination of boundary conditions in extended thermodynamics via fluctuation theory. Contin. Mech. Thermodyn. 16:411–25 [Google Scholar]
  14. Bird GA. 1998. Molecular Gas Dynamics and the Direct Simulation of Gas Flows New York: Oxford Univ. Press, 2nd ed.. [Google Scholar]
  15. Bobylev AV. 1982. The Chapman-Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27:29–31 [Google Scholar]
  16. Bobylev AV. 2006. Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations. J. Stat. Phys. 124:371–99 [Google Scholar]
  17. Boltzmann L. 1872. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Wien. Ber. 66:275–370 [Google Scholar]
  18. Brini F. 2001. Hyperbolicity region in extended thermodynamics with 14 moments. Contin. Mech. Thermodyn. 13:1–8 [Google Scholar]
  19. Brown S. 1999. Approximate Riemann solvers for moment models of dilute gases PhD Thesis, Univ. Mich., Ann Arbor [Google Scholar]
  20. Burnett D. 1935. The distribution of velocities in a slightly non-uniform gas. Proc. Lond. Math. Soc. 39:385–430 [Google Scholar]
  21. Cai Z, Fan YW, Li R. 2014. Globally hyperbolic regularization of Grad's moment system. Commun. Pure Appl. Math. 64:464–518 [Google Scholar]
  22. Cai Z, Li R. 2010. Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation. SIAM J. Sci. Comput. 32:2875–907 [Google Scholar]
  23. Cai Z, Li R, Qiao Z. 2012. NRxx simulation of microflows with Shakhov model. SIAM J. Sci. Comput. 34:A339–69 [Google Scholar]
  24. Cercignani C. 1975. Theory and Applications of the Boltzmann Equation Edinburgh: Scott. Acad. [Google Scholar]
  25. Cercignani C, Lampis M. 1971. Kinetic models for gas-surface interactions. Transp. Theory Stat. Phys. 1:101–14 [Google Scholar]
  26. Chapman S, Cowling TG. 1970. The Mathematical Theory of Non-Uniform Gases Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  27. Deissler RG. 1964. An analysis of second order slip flow and temperature jump boundary conditions for rarefied gases. Int. J. Heat Mass Transf. 7:681–94 [Google Scholar]
  28. Dreyer W. 1987. Maximization of the entropy in non-equilibrium. J. Phys. A 20:6505–17 [Google Scholar]
  29. Dreyer W, Junk M, Kunik M. 2001. On the approximation of the Fokker-Planck equation by moment systems. Nonlinearity 14:881–906 [Google Scholar]
  30. Gad-el-Hak M. 2002. The MEMS Handbook. Boca Raton, FL: CRC [Google Scholar]
  31. Eu BC. 1980. A modified moment method and irreversible thermodynamics. J. Chem. Phys. 73:2958–69 [Google Scholar]
  32. Ferziger JH, Kaper HG. 1972. Mathematical Theory of Transport Processes in Gases Amsterdam: North-Holland [Google Scholar]
  33. Fiscko KA, Chapman DR. 1989. Comparison of Burnett, super-Burnett and Monte Carlo solutions for hypersonic shock structure. Proc. 16th Int. Symp. Rarefied Gas Dynamics EP Muntz, D Weaver, D Campbell 374–95 Reston, VA: AIAA [Google Scholar]
  34. Fox RO. 2008. A quadrature-based third-order moment method for dilute gas-particle flows. J. Comput. Phys. 227:6313–50 [Google Scholar]
  35. Garcia-Colin LS, Velasco RM, Uribe FJ. 2008. Beyond the Navier-Stokes equations: Burnett hydrodynamics. Phys. Rep. 465:149–89 [Google Scholar]
  36. Gilberg D, Paolucci D. 1953. The structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2:617–42 [Google Scholar]
  37. Goldberg R. 1954. The slow flow of a rarefied gas past a spherical obstacle PhD Thesis, New York Univ. [Google Scholar]
  38. Grad H. 1949. On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2:331–407 [Google Scholar]
  39. Grad H. 1952. The profile of a steady plane shock wave. Commun. Pure Appl. Math. 5:257–300 [Google Scholar]
  40. Grad H. 1958. Principles of the kinetic theory of gases. Thermodynamics of Gases S Flügge 205–94 Berlin: Springer-Verlag [Google Scholar]
