Effect of abrupt change of the wall temperature in the kinetic theory

Research output: Contribution to journalArticle

Abstract

We consider a semi-infinite expanse of a rarefied gas bounded by an infinite plane wall. The temperature of the wall is T0, and the gas is initially in equilibrium with density ρ0 and temperature T0. The temperature of the wall is suddenly changed to Tw at time t = 0 and is kept at Tw afterward. We study the quantitative short time behavior of the gas in response to the abrupt change of the wall temperature on the basis of the linearized Boltzmann equation. Our approach is based on a straightforward calculation of the exact formulas derived by Duhamel's integral. Our method allows us to establish the pointwise estimates of the microscopic distribution and the macroscopic variables in short time. We show that the short-time solution consists of the free molecular ow and its perturbation, which exhibits logarithmic singularities along the characteristic line and on the boundary.

Original languageEnglish
Pages (from-to)765-789
Number of pages25
JournalKinetic and Related Models
Volume12
Issue number4
DOIs
Publication statusPublished - 2019 Jan 1

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Kinetic theory
Kinetic Theory
Gases
Rarefied Gas
Temperature
Pointwise Estimates
Boltzmann equation
Boltzmann Equation
Logarithmic
Singularity
Perturbation
Line
Gas

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modelling and Simulation

Cite this

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title = "Effect of abrupt change of the wall temperature in the kinetic theory",
abstract = "We consider a semi-infinite expanse of a rarefied gas bounded by an infinite plane wall. The temperature of the wall is T0, and the gas is initially in equilibrium with density ρ0 and temperature T0. The temperature of the wall is suddenly changed to Tw at time t = 0 and is kept at Tw afterward. We study the quantitative short time behavior of the gas in response to the abrupt change of the wall temperature on the basis of the linearized Boltzmann equation. Our approach is based on a straightforward calculation of the exact formulas derived by Duhamel's integral. Our method allows us to establish the pointwise estimates of the microscopic distribution and the macroscopic variables in short time. We show that the short-time solution consists of the free molecular ow and its perturbation, which exhibits logarithmic singularities along the characteristic line and on the boundary.",
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Effect of abrupt change of the wall temperature in the kinetic theory. / Kuo, Hung-Wen.

In: Kinetic and Related Models, Vol. 12, No. 4, 01.01.2019, p. 765-789.

Research output: Contribution to journalArticle

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AB - We consider a semi-infinite expanse of a rarefied gas bounded by an infinite plane wall. The temperature of the wall is T0, and the gas is initially in equilibrium with density ρ0 and temperature T0. The temperature of the wall is suddenly changed to Tw at time t = 0 and is kept at Tw afterward. We study the quantitative short time behavior of the gas in response to the abrupt change of the wall temperature on the basis of the linearized Boltzmann equation. Our approach is based on a straightforward calculation of the exact formulas derived by Duhamel's integral. Our method allows us to establish the pointwise estimates of the microscopic distribution and the macroscopic variables in short time. We show that the short-time solution consists of the free molecular ow and its perturbation, which exhibits logarithmic singularities along the characteristic line and on the boundary.

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