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Boundary layers for the nonlinear discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas
Karlstad University, Faculty of Technology and Science, Department of Mathematics. (Kinetisk teori)ORCID iD: 0000-0003-1232-3272
2012 (English)In: Proceedings of 28th International Symposium on Rarefied Gas Dynamics 2012 / [ed] Michel Mareschal, AndrĂ©s Santos, Melville, New York: American Institute of Physics (AIP), 2012, 1, p. 223-230Conference paper, Published paper (Refereed)
Abstract [en]

Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behaviorof the solutions of boundary value problems of the Boltzmann equation for small Knudsen numbers. Half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. Here we consider a half-space problem of condensation for apure vapor in the presence of a non-condensable gas by using discrete velocity models (DVMs) of the Boltzmann equation. The Boltzmann equation can be approximated by DVMs up to any order, and these DVMs can be applied for numerical methods,but also for mathematical studies to bring deeper understanding and new ideas. For one-dimensional half-space problems,the discrete Boltzmann equation (the general DVM) reduces to a system of ODEs. We obtain that the number of parametersto be specified in the boundary conditions depends on whether the condensing vapor flow is subsonic or supersonic. Thisbehavior has earlier been found numerically. We want to stress that our results are valid for any finite number of velocities.This is an extension of known results for single-component gases (and for binary mixtures of two vapors) to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian, with a flow velocity towards thecondensed phase, at infinity, while the non-condensable gas tends to zero at infinity. Steady condensation of the vapor takes place at the condensed phase, which is held at a constant temperature. We assume that the vapor is completely absorbed, that the non-condensable gas is diffusively reflected at the condensed phase, and that vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution at the condensed phase, needed for the existence of a unique solution of the problem are investigated, assuming that the given distribution at the condensed phase is sufficiently close to the Maxwellian at infinity and that the total mass of the non-condensable gas is sufficiently small. Exact solutions and solvability conditions are found for a specific simplified discrete velocity model (with few velocities).

Place, publisher, year, edition, pages
Melville, New York: American Institute of Physics (AIP), 2012, 1. p. 223-230
Series
AIP Conference Proceedings, ISSN 0094-243X ; 1501
Keywords [en]
Boltzmann equation, boundary layers, discrete velocity models, half-space problem, non-condensable gas
Mathematics
Mathematics
Identifiers
ISI: 000312411200028ISBN: 978-0-7354-1115-9 (print)OAI: oai:DiVA.org:kau-16034DiVA, id: diva2:573563
Conference
28th International Symposium on Rarefied Gas Dynamics 2012, July 9 - 13, Zaragoza
Available from: 2012-12-02 Created: 2012-12-02 Last updated: 2018-12-17Bibliographically approved

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Bernhoff, Niclas

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Cite
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