  41. Gu XJ, Emerson D. 2007. A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions. J. Comput. Phys. 225:263–83 [Google Scholar]
  42. Gu XJ, Emerson D. 2009. A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J. Fluid Mech. 636:177–216 [Google Scholar]
  43. Gu XJ, Emerson D, Tang GH. 2010. Analysis of the slip coefficient and defect velocity in the Knudsen layer of a rarefied gas using the linearized moment equations. Phys. Rev. E 81:016313 [Google Scholar]
  44. Gupta VK, Torrilhon M. 2012. Automated Boltzmann collision integrals for moment equations. AIP Conf. Proc. 1501:67–74 [Google Scholar]
  45. Gupta VK, Torrilhon M. 2015. Higher order moment equations for mixtures of rarefied gases. Proc. R. Soc. A 471:20140754 [Google Scholar]
  46. Hadjiconstantinou NG. 2006. Comment on Cercignani's second-order slip coefficient. Phys. Fluids 15:2352–54 [Google Scholar]
  47. Hickey KA, Loyalka SK. 1990. Plane Poiseuille flow: rigid sphere gas. J. Vac. Sci. Technol. A 8:957–60 [Google Scholar]
  48. Holian BL, Patterson CW, Mareschal M, Salomons E. 1993. Modeling shock waves in an ideal gas: going beyond the Navier-Stokes level. Phys. Rev. E 47:R24–27 [Google Scholar]
  49. Ivanov IE, Kryukov IA, Timokhin MY. 2013. Application of moment equations to the mathematical simulation of gas microflows. Comput. Math. Math. Phys. 53:1543–50 [Google Scholar]
  50. Ivanov IE, Kryukov IA, Timokhin MY, Bondar YA, Kokhanchik AA, Ivanov MS. 2012. Stud of the shock wave structure by regularized Grad's set of equations. AIP Conf. Proc. 1501:215–22 [Google Scholar]
  51. Jin S, Slemrod M. 2001. Regularization of the Burnett equations via relaxation. J. Stat. Phys. 103:1009–33 [Google Scholar]
  52. Jou D, Casas-Vazquez J, Lebon G. 1996. Extended Irreversible Thermodynamics Berlin: Springer-Verlag, 2nd ed.. [Google Scholar]
  53. Junk M. 1998. Domain of definition of Levermore's five-moment system. J. Stat. Phys. 93:1143–67 [Google Scholar]
  54. Junk M. 2002. Maximum entropy moment systems and Galilean invariance. Contin. Mech. Thermodyn. 14:563–76 [Google Scholar]
  55. Karlin IV, Gorban AN, Dukek G, Nonnenmacher TF. 1998. Dynamic correction to moment approximations. Phys. Rev. E 57:1668–72 [Google Scholar]
  56. Karniadakis GE, Beskok A. 2001. Micro Flows: Fundamentals and Simulation New York: Springer [Google Scholar]
  57. Kauf P, Torrilhon M, Junk M. 2010. Scale-induced closure for approximations of kinetic equations. J. Stat. Phys. 41:848–88 [Google Scholar]
  58. Knudsen M. 1909. Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren. Ann. Phys. 333:75–130 [Google Scholar]
  59. Kogan MN. 1969. Rarefied Gas Dynamics New York: Plenum [Google Scholar]
  60. Köllermeier J, Schärer RP, Torrilhon M. 2014. A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods. Kinet. Rel. Models 7:531–49 [Google Scholar]
  61. Levermore CD. 1996. Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83:1021–65 [Google Scholar]
  62. Levermore CD, Morokoff WJ. 1998. The Gaussian moment closure for gas dynamics. SIAM J. Appl. Math. 59:72–96 [Google Scholar]
  63. Lockerby DA, Reese JM. 2003. High-resolution Burnett simulations of micro Couette flow and heat transfer. J. Comput. Phys. 188:333–47 [Google Scholar]
  64. Lockerby DA, Reese JM. 2008. On the modeling of isothermal gas flows at the microscale. J. Fluid Mech. 604:235–61 [Google Scholar]
  65. Lockerby DA, Reese JM, Emerson DR, Barber RW. 2004. Velocity boundary condition at solid walls in rarefied gas calculations. Phys. Rev. E 70:017303 [Google Scholar]
  66. Magin TE, Martins G, Torrilhon M. 2010. Regularized Grad equations for multicomponent plasmas. AIP Conf. Proc. 1333:99–104 [Google Scholar]
