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Numerical and asymptotic analyses of the Boltzmann equation-Their engineering and mathematical aspects-

Research Project

Project/Area Number 10045040
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Engineering fundamentals
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

AOKI Kazuo  Kyoto University, Graduate School of Engineering, Professor, 工学研究科, 教授 (10115777)

Co-Investigator(Kenkyū-buntansha) ASANO Kiyoshi  Kyoto University, Graduate School of Human and Environmental Studies, Professor, 人間環境学研究科, 教授 (90026774)
TAKATA Shigeru  Kyoto University, Graduate School of Engineering, Associate Professor, 工学研究科, 助教授 (60271011)
SONE Yoshio  Kyoto University, Graduate School of Engineering, Professor Emeritus, 工学研究科, 名誉教授 (80025923)
GOLSE Franco  パリ第7大学, 数学科, 教授
BARDOS Claud  パリ第7大学, 数学科, 教授
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥4,400,000 (Direct Cost: ¥4,400,000)
Fiscal Year 2000: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1999: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1998: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsBoltzmann equation / Euler equation / boundary-layer equation / Liouville equation / Knudsen layer / rarefied gas flows / flow bifurcation / flow stability / 多成分混合気体 / 速度分布関数 / 輸送方程式 / 希薄化効果
Research Abstract

1. With the help of the French group (FG), the Japanese group (JG) has established the asymptotic theory (the fluid-dynamic equations, the slip boundary conditions, and the Knudsen-layer corrections near the boundary) describing the general behavior of slightly rarefied gas flows with finite Mach numbers by means of a systematic asymptotic analysis of the Boltzmann equation. This theory also contains the systematic derivation of the viscous boundary-layer equation in fluid dynamics from the Boltzmann equation. The results inspired the mathematical study of FG on the boundary-layer equation.
2. JG and FG have jointly worked on the direct derivation of the fluid-dynamic equations (the Euler system of equations) from the many-particle system (the Liouville equation). Using different approaches, each group has succeeded in the formal derivation. In addition, JG reflected on the formal derivation of the Boltzmann equation from the many-particle system and provided FG with a clear and systema … More tic derivation. This leads to a possibility of a new mathematical proof of the derivation of the Boltzmann equation.
3. Cooperating with FG, JG investigated the rarefied gas flow induced by the discontinuity in the boundary temperature by means of a numerical analysis and clarified the behavior of the flow for a wide range of gas rarefaction. In this flow, the discontinuity of the molecular velocity distribution function propagates from the boundary into the gas. As the first step of mathematical study, on the basis of a linear transport equation that possesses a similar property to the Boltzmann equation concerning the propagation of discontinuity but has a much simpler structure, JG and FG jointly analyzed, with mathematical rigor, the propagation of discontinuity in the medium.
4. JG constructed a systematic theory for the kinetic solution scheme of the general conservation equations including fluid-dynamic equations (e.g., the standard Euler and Navier-Stokes equations) and provided FG with the theory for further mathematical study. In addition, JG, with the help of FG, succeeded in obtaining mathematically rigorous bounds of the boundary conditions for the fluid-dynamic equations on the interface where evaporation or condensation is taking place.
5. JG provided FG with the system of fluid-dynamic type equations that describe the ghost effect for gaseous mixtures, and FG investigated its mathematical properties. Less

