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Invariance correction to Grad’s equations: where to go beyond approximations?

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Abstract

We review some recent developments of Grad’s approach to solving the Boltzmann equation and creating a reduced description. The method of the invariant manifold is put forward as a unified principle to establish corrections to Grad’s equations. A consistent derivation of regularized Grad’s equations in the framework of the method of the invariant manifold is given. A new class of kinetic models to lift the finite-moment description to a kinetic theory in the whole space is established. Relations of Grad’s approach to modern mesoscopic integrators such as the entropic lattice Boltzmann method are also discussed.

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References

  1. Grad, H.: On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331–407 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beskok, A., Karniadakis, G.E.: Microflows: Fundamentals and Simulation. Springer, Berlin (2001)

    Google Scholar 

  3. Kogan, A.M.: Derivation of Grad-type equations, study of their properties by the method of entropy maximization. Prikl. Math. Mech. 29, 122–133 (1965)

    Google Scholar 

  4. Lewis, R.M.: A unified principle in statistical mechanics. J. Math. Phys. 8, 1448–1459 (1967)

    Article  MATH  Google Scholar 

  5. Gorban, A.N.: Equilibrium Encircling. Equations of chemical kinetics, and their thermodynamic analysis. Nauka, Novosibirsk (1984)

    Google Scholar 

  6. Karlin, I.V.: Relaxation of chemical reaction rates under translationally nonequilibrium conditions. In: Proc. VIII USSR Symp. on Burning, Combustion, 97–99. Chernogolovka, Inst. Chem. Phys. (1986)

  7. Gorban, A.N., Karlin, I.V.: Quasi-equilibrium approximations and non-standard expansions in the theory of the Boltzmann kinetic equation. In: Khlebopros, R.G. (Ed.): Mathematical Modeling in Biology and Chemistry (New Approaches), pp.~69–117. Nauka, Novosibirsk (1991). English translation of the first part of this paper (triangle entropy method): http://arXiv.org/abs/cond-mat/0305599

  8. Gorban, A.N., Karlin, I.V.: Scattering rates versus moments: alternative grad equations. Phys. Rev. E 54, R3109–R3112 (1996)

    Article  Google Scholar 

  9. Gorban, A.N., Karlin, I.V.: Thermodynamic parameterization. Physica A 190, 393–404 (1992)

    Article  MathSciNet  Google Scholar 

  10. Levermore, C.D.: Moment closure hierarchies. J. Stat. Phys. 83, 1021 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ilg, P., Karlin, I.V., Öttinger, H.C.: Canonical distribution functions in polymer dynamics: I. Dilute solutions of flexible polymers. Physica A 315(3–4), 318–336 (2002)

    Article  Google Scholar 

  12. Ilg, P., Karlin, I.V., Kröger, M., Öttinger, H.C.: Canonical distribution functions in polymer dynamics: II. Liquid-crystalline polymers. Physica A 319, 134–150 (2003)

    Article  MATH  Google Scholar 

  13. Gorban, A.N., Gorban, P.A., Karlin, I.V.: Legendre integrators, postprocessing and quasiequilibrium. J. Non-Newtonian Fluid Mech. 120, 149–167 (2004)

    Article  MATH  Google Scholar 

  14. Müller, I., Ruggeri, T.: Extended Thermodynamics. Springer, Berlin (1993)

    MATH  Google Scholar 

  15. Bobylev, A.V.: On the Chapman-Enskog and Grad methods. Dokl. Acad. Nauk SSSR 262, 71 (1982)

    MathSciNet  Google Scholar 

  16. Karlin, I.V.: Method of Invariant Manifold in Kinetic Theory. PhD thesis, AMSE University, Tassin (1992)

    Google Scholar 

  17. Gorban, A.N., Karlin, I.V.: Method of invariant manifolds and regularization of acoustic spectra. Transport Theory and Stat. Phys. 23, 559–632 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gorban, A.N., Karlin, I.V.: New Methods for Solving the Boltzmann Equations, vol.~10 [Physical Kinetics] of Scientific Siberian A. AMSE Press, Tassin (1993)

    Google Scholar 

  19. Gorban, A.N., Karlin, I.V., Zinovyev, A.Yu.: Constructive methods of invariant manifolds for kinetic problems. Physics Reports 396, 197–403 (2004)

    Article  MathSciNet  Google Scholar 

  20. Gorban, A.N., Karlin, I.V.: Invariant Manifolds for Physical and Chemical Kinetics, vol.~660 of Lect. Notes Phys. Springer, Berlin Heidelberg (2005)