  67. Mansour MM, Babras F, Garcia AL. 1997. On the validity of hydrodynamics in plane Poiseuille flows. Physica A 240:255–67 [Google Scholar]
  68. Marquez W Jr, Kremer GM. 2001. Couette flow from a thirteen field theory with jump and slip boundary conditions. Contin. Mech. Thermodyn. 13:207–17 [Google Scholar]
  69. Maxwell JC. 1879. On stresses in rarefied gases arising from inequalities of temperature. Philos. Trans. R. Soc. 170:231–56 [Google Scholar]
  70. McDonald JG. 2010. Extended fluid-dynamic modelling for numerical solution of micro-scale flows PhD Thesis, Univ. Toronto [Google Scholar]
  71. McDonald JG, Groth CPT. 2008. Extended fluid dynamic model for micron-scale flows based on Gaussian moment closure Presented at AIAA Aerosp. Sci. Meet., 46th, Reno, NV, AIAA Pap 2008–691 [Google Scholar]
  72. McDonald JG, Groth CPT. 2009. Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution. Contin. Mech. Thermodyn. 21:467–93 [Google Scholar]
  73. McDonald JG, Torrilhon M. 2013. Affordable robust moment closures for CFD based on the maximum-entropy hierarchy. J. Comput. Phys. 251:500–23 [Google Scholar]
  74. McGraw R. 1997. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. Technol. 27:255–65 [Google Scholar]
  75. Müller I, Reitebuch D, Weiss W. 2003. Extended thermodynamics: consistent in order of magnitude. Contin. Mech. Thermodyn. 15:411–25 [Google Scholar]
  76. Müller I, Ruggeri T. 1998. Rational Extended Thermodynamics New York: Springer, 2nd ed.. [Google Scholar]
  77. Myong RS. 2001. A computational method for Eu's generalized hydrodynamic equations of rarefied and microscale gasdynamics. J. Comput. Phys. 168:47–72 [Google Scholar]
  78. Myong RS. 2011. A full analytical solution for the force-driven compressible Poiseuille gas flow based on a nonlinear coupled constitutive relation. Phys. Fluids 23:012002 [Google Scholar]
  79. Ohwada T, Sone Y, Aoki K. 1989. 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 1:2042–49 [Google Scholar]
  80. Öttinger HC. 2005. Beyond Equilibrium Thermodynamics New York: Wiley [Google Scholar]
  81. Öttinger HC. 2010a. Öttinger replies. Phys. Rev. Lett. 105:128902 [Google Scholar]
  82. Öttinger HC. 2010b. Thermodynamically admissible 13 moment equations from the Boltzmann equation. Phys. Rev. Lett. 104:120601 [Google Scholar]
  83. Pham-Van-Diep GC, Erwin DA, Muntz EP. 1991. Testing continuum descriptions of low-Mach-number shock structures. J. Fluid Mech. 232:403–13 [Google Scholar]
  84. Rahimi B, Struchtrup H. 2014. Capturing non-equilibrium phenomena in rarefied polyatomic gases: a high-order macroscopic model. Phys. Fluids 26:052001 [Google Scholar]
  85. Rana AS, Mohammadzadeh A, Struchtrup H. 2015. A numerical study of the heat transfer through a rarefied gas confined in a microcavity. Contin. Mech. Thermodyn. 27:433–46 [Google Scholar]
  86. Rana AS, Torrilhon M, Struchtrup H. 2013. A robust numerical method for the R13 equations of rarefied gas dynamics: application to lid driven cavity. J. Comput. Phys. 236:169–86 [Google Scholar]
  87. Reese JM, Zheng Y, Lockerby DA. 2007. Computing the near-wall region in gas micro- and nanofluidics: critical Knudsen layer phenomena. J. Comput. Theor. Nanosci. 4:807–13 [Google Scholar]
  88. Reinecke S, Kremer GM. 1990. Methods of moments of Grad. Phys. Rev. A 42:815–20 [Google Scholar]
  89. Reitebuch D, Weiss W. 1999. Application of high moment theory to the plane Couette flow. Contin. Mech. Thermodyn. 11:217–25 [Google Scholar]
  90. Risso D, Cordero P. 1998. Generalized hydrodynamics for a Poiseuille flow: theory and simulations. Phys. Rev. E 58:546–53 [Google Scholar]
  91. Rosenau P. 1989. Extending hydrodynamics via the regularization of the Chapman-Enskog solution. Phys. Rev. A 40:7193–96 [Google Scholar]
