Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics. (English) Zbl 1184.76844

Summary: Four basic flow configurations are employed to investigate steady and unsteady rarefaction effects in monatomic ideal gas flows. Internal and external flows in planar geometry, namely, viscous slip (Kramer’s problem), thermal creep, oscillatory Couette, and pulsating Poiseuille flows are considered. A characteristic feature of the selected problems is the formation of the Knudsen boundary layers, where non-Newtonian stress and non-Fourier heat conduction exist. The linearized Navier-Stokes-Fourier and regularized 13-moment equations are utilized to analytically represent the rarefaction effects in these boundary-value problems. It is shown that the regularized 13-moment system correctly estimates the structure of Knudsen layers, compared to the linearized Boltzmann equation data.


76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
Full Text: DOI


[1] Cercignani C.: Theory and application of the Boltzmann equation. Scottish Academic Press, Edinburgh (1975) · Zbl 0403.76065
[2] de Groot S.R., Mazur P.: Non-equilibrium thermodynamics. Dover, New York (1984) · Zbl 0041.58401
[3] Chapman S., Cowling T.G.: The mathematical theory of non-uniform gases. Cambridge University Press, Cambridge (1970) · Zbl 0063.00782
[4] Grad H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949) · Zbl 0037.13104
[5] Grad H.: Principles of the kinetic theory of gases. In: Flügge, S. (eds) Handbuch der Physik, Springer, Berlin (1958)
[6] Bobylev A.V.: The Chapman-Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys. Dokl. 27, 29–31 (1982)
[7] Rosenau P.: Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. Phys. Rev. A 40, 7193–7196 (1989)
[8] Zhong X., MacCormack R.W., Chapman D.R.: Stabilization of the Burnett equations and applications to hypersonic flows. AIAA J. 31, 1036–1043 (1993) · Zbl 0774.76070
[9] Jin S., Slemrod M.: Regularization of the Burnett equations via relaxation. J. Stat. Phys. 103, 1009–1033 (2001) · Zbl 1019.82018
[10] Müller I., Reitebuch D., Weiss W.: Extended thermodynamics–consistent in order of magnitude. Contin. Mech. Thermodyn. 15, 113–146 (2003) · Zbl 1057.80002
[11] Bobylev A.V.: Instabilities in the Chapman-Enskog expansion and Hyperbolic Burnett equations. J. Stat. Phys. 124, 371–399 (2006) · Zbl 1134.82031
[12] Söderholm L.H.: Hybrid Burnett Equations: a new method of stabilizing. Transp. Theory Stat. Phys. 36, 495–512 (2007) · Zbl 1183.82070
[13] Struchtrup H.: Macroscopic transport equations for rarefied gas flows. Springer, New York (2005) · Zbl 1119.76002
[14] Struchtrup H., Torrilhon M.: Regularization of Grad’s 13-moment equations: derivation and linear analysis. Phys. Fluids 15, 2668–2680 (2003) · Zbl 1186.76504
[15] Struchtrup H.: Stable transport equations for rarefied gases at high orders in the Knudsen number. Phys. Fluids 16, 3921–3934 (2004) · Zbl 1187.76505
[16] Torrilhon M., Struchtrup H.: Regularized 13-moment-equations: shock structure calculations and comparison to Burnett models. J. Fluid Mech. 513, 171–198 (2004) · Zbl 1107.76069
[17] Struchtrup H., Thatcher T.: Bulk equations and Knudsen layers for the regularized 13 moment equations. Contin. Mech. Thermodyn. 19, 177–189 (2007) · Zbl 1160.76407
[18] Struchtrup H., Torrilhon M.: H theorem, regularization, and boundary conditions for linearized 13-moment equations. Phys. Rev. Lett. 99, 014502 (2007) · Zbl 1284.76332
[19] Struchtrup H.: Linear kinetic heat transfer: moment equations, boundary conditions, and Knudsen layers. Phys. A 387, 1750–1766 (2008) · Zbl 1395.82210
[20] Torrilhon M., Struchtrup H.: Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227, 1982–2011 (2008) · Zbl 1132.76049
[21] Struchtrup H., Torrilhon M.: High order effects in rarefied channel flows. Phys. Rev. E 78, 046301 (2008) · Zbl 1132.76049
[22] Taheri P., Torrilhon M., Struchtrup H.: Couette and Poiseuille microflows: analytical solutions for regularized 13-moment equations. Phys. Fluids 21, 017102 (2009) · Zbl 1183.76503
[23] Taheri, P., Struchtrup, H.: Rarefaction effects in thermally-driven microflows (2009, submitted)
[24] Gu X.J., Emerson D.R.: A computational strategy for the regularized 13-moment equations with enhanced wall-boundary conditions. J. Comput. Phys. 225, 263–283 (2007) · Zbl 1201.76127