    Google Scholar 

  21. Gorban, A.N., Karlin, I.V.: Uniqueness of thermodynamic projector and kinetic basis of molecular individualism. Physica A 336(3–4), 391–432 (2004) Preprint online: http://arxiv.org/abs/cond-mat/0309638

    Article  Google Scholar 

  22. Slemrod, M.: Renormalization of the Chapman-Enskog expansion: Isothermal fluid flow, Rosenau saturation. J. Stat. Phys. 91, 285–305 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Karlin, I.V., Gorban, A.N., Dukek, G., Nonnenmacher, T.: Dynamic correction to moment approximations. Phys. Rev. E 57, 1668–1672 (1998)

    Article  Google Scholar 

  24. Zmievskii, V.B., Karlin, I.V., Deville, M.: The universal limit in dynamics of dilute polymeric solutions. Physica A 275(1–2), 152–177 (2000)

    Article  Google Scholar 

  25. Gorban, A.N., Karlin, I.V.: Structure and approximations of the Chapman-Enskog expansion for Grad linearized equations. Sov. Phys. JETP 73(4), 637–641 (1991)

    Google Scholar 

  26. Struchtrup, H., Torrilhon, M.: Regularization of Grad’s 13 moment equations: Derivation, linear analysis. Phys. Fluids 15, 2668–2680 (2003)

    Article  MathSciNet  Google Scholar 

  27. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge (1970)

    Google Scholar 

  28. Gorban, A.N., Karlin, I.V., Zmievskii, V.B., Dymova, S.V.: Reduced description in reaction kinetics. Physica A 275, 361–379 (2000)

    Article  Google Scholar 

  29. Gorban, A.N., Karlin, I.V.: Method of invariant manifold for chemical kinetics. Chem. Eng. Sci. 58, 4751–4768 (2003)

    Article  Google Scholar 

  30. Gorban, A.N., Karlin, I.V., Zinovyev, A.Yu.: Invariant grids for reaction kinetics. Physica A 333, 106–154 (2004)

    Article  Google Scholar 

  31. Gorban, A.N., Karlin, I.V.: Short-wave limit of hydrodynamics: A soluble example. Phys. Rev. Lett. 77, 282–285 (1996)

    Article  PubMed  Google Scholar 

  32. Karlin, I.V., Gorban, A.N.: Hydrodynamics from Grad’s equations: What can we learn from exact solutions? Ann. Phys. (Leipzig) 11(10–11), 783–833 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Karlin, I.V., Dukek, G., Nonnenmacher, T.: Invariance principle for extension of hydrodynamics: Nonlinear viscosity. Phys. Rev. E 55(2), 1573–1576 (1997)

    Article  Google Scholar 

  34. Santos, A.: Nonlinear viscosity, velocity distribution function in a simple longitudinal flow. Phys. Rev. E 62, 4747–4751 (2000)

    Article  Google Scholar 

  35. Gorban, A.N., Karlin, I.V.: General approach to constructing models of the Boltzmann equation. Physica A 206, 401–420 (1994)

    Article  Google Scholar 

  36. Andries, P., Aoki, K., Perthame, B.: A consistent BGK-type model for gas mixtures. J. Stat. Phys. 106, 993–1018 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  37. Shan, X., He, X.: Discretization of the velocity space in the solution of the Boltzmann equation. Phys. Rev. Lett. 80, 65–68 (1998)

    Article  Google Scholar 

  38. Ansumali, S., Karlin, I.V.: Entropy function approach to the lattice Boltzmann method. J. Stat. Phys. 107, 291–308 (2002)

    Article  MATH  Google Scholar 

  39. Ansumali, S., Karlin, I.V.: Single relaxation time model for entropic lattice Boltzmann methods. Phys. Rev. E 65(1–9), 056312 (2002)

    Article  MathSciNet  Google Scholar 

  40. Karlin, I.V., Gorban, A.N., Succi, S., Boffi, V.: Maximum entropy principle for lattice kinetic equations. Phys. Rev. Lett. 81, 6–9 (1998)

    Article  Google Scholar 

  41. Karlin, I.V., Ferrante, A., Öttinger, H.C.: Perfect entropy functions of the lattice Boltzmann method. Europhys. Lett. 47, 182–188 (1999)

    Article  Google Scholar 

  42. Boghosian, B.M., Yepez, J., Coveney, P.V., Wagner, A.J.: Entropic lattice Boltzmann methods. Proc. Roy. Soc. Lond. A 457, 717–766 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  43. Ansumali, S., Karlin, I.V.: Kinetic boundary condition for the lattice Boltzmann method. Phys. Rev. E 66(1–6), 026311 (2002)