  92. Ruggeri T. 1993. Breakdown of shock wave structure solutions. Phys. Rev. E 47:4135–40 [Google Scholar]
  93. Schärer RP, Torrilhon M. 2015. On singular closures for the 5-moment system in kinetic gas theory. Commun. Comput. Phys. 17:371–400 [Google Scholar]
  94. Seeger S, Hoffmann H. 2000. The cumulant method in computational kinetic theory. Contin. Mech. Therm. 12:403–21 [Google Scholar]
  95. Sharipov F, Seleznev V. 1998. Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data 27:657–706 [Google Scholar]
  96. Shavaliyev MS. 1993. Super-Burnett corrections to the stress tensor and the heat flux in a gas of Maxwellian molecules. J. Appl. Math. Mech. 57:573–76 [Google Scholar]
  97. Söderholm LHS. 2007. Hybrid Burnett equations: a new method of stabilizing. Trans. Theory Stat. Phys. 36:495–512 [Google Scholar]
  98. Sone Y. 2002. Kinetic Theory and Fluid Dynamics Basel: Birkhauser [Google Scholar]
  99. Struchtrup H. 2002. Heat transfer in the transition regime: solution of boundary value problems for Grad's moment equations via kinetic schemes. Phys. Rev. E 65:041204 [Google Scholar]
  100. Struchtrup H. 2004. Stable transport equations for rarefied gases at high orders in the Knudsen number. Phys. Fluids 16:3921–34 [Google Scholar]
  101. Struchtrup H. 2005a. Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials. Multiscale Model. Simul. 3:221–43 [Google Scholar]
  102. Struchtrup H. 2005b. Failures of the Burnett and super-Burnett equations in steady state processes. Contin. Mech. Thermodyn. 17:43–50 [Google Scholar]
  103. Struchtrup H. 2005c. Macroscopic Transport Equations for Rarefied Gas Flows New York: Springer [Google Scholar]
  104. Struchtrup H. 2008. Linear kinetic heat transfer: moment equations, boundary conditions, and Knudsen layers. Physica A 387:1750–66 [Google Scholar]
  105. Struchtrup H, Taheri P. 2011. Macroscopic transport models for rarefied gas flows: a brief review. IMA J. Appl. Math. 76:672–97 [Google Scholar]
  106. Struchtrup H, Thatcher T. 2007. Bulk equations and Knudsen layers for the regularized 13 moment equations. Contin. Mech. Thermodyn. 19:177–89 [Google Scholar]
  107. Struchtrup H, Torrilhon M. 2003. Regularization of Grad's 13 moment equations: derivation and linear analysis. Phys. Fluids 15:2668–80 [Google Scholar]
  108. Struchtrup H, Torrilhon M. 2007. H-theorem, regularization, and boundary conditions for linearized 13 moment equations. Phys. Rev. Lett. 99:014502 [Google Scholar]
  109. Struchtrup H, Torrilhon M. 2008. High order effects in rarefied channel flows. Phys. Rev. E 78:046301 Erratum. 2008 Phys. Rev. E 78:069903E [Google Scholar]
  110. Struchtrup H, Torrilhon M. 2010. Comment on “Thermodynamically admissible 13 moment equations from the Boltzmann equation.”. Phys. Rev. Lett. 105:128901 [Google Scholar]
  111. Struchtrup H, Torrilhon M. 2013. Regularized 13 moment equations for hard sphere molecules: linear bulk equations. Phys. Fluids 25:052001 [Google Scholar]
  112. Taheri P, Rana AS, Torrilhon M, Struchtrup H. 2009a. Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics. Contin. Mech. Thermodyn. 21:423–43 [Google Scholar]
  113. Taheri P, Struchtrup H. 2010a. An extended macroscopic transport model for rarefied gas flows in long capillaries with circular cross section. Phys. Fluids 22:112004 [Google Scholar]
  114. Taheri P, Struchtrup H. 2010b. Rarefaction effects in thermally-driven microflows. Physica A 389:3069–80 [Google Scholar]
  115. Taheri P, Torrilhon M, Struchtrup H. 2009b. Couette and Poiseuille microflows: analytical solutions for regularized 13-moment equations. Phys. Fluids 21:017102 [Google Scholar]
  116. 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 5:716–37 [Google Scholar]
