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Gas-surface interaction and boundary conditions for the Boltzmann equation. (English) Zbl 1304.82062
Summary: In this paper we revisit the derivation of boundary conditions for the Boltzmann Equation. The interaction between the wall atoms and the gas molecules within a thin surface layer is described by a kinetic equation introduced in [V. D. Borman, S.L. Krylov and A. V. Prosianov, “The theory of nonequilibrium phenomena at the gas-solid interface”, Zh. Ehksper. Teor. Fiz. 94, 271–289 (1988)] and used in [K. Aoki et al., Kinet. Relat. Models 4, No. 1, 53–85 (2011; Zbl 1218.82029)]. This equation includes a Vlasov term and a linear molecule-phonon collision term and is coupled with the Boltzmann equation describing the evolution of the gas in the bulk flow. Boundary conditions are formally derived from this model by using classical tools of kinetic theory such as scaling and systematic asymptotic expansion. In a first step this method is applied to the simplified case of a flat wall. Then it is extented to walls with nanoscale roughness allowing to obtain more complex scattering patterns related to the morphology of the wall. It is proved that the obtained scattering kernels satisfy the classical imposed properties of non-negativeness, normalization and reciprocity introduced by C. Cercignani [The Boltzmann equation and its applications. New York etc.: Springer-Verlag (1988; Zbl 0646.76001)].

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
82D05 Statistical mechanical studies of gases
74A25 Molecular, statistical, and kinetic theories in solid mechanics
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