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Computing Qualitatively Correct Approximations of Balance Laws pp 269–293Cite as

Linearized BGK Model of Heat Transfer

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Part of the book series: SIMAI Springer Series ((SEMA SIMAI,volume 2))

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.

There are basically two approaches for the mathematical study of Boltzmann equation. The French School excelled in the study of weak solutions in the sense of Leray. Another approach aims at more quantitative understanding of physical phenomena. The latter is being revived in recent years.

Tai-Ping Liu, University of Singapore, 2008

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Notes

  1. 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. 2.

    k = R, the gas constant when the particle’s mass equals 1.

  3. 3.

    Up to the wrong value of Prandtl’s number of 1 against 0.7 for air.

  4. 4.

    Zero is excluded.

  5. 5.

    5 If the macroscopic mass flux vanishes, the walls are said to have “thermalized” the gas [27].

  6. 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. 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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    Laurent Gosse

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Gosse, L. (2013). Linearized BGK Model of Heat Transfer. In: Computing Qualitatively Correct Approximations of Balance Laws. SIMAI Springer Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2892-0_14

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