Non-Newtonian phenomena and kinetic theory of gases

There are two basic limitations of the Navier-Stokes equations. One is that they are parabolic in time: some wave modes are lacking. The other one is that they are valid only if the mean free path is at least a factor 10 smaller than the macroscopic scale. This is important for high altitude flight. It is also important for flow around very small objects, an area which is becoming increasingly important at present.

General flow equations for gases can be obtained from the Boltzmann equation via the Chapman-Enskog method and the Grad method. For slow, already relaxed, flows, the Chapman-Enskog method gives more accurate results than the corresponding Grad approximation. But for the (fast) relaxation, the Grad method is more reliable than the Chapman-Enskog method.

An intermediate method is developed, in order to combine the advantages of both. A new 13 moments system of equations, which has been obtained, is now studied. The system is hyberbolic like the Grad system (so that a full set of wave modes is included), but gives the correct values for viscocity and heat conduction, in contrast with the Grad system. A new type of regularization of the Burnett equations is also studied. 

Article:
  • 13 Moment Equations Based on First Order Chapman-Enskog Solution.                                                                    Transport Theory and Statistical Physics 27, 681-90 (1998).

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    Abstract
    A modified method of 13 moments based on the first order Chapman-Enskog
    solution rather than the corresponding Hermite polynomials is proposed.
    Contrary to the Grad 13 moments equations the equations are correct to first
    order in the Knudsen number, having correct heat conductivity and viscosity
    and thus Prandtl number. The relaxation times and coupling coefficient
    between heat current and viscous pressure are calculated.

    Conference contributions:
  • Relaxing gases beyond the Navier-Stokes equations. J. Techn. Phys. 37, 567-70 (1996).
  • 13 Moments Equations Based on First-Order Chapman-Enskog Solution. Rarefied Gas Dynamics 20, 124-7. Peking University Press, Beijing (1997). 
    Cooperation with the gaskinetic research group

    • at the Department of Aeronautics and Astronautics at Kyoto University

     
  • Y. Sone, Notes on kinetic-equation approach of fluid-dynamic equations. Mekanik-KTH 2000:09. 

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    Last updated December 8, 2000.