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Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation. (English) Zbl 1097.76061
Summary: We present in detail a theoretical framework for representing hydrodynamic systems through a systematic discretization of the Boltzmann kinetic equation. The work is an extension of a previously proposed formulation [X. Shan and X. He, Phys. Rev. Lett. 80, 65 ff (1998)]. Conventional lattice Boltzmann models can be shown to be directly derivable from this systematic approach. Furthermore, we provide here a clear and rigorous procedure for obtaining higher-order approximations to the continuum Boltzmann equation. The resulting macroscopic moment equations at each level of the systematic discretization give rise to the Navier-Stokes hydrodynamics and those beyond. In addition, theoretical indications to the order of accuracy requirements are given for each discrete approximation, for thermohydrodynamic systems, and for fluid systems involving long-range interactions. All these are important for complex and micro-scale flows and are missing in the conventional Navier-Stokes order descriptions. The resulting discrete Boltzmann models are based on a kinetic representation of the fluid dynamics, hence the drawbacks in conventional higher-order hydrodynamic formulations can be avoided.

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76A02 Foundations of fluid mechanics
82D15 Statistical mechanics of liquids
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