  117. Tij M, Sabbane M, Santos A. 1998. Nonlinear Poiseuille flow in a gas. Phys. Fluids 10:1021–27 [Google Scholar]
  118. Tij M, Santos A. 1994. Perturbation analysis of a stationary nonequilibrium flow generated by external force. J. Stat. Phys. 76:1399–414 [Google Scholar]
  119. Todd BD, Evans DJ. 1995. The heat-flux vector for highly inhomogeneous nonequilibrium fluids in very narrow pores. J. Chem. Phys. 103:9804–9 [Google Scholar]
  120. Todd BD, Evans DJ. 1997. Temperature profiles for Poiseuille flow. Phys. Rev. E 55:2800–7 [Google Scholar]
  121. Torrilhon M. 2000. Characteristic waves and dissipation in the 13-moment-case. Contin. Mech. Thermodyn. 12:289–301 [Google Scholar]
  122. Torrilhon M. 2006. Two-dimensional bulk microflow simulations based on regularized 13-moment equations. SIAM Multiscale Model. Simul. 5:695–728 [Google Scholar]
  123. Torrilhon M. 2009. Editorial: special issue on moment methods in kinetic gas theory. Contin. Mech. Thermodyn. 21:341–43 [Google Scholar]
  124. Torrilhon M. 2010a. Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions. Commun. Comput. Phys. 7:639–73 [Google Scholar]
  125. Torrilhon M. 2010b. Slow rarefied flow past a sphere: analytical solutions based on moment equations. Phys. Fluids 22:072001 [Google Scholar]
  126. Torrilhon M. 2012. H-theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic Relat. Models 5:185–201 [Google Scholar]
  127. Torrilhon M, Struchtrup H. 2004. Regularized 13-moment equations: shock structure calculations and comparison to Burnett models. J. Fluid Mech. 513:171–98 [Google Scholar]
  128. Torrilhon M, Struchtrup H. 2008a. Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227:1982–2011 [Google Scholar]
  129. Torrilhon M, Struchtrup H. 2008b. Modeling micro-mass and -heat transfer for gases based on extended continuum models. ASME J. Heat Transfer 131:033103 [Google Scholar]
  130. Uribe FJ, Garcia AL. 1999. Burnett description for plane Poiseuille flow. Phys. Rev. E 60:4063–78 [Google Scholar]
  131. Uribe FJ, Velasco RM, Garcia-Colin LS. 1998. Burnett description of strong shock waves. Phys. Rev. Lett. 81:2044–47 [Google Scholar]
  132. Uribe FJ, Velasco RM, Garcia-Colin LS. 2000. Bobylev's instability. Phys. Rev. E 62:5835–38 [Google Scholar]
  133. Vincenti WG, Kruger CH. 1965. Introduction to Physical Gas Dynamics New York: Wiley [Google Scholar]
  134. Weiss W. 1995. Continuous shock structures in extended thermodynamics. Phys. Rev. E 52:R5760–63 [Google Scholar]
  135. Westerkamp A, Torrilhon M. 2012. Slow rarefied gas flow past a cylinder: analytical solution in comparison to the sphere. AIP Conf. Proc. 1501:207–14 [Google Scholar]
  136. Xu K. 2003. Super-Burnett solutions for Poiseuille flow. Phys. Fluids 15:2077–80 [Google Scholar]
  137. Xu K, Li ZH. 2004. Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions. J. Fluid Mech. 513:87–110 [Google Scholar]
  138. Young JB. 2011. Calculation of Knudsen layers and jump conditions using the linearised G13 and R13 moment methods. Int. J. Heat Mass Transfer 54:2902–12 [Google Scholar]
  139. Yuan C, Fox RO. 2011. Conditional quadrature method of moments for kinetic equations. J. Comput. Phys. 230:8216–46 [Google Scholar]
  140. Zhdanov VM. 2002. Transport Processes in Multicomponent Plasma London: Taylor & Francis [Google Scholar]
  141. Zheng Y, Garcia AL, Alder JB. 2002. Comparison of kinetic theory and hydrodynamics for Poiseuille flow. J. Stat. Phys. 109:495–505 [Google Scholar]
  142. Zhong X, MacCormack RW, Chapman DR. 1993. Stabilization of the Burnett equations and applications to hypersonic flows. AIAA J. 31:1036–43 [Google Scholar]

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