    Article  MathSciNet  Google Scholar 

  44. Ansumali, S., Karlin, I.V., Öttinger, H.C.: Minimal entropic kinetic models for simulating hydrodynamics. Europhys. Lett. 63, 798–804 (2003)

    Article  Google Scholar 

  45. Cercignani, C.: Theory and Application of the Boltzmann Equation. Scottish Academic Press, Edinburgh (1975)

    MATH  Google Scholar 

  46. Ansumali, S., Chikatamarla, S.S., Frouzakis, C.M., Boulouchos, K.: Entropic lattice Boltzmann simulation of the flow past square cylinder. Int. J. Mod. Phys. C 15(3), 435–445 (2004)

    Article  MATH  Google Scholar 

  47. Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics, Beyond. Oxford University Press, Oxford (2001)

    Google Scholar 

  48. Karlin, I.V., Ansumali, S., De Angelis, E., Öttinger, H.C., Succi, S.: Entropic lattice Boltzmann method for large scale turbulence simulation. http://xxx.lanl.gov/abs/cond-mat/0306003 (2003)

  49. Ansumali, S.: Minimal kinetic modeling of hydrodynamics. PhD thesis, Swiss Federal Inst. of Tech. Zürich, 15534 (2004)

    Google Scholar 

  50. Broadwell, J.E.: Study of rarefied shear flow by the discrte velocity method. J. Fluid Mech. 19, 401–414 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  51. Ho, C.M., Tai, Y.C.: Micro-electro-mechanical-systems(MEMS), fluid flows. Annu. Rev. Fluid Mech. 30, 579–612 (1998)

    Article  Google Scholar 

  52. Sone, Y.: Kinetic Theory, Fluid Dynamics. Birkhäuser, Basel (2002)

    Google Scholar 

  53. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of gas flows. Theory and Application of the Boltzmann Equation. Clarendon Press, Oxford (1994)

    Google Scholar 

  54. Oran, E.S., Oh, C.K., Cybyk, B.Z.: Direct simulation Monte Carlo: Recent advances, applications. Annu. Rev. Fluid Mech. 30, 403–441 (1998)

    Article  MathSciNet  Google Scholar 

  55. Grmela, M., Karlin, I.V., Zmievski, V.B.: Boundary layer variational principle: A case study. Phys. Rev. E 66(1–12), 011201 (2002)

    Article  Google Scholar 

  56. Nie, X., Doolen, G., Chen, S.: Lattice-Boltzmann simulations of fluid flows in MEMS. J. Stat. Phys. 107, 279–289 (2002)

    Article  MATH  Google Scholar 

  57. Lim, C.Y., Shu, C., Niu, X.D., Chew, Y.T.: Application of lattice Boltzmann method to simulate microchannel flows. Phys. Fluid 107, 2299–2308 (2002)

    Article  Google Scholar 

  58. Succi, S.: Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis. Phys. Rev. Lett. 89, 064502 (2002)

    Article  PubMed  Google Scholar 

  59. Li, B., Kwok, D.: Discrete Boltzmann equation for microfluidics. Phys. Rev. Lett. 90, 124502 (2003)

    Article  PubMed  Google Scholar 

  60. Niu, X.D., Shu, C., Chew, Y.T.: Lattice Boltzmann BGK model for simulation of micro flows. Euro. Phys. Lett. 67, 600–606 (2004)

    Article  Google Scholar 

  61. Ansumali, S., Karlin, I.V., Frouzakis, Ch. E., Boulouchos, K.B.: Entropic lattice Boltzmann method for microflows. http://xxx.lanl.gov/abs/cond-mat/0412555 (2004)

  62. Ansumali, S., Frouzakis, Ch. E., Karlin, I.V., Kevrekidis, I.G.: Exploring hydrodynamic closures for the lid-driven micro-cavity. http://arxiv.org/abs/cond-mat/0502018 (2005)

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, LE1 7RH, Leicester, UK

    Alexander N. Gorban

  2. ETH Zürich, Institute of Energy Technology, CH-8092, Zürich, Switzerland

    Iliya V. Karlin

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  1. Alexander N. Gorban
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  2. Iliya V. Karlin
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Correspondence to Iliya V. Karlin.

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Communicated by H. Struchtrup

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Gorban, A.N., Karlin, I.V. Invariance correction to Grad’s equations: where to go beyond approximations?. Continuum Mech. Thermodyn. 17, 311–335 (2005). https://doi.org/10.1007/s00161-005-0202-z

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