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Erschienen in:
Introduction and Chronological Perspective Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

14. Linearized BGK Model of Heat Transfer

verfasst von : Laurent Gosse

Erschienen in: Computing Qualitatively Correct Approximations of Balance Laws

Verlag: Springer Milan

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Abstract

In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear flow effects follow a scalar integro -differential equation whereas the heat transfer is described by a 2 × 2 coupled system.

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Vorheriges Kapitel A Model for Scattering of Forward-Peaked Beams
Nächstes Kapitel Balances in Two Dimensions: Kinetic Semiconductor Equations Again
Fußnoten
1
The notation M(v) refers to an “uniform Maxwellian” which doesn’t depend on the variables t, x. On the other hand, a “local Maxwellian” would be denoted by M (t, x, v).
 
2
k = R, the gas constant when the particle’s mass equals 1.
 
3
Up to the wrong value of Prandtl’s number of 1 against 0.7 for air.
 
4
Zero is excluded.
 
5
5 If the macroscopic mass flux vanishes, the walls are said to have “thermalized” the gas [27].
 
6
The normalization factor is \( \frac{\alpha_1{\alpha}_2\left({\delta}_1-{\delta}_2\right)}{\sqrt{\pi}\left({\alpha}_1+{\alpha}_2-{\alpha}_1{\alpha}_2\right)} \), see [6].
 
7
Actually, since themacroscopic flux term is of the order of 0.3 ≃ 10Δ x, there is only little difference between these results obtained by means of the modified matrices and the ones one can obtain with the matrices written in §3.​3. However, discrepancies may worsen as Δ x is decreased.
 
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Metadaten
Titel
Linearized BGK Model of Heat Transfer
verfasst von
Laurent Gosse
Copyright-Jahr
2013
Verlag
Springer Milan
Buch
Computing Qualitatively Correct Approximations of Balance Laws

Print ISBN: 978-88-470-2891-3

Electronic ISBN: 978-88-470-2892-0

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-88-470-2892-0_14

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