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] Kazuo Aoki: "Numerical analysis of the Taylor-Couette problem for a rarefied gas by the direct simulation Monte Carlo method"Rarefied Gas Dynamics (Cepadues,Toulouse,France edited by R.Brun et.al.). Vol.2. 109-116 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kazuo Aoki: "Derivation of hydrodynamic limits from either the Liouville equation or kinetic models : Study of an example"京都大学数理解析研究所講究録. 1146. 154-181 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Yoshio Sone: "Asymptotic theory of the Boltzmann System, for a steady flow of a slightly rarefied gas with a finite Math number : General theory"European Journal of Mechanics,B/Fluids. 19. 325-360 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Yoshio Sone: "Notes on the boundary conditions for fluid-dynamic equations on the interface of a gas and its condensed phase"physics of Fluids. 13. 324-334 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shigeru Takata: "The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases : Asymptotic analysis of the Boltzmann equation"Transport Theory and Statistical Physics. (印刷中). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kazuo Aoki: "The behavior of a vapor-gas mixture in the continuum limit! Asymptotic analysis based on the Boltzmann equation"Rarefied Gas Dynamics (AIP,Melville,edited by T.J.Bartel et.al.). (印刷中). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Aoki: "Numerical analysis of the Taylor-Couette problem for a rarefied gas by the direct simulation Monte Carlo method"Rarefied Gas Dynamica (Cepadues, Toulouse, 1999). Vol.2. 109-116

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Aoki: "Derivation of hydrodynamic limits from either the Liouville equation or kinetic models : Study of an example"RIMS Kokyuroku. 1146. 154-181 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Sone: "Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number : General theory"Eur.J.Mech, B/Fluids. 19. 325-360 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Sone: "Notes on the boundary conditions for fluid-dynamic equations on the interface of a gas and its condensed phase"Phys.Fluids. 13. 324-334 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Takata: "The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases : Asymptotic analysis of the Boltzmann equation"Transp.Theory Stat.Phys.. 30 (in print). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Aoki: "The behavior of a vapor-gas mixture in the continuum limit : Asymptotic analysis based on the Boltzmann equation"Rarefied Gas Dynamics (AIP, Melville, 2001) (in print).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kazuo Aoki: "Derivation of hydrodynamic limits from either the Liouville equation or Kinetic models : Study of an example"京都大学数理解析研究所講究録. 1146. 154-181 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kazuo Aoki: "Dynamics of rarefied gas flows; Asymptotic and numerical analyses of the Boltzmann equation"AIAA Paper. 2001・0874. 1-15 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Yoshio Sone: "Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number : General theory"European Journal of Mechanics, B/Fluids. 19(3). 325-360 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Yoshio Sone: "Notes on the boundary conditions for fluid-dynamic equations on the interface of a gas and its condensed phase"Physics of Fluids. 13(1). 324-334 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Shingo Kosuge: " Shock-wave structure for a binary gas mixture : finite-difference analysis of the Boltzmann equation for hard-sphere molecules"European Journal of Mechanics, B/Fluids. 20(1). 87-126 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Shigeru Takata: "The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases : Asymptotic analysis of the Boltzmann equation"Transport Theory and Statistical Physics. (印刷中). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kazuo Aoki: "Numerical analysis of the Taylor-Couette problem for a rarefied gas by the direct simulation Monte CaHo method"Rarefied Gas Dynamics (edited by R. Brun. Et al. Cepadues,Toulouse, France.). Vol.2. 109-116 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Shigeru Takata: "The ghost effect in the continuum limit for a vapor-gas mixture around condensed phases: Asymptotic analysis of the Boltzmann equation"Transport Theory and Statistical Physics. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Yoshio Sone: "Asymptotic theory of the Boltzmann system for a steady flow of a Slightly rarefied gas with a finite Mach number: General theory"European Journal of Mechanics, B/Fluids. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Kazuo Aoki: "Derivation of hydrodynamic limits from either the Liouville equation or kinetic models: Study of an example"京都大学数理解析研究所講究録. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Kazuo Aoki: "Numerical analysis of the Taylor-Couette problem fora rarefied gas by the direct simulation Monte Carle method" Rarefied Gas Dynamics(Cepadues,Toulouse,France). (印刷中). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Shigeru Takata: "Behavior of a vapor-gas mixture between two parallel plane condensed phases in the continuum limit" Rarefied Gas Dynamics(Cepadues,Toulouse,France). (印刷中). (1999)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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