Resolving Knudsen layer by high-order moment expansion

  • Yuwei Fan
  • Jun Li
  • Ruo LiEmail author
  • Zhonghua Qiao
Original Article


We model the Knudsen layer in Kramers’ problem by the linearized high-order hyperbolic moment system. Thanks to the hyperbolicity of the moment system, its boundary conditions are properly reduced from the kinetic boundary condition. For the Kramers’ problem, we present the analytical solutions of the linearized moment systems. The velocity profile in the Knudsen layer is captured with improved accuracy for a wide range of accommodation coefficients. With the order of the moment system increasing, the velocity profile approaches to that of the linearized Boltzmann–BGK equation.


Knudsen layer Kramers’ problem Hyperbolic moment equations Boltzmann equation 


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We thank the anonymous referees for their valuable comments and suggestions on the paper. The research of J. Li is partially supported by the Hong Kong Research Council ECS Grant No. 509213 during her visit periods at the Hong Kong Polytechnic University. The research of J. Li and R. Li is supported by the National Natural Science Foundation of China (11325102, 11421110001, 91630310). The research of Z.-H. Qiao is partially supported by the Hong Kong Research Council ECS Grant No. 509213 and the Hong Kong Polytechnic University research fund G-YBKP.


  1. 1.
    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(3), 511–525 (1954)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford (1994)Google Scholar
  3. 3.
    Boltzmann, L.: Weitere studien über das wärmegleichgewicht unter gas-molekülen. Wien. Ber. 66, 275–370 (1872)zbMATHGoogle Scholar
  4. 4.
    Cai, Z., Fan, Y., Li, R.: Globally hyperbolic regularization of Grad’s moment system in one dimensional space. Commun. Math. Sci. 11(2), 547–571 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cai, Z., Fan, Y., Li, R.: Globally hyperbolic regularization of Grad’s moment system. Commun. Pure Appl. Math. 67(3), 464–518 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cai, Z., Fan, Y., Li, R.: On hyperbolicity of 13-moment system. Kinet. Relat. Models 7(3), 415–432 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cai, Z., Fan, Y., Li, R., Qiao, Z.: Dimension-reduced hyperbolic moment method for the Boltzmann equation with BGK-type collision. Commun. Comput. Phys. 15(5), 1368–1406 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cai, Z., Li, R., Qiao, Z.: NR\(xx\) simulation of microflows with Shakhov model. SIAM J. Sci. Comput. 34(1), A339–A369 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cai, Z., Li, R., Qiao, Z.: Globally hyperbolic regularized moment method with applications to microflow simulation. Comput. Fluids 81, 95–109 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cercignani, C.: Mathematical Methods in Kinetic Theory. Springer, New York (1969)CrossRefzbMATHGoogle Scholar
  11. 11.
    Chapman, S.: On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas. Philos. Trans. R. Soc. A 216(538–548), 279–348 (1916)ADSCrossRefGoogle Scholar
  12. 12.
    Dongari, N., Sambasivam, R., Durst, F.: Extended Navier–Stokes equations and treatments of micro-channel gas flows. J. Fluid Sci. Technol. 4(2), 454–467 (2009)CrossRefGoogle Scholar
  13. 13.
    Fan, Y., Li, R.: Globally hyperbolic moment system by generalized Hermite expansion. Scientia Sinica Mathematica 45(10), 1635–1676 (2015)Google Scholar
  14. 14.
    Garcia, R.D.M., Siewert, C.E.: The linearized Boltzmann equation with Cercignani–Lampis boundary conditions: basic flow problems in a plane channel. Eur. J. Mech. B Fluids 28(3), 387–396 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331–407 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grucelski, A., Pozorski, J.: Lattice Boltzmann simulations of flow past a circular cylinder and in simple porous media. Comput. Fluids 71, 406–416 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gu, X.J., Emerson, D.R., Tang, G.H.: Kramers’ problem and the Knudsen minimum: a theoretical analysis using a linearized 26-moment approach. Contin. Mech. Thermodyn. 21, 345–360 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gu, X.J., Emerson, D.R., Tang, G.H.: 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 (2010)ADSCrossRefGoogle Scholar
  19. 19.