[25] Ohwada T., Sone Y., Aoki K.: 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, 2042–2049 (1989) · Zbl 0696.76092
[26] Landau L.D., Lifshitz E.M.: Fluid mechanics. Pergamon, Oxford (1987) · Zbl 0146.22405
[27] Gad-el-Hak, M. (eds): The MEMS handbook: introduction and fundamentals. CRC, London (2005)
[28] Bahukudumbi P., Park J.H., Beskok A.: A unified engineering model for steady and quasi-steady shear-driven gas microflows. Microscale Thermophys. Eng. 7, 291–315 (2003)
[29] Park J.H., Bahukudumbi P., Beskok A.: Rarefaction effects on shear driven oscillatory gas flows: A direct simulation Monte Carlo study in the entire Knudsen regime. Phys. Fluids 16, 317–330 (2004) · Zbl 1186.76413
[30] Hadjiconstantinou N.G.: Oscillatory shear-driven gas flow in the transition and free-molecular-flow regimes. Phys. Fluids 17, 100611 (2005) · Zbl 1187.76199
[31] Sharipov F., Kalempa D.: Oscillatory Couette flow at arbitrary oscillation frequency over the whole range of the Knudsen number. Microfluid. Nanofluid. 4, 363–374 (2008)
[32] Maxwell J.C.: On stresses in rarefied gases arising from inequalities of temperature. Philos. Trans. R. Soc. Lond. 170, 231–256 (1879) · JFM 11.0777.01
[33] Lockerby D.A., Reese J.M., Emerson D.R., Barber R.W.: Velocity boundary condition at solid walls in rarefied gas calculations. Phys. Rev. E 70, 017303 (2004)
[34] Deissler R.G.: An analysis of second order slip flow and temperature jump boundary conditions for rarefied gases. Int. J. Heat Mass Transf. 7, 681–694 (1964) · Zbl 0141.43604
[35] Hadjiconstantinou N.G.: Comment on Cercignani’s second-order slip coefficient. Phys. Fluids 15, 2352–2354 (2003)
[36] Loyalka S.K.: Velocity profile in the Knudsen layer for the Kramer’s problem. Phys. Fluids 18, 1666–1669 (1975) · Zbl 0337.76029
[37] Loyalka S.K., Petrellis N., Storvick T.S.: Some numerical results for the BGK model: thermal creep and viscous slip problems with arbitrary accommodation at the surface. Phys. Fluids 18, 1094–1099 (1975) · Zbl 0315.76037
[38] Loyalka S.K., Ferziger H.: Model dependence of the slip coefficient. Phys. Fluids 10, 1833–1839 (1967) · Zbl 0204.27901
[39] Loyalka S.K., Hickey K.A.: Velocity slip and defect: hard sphere gas. Phys. Fluids A 1, 612–614 (1989) · Zbl 0659.76085
[40] Ohwada T., Sone Y., Aoki K.: 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, 1588–1599 (1989) · Zbl 0695.76032
[41] Loyalka S.K., Hickey K.A.: The Kramers problem: velocity slip and defect for a hard sphere gas with arbitrary accommodation. J. Appl. Math. Phys. (ZAMP) 41, 245–253 (1990) · Zbl 0699.76085
[42] Barichello L.B., Camargo M., Rodrigues P., Siewert C.E.: Unified solutions to classical flow problems based on the BGK model. Z. Angew. Math. Phys. (ZAMP) 52, 517–534 (2001) · Zbl 1017.76078
[43] Lockerby D.A., Reese J.M., Gallis M.A.: The usefulness of higher-order constitutive relations for describing the Knudsen layer. Phys. Fluids 17, 100609 (2005) · Zbl 1187.76320
[44] Lilley C.R., Sader J.E.: Velocity gradient singularity and structure of the velocity profile in the Knudsen layer according to the Boltzmann equation. Phys. Rev. E 76, 026315 (2007) · Zbl 1145.76353
[45] Loyalka S.K., Cipolla J.W.: Thermal creep slip with arbitrary accommodation at the surface. Phys. Fluids 14, 1656–1661 (1971)
[46] Kanki T., Iuchi S.: Poiseuille flow and thermal creep of a rarefied gas between parallel plates. Phys. Fluids 16, 594–599 (1973) · Zbl 0267.76060
[47] Loyalka S.K.: Comments on Poiseuille flow and thermal creep of a rarefied gas between parallel plates. Phys. Fluids 17, 1053–1055 (1974) · Zbl 0291.76026
[48] Loyalka S.K., Petrellis N., Storvick T.S.: Some exact numerical results for the BGK model: Couette, Poiseuille and thermal creep flow between parallel plates. Z. Angew. Math. Phys. (ZAMP) 30, 514–521 (1979) · Zbl 0404.76063
[49] Loyalka S.K.: Temperature jump and thermal creep slip: rigid sphere gas. Phys. Fluids A1, 403–408 (1989) · Zbl 0661.76081
[50] Sone Y.: Kinetic theory and fluid dynamics. Birkhäuser, Boston (2002) · Zbl 1021.76002
[51] Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954) · Zbl 0055.23609
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.