    Guo, Z., Zhao, T.S., Shi, Y.: Generalized hydrodynamic model for fluid flows: from nanoscale to macroscale. Phys. Fluids 18(6), 067107 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Higuera, F.J., Succi, S.: Simulating the flow around a circular cylinder with a lattice Boltzmann equation. Europhys. Lett. 8(6), 517 (1989)ADSCrossRefGoogle Scholar
  21. 21.
    Karniadakis, G.E., Beskok, A., Aluru, N.: Microflows: Fundamentals and Simulation. Springer, New York (2002)zbMATHGoogle Scholar
  22. 22.
    Klinc, T., Kuscer, I.: Slip coefficients for general gas–surface interaction. Phys. Fluids 15, 1018 (1972)ADSCrossRefGoogle Scholar
  23. 23.
    Kramers, H.A.: On the behaviour of a gas near a wall. Il Nuovo Cimento (1943–1954) 6(2), 297–304 (1949)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211, Third edn. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  25. 25.
    Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83(5–6), 1021–1065 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)ADSCrossRefGoogle Scholar
  27. 27.
    Lockerby, D.A., Reese, J.M.: On the modelling of isothermal gas flows at the microscale. J. Fluid Mech. 604, 235–261 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Loyalka, S.K., Ferziger, J.H.: Model dependence of the slip coefficient. Phys. Fluids 10, 1833 (1967)ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Loyalka, S.K., Hickey, K.A.: The Kramers problem: velocity slip and defect for a hard sphere gas with arbitrary accommodation. Z. Angew. Math. Phys. 41, 245 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Loyalka, S.K., Naturforsch, Z.: Approximate method in kinetic theory. Phys. Fluids 14, 2291–2294 (1971)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Loyalka, S.K., Petrellis, N., Storvick, T.S.: Some numerical results for the BGK model: thermal creep and viscous slip problems with arbitrary accomodation at the surface. Phys. Fluids 18(9), 1094–1099 (1975)ADSCrossRefzbMATHGoogle Scholar
  32. 32.
    Marques Jr., W., Kremer, G.M.: Couette flow from a thirteen field theory with slip and jump boundary conditions. Contin. Mech. Thermodyn. 13(3), 207–217 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Maxwell, J.C.: On stresses in rarefied gases arising from inequalities of temperature. Proc. R. Soc. Lond. 27(185–189), 304–308 (1878)zbMATHGoogle Scholar
  34. 34.
    Mizzi, S., Barber, R.W., Emerson, D.R., Reese, J.M., Stefanov, S.K.: A phenomenological and extended continuum approach for modelling non-equilibrium flows. Contin. Mech. Thermodyn. 19(5), 273–283 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer Tracts in Natural Philosophy, vol. 37, Second edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  36. 36.
    Reese, J.M., Gallis, M.A., Lockerby, D.A.: New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 361(1813), 2967–2988 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Siewert, C.E.: Kramers’ problem for a variable collision frequency model. Eur. J. Appl. Math. 12, 179–191 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Struchtrup, H.: Kinetic schemes and boundary conditions for moment equations. Z. Angew. Math. Phys. 51(3), 346–365 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Struchtrup, H.: Grad’s moment equations for microscale flows. In: Ketsdever, A.D., Muntz, E.P. (eds.) Rarefied Gas Dynamics: 23rd International Symposium, vol. 663, pp. 792–799. AIP (2003)Google Scholar
  40. 40.
    Torrilhon, M.: Special issues on moment methods in kinetic gas theory. Contin. Mech. Thermodyn. 21(5), 341–343 (2009)CrossRefGoogle Scholar
  41. 41.
    Torrilhon, M., Struchtrup, H.: Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227(3), 1982–2011 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Williams, M.M.R.: A review of the rarefied gas dynamics theory associated with some classical problems in flow and heat transfer. Z. Angew. Math. Phys. ZAMP 52(3), 500–516 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zhang, W.M., Meng, G., Wei, X.Y.: A review on slip models for gas microflows. Microfluid. Nanofluidics 13(6), 845–882 (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.School of Mathematical SciencesPeking UniversityBeijingChina
  3. 3.CAPT, LMAM & School of Mathematical SciencesPeking UniversityBeijingChina
  4. 4.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung HomHong Kong

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