Mathematical analysis of fluid flow on evolving surfaces

• Project/Area Number: 15K17580
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Osaka University (2016-2018); Waseda University (2015)
• Principal Investigator: Koba Hajime 大阪大学, 基礎工学研究科, 助教 (80707729)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 数理モデリング / 流れ / 流体 / 時間発展する曲面 / エネルギー / 圧縮性流体 / 非圧縮性流体 / 非ニュートン流体 / 時間発展曲面 / エネルギー変分法 / 拡散方程式 / ニュートン流体 / 多様体 / 曲率 / 支配方程式の導出

Outline of Final Research Achievements

We are interested in the dominant equations for the motion of fluid on an evolving surface. An evolving surface means that the surface is moving or the shape of the surface is changing along with the time. Applying our energetic variational approaches, we make several mathematical models for fluid flow on evolving surfaces. Moreover, we investigate the conservative forms and conservation laws of our fluid systems on the evolving surface.

Academic Significance and Societal Importance of the Research Achievements

本研究により、時間発展する曲面上における流体の流れ、熱の伝わりや濃度拡散などを支配する方程式が数理的手法によって得られ、地球表面上の雲の流れや空中を浮遊するシャボン玉内部の表面流の流れが理論的に取り扱えるようになった。また、ケッチャプやマヨネーズなどの高粘性流体(非ニュートン流体)の数理モデリングも行い、従来構築されていた幾つかの非ニュートン流体や曲面上の流体の流れに関するモデル方程式に数理的妥当性を与えることに成功した。

Research Products

• [Journal Article] Energetic variational approaches for non-Newtonian fluid systems (2018)
  • Author(s): Koba Hajime、Sato Kazuki
  • Journal Title: Zeitschrift fur angewandte Mathematik und Physik Volume: 69 Issue: 6 Pages: 1-28
  • DOI: 10.1007/s00033-018-1039-1
  • Peer Reviewed
• [Journal Article] On derivation of compressible fluid systems on an evolving surface (2018)
  • Author(s): Koba Hajime
  • Journal Title: Quarterly of Applied Mathematics Volume: 76 Issue: 2 Pages: 303-359
  • DOI: 10.1090/qam/1491
  • Peer Reviewed
• [Journal Article] Energetic variational approaches for incompressible fluid systems on an evolving surface (2017)
  • Author(s): Hajime Koba, Chun Liu, and Yoshikazu Giga
  • Journal Title: Quarterly of Applied Mathematics Volume: 75 Issue: 2 Pages: 359-389
  • DOI: 10.1090/qam/1452
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] On compressible fluid system on an evolving surface with boundaries (2018)
  • Author(s): Hajime Koba
  • Organizer: Conference on Mathematical Fluid Dynamics, Evangelische Akademie Bad Boll
  • Int'l Joint Research / Invited
• [Presentation] Double bubble 内の濃度拡散の数理モデリング (2018)
  • Author(s): 古場一
  • Organizer: 日本数学会2018年度秋季総合分科会岡山大学
• [Presentation] 境界を持つ時間発展する曲面上における熱の伝わりや濃度拡散の数理モデリング (2018)
  • Author(s): 古場一
  • Organizer: 日本数学会2018年度秋季総合分科会岡山大学
• [Presentation] 境界を持つ時間発展する曲面上における圧縮性流体の流れの数理モデリング (2018)
  • Author(s): 古場一
  • Organizer: 日本数学会2018年度秋季総合分科会岡山大学
• [Presentation] 境界を持つ時間発展する曲面上における圧縮性流体の流れの数理モデリング (2018)
  • Author(s): 古場一
  • Organizer: 日本流体力学会年会2018大阪大学
• [Presentation] On the Helmholtz-Weyl decomposition on surfaces and its application to multiphase flow (2018)
  • Author(s): Hajime Koba
  • Organizer: The 15th Japanese-German International Workshop on Mathematical Fluid Dynamics
  • Int'l Joint Research / Invited
• [Presentation] 時間発展する曲面上における圧縮性流体方程式の保存形や保存則に関して (2017)
  • Author(s): 古場一
  • Organizer: 日本数学会 2017年度会
  • Place of Presentation: 首都大学東京(東京都・八王子)
  • Year and Date: 2017-03-26
• [Presentation] Energetic variational approaches for PDEs (2017)
  • Author(s): Hajime Koba
  • Organizer: HeKKSaGOn Working Group Seeds in Mathematics versus Needs outside Mathematics Winter School in Osaka 2017
  • Place of Presentation: Osaka Univeristy(大阪府・豊中市)
  • Year and Date: 2017-03-06
  • Invited
• [Presentation] On the Helmholtz-Weyl decomposition on surfaces and its application to fluid dynamics (2017)
  • Author(s): Hajime Koba
  • Organizer: 2017 Fall Program on Analysis of PDE
  • Int'l Joint Research / Invited
• [Presentation] On derivation of compressible fluid systems on an evolving surface (2017)
  • Author(s): Hajime Koba
  • Organizer: The sixth Japan-China Workshop on Mathematical Topics from Fluid Mechanics
  • Int'l Joint Research / Invited
• [Presentation] On fluid flow on an evolving surface (2016)
  • Author(s): Hajime Koba
  • Organizer: 2016 NCTS Workshop on Complex and Biological Fluid Dynamics with Applications
  • Place of Presentation: National Taiwan University [Taiwan (Taipei)]
  • Year and Date: 2016-12-20
  • Invited
• [Presentation] On derivation of incompressible fluid systems with heat equation (2016)
  • Author(s): Hajime Koba
  • Organizer: RIMS Workshop on Mathematical Analysis of Viscous Incompressible Fluid
  • Place of Presentation: RIMS Kyoto University(京都府・京都市)
  • Year and Date: 2016-11-16
  • Invited
• [Presentation] 時間発展する曲面上における非圧縮性流体方程式の導出 (2016)
  • Author(s): 古場一
  • Organizer: 日本流体力学会年会2016
  • Place of Presentation: 名古屋工業大学(愛知県・名古屋市)
  • Year and Date: 2016-09-26
• [Presentation] 時間発展する曲面上における圧縮性流体方程式の導出 (2016)
  • Author(s): 古場 一
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-18
• [Presentation] 時間発展する曲面上における熱の影響を考慮した非圧縮性流体方程式の導出 (2016)
  • Author(s): 古場 一
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-18
• [Presentation] Introduction to mathematical modeling of soap bubble in air in the sun (2016)
  • Author(s): 古場 一
  • Organizer: Ｇｅｏｍｅｔｒｉｃ ｆｌｏｗｓ ａｎｄ ｒｅｌａｔｅｄ problemｓ
  • Place of Presentation: 東京工業大学
  • Year and Date: 2016-03-03
  • Invited
• [Presentation] 動く曲面上における流体の流れに関して (2015)
  • Author(s): 古場 一
  • Organizer: 部分多様体幾何とリー群作用2015
  • Place of Presentation: 東京理科大学
  • Year and Date: 2015-09-07
  • Invited
• [Presentation] 動く曲面上における非圧縮性流体の流れに関して (2015)
  • Author(s): 古場 一
  • Organizer: 第62 回幾何学シンポジウム
  • Place of Presentation: 東京理科大学
  • Year and Date: 2015-08-30
  • Invited
• [Presentation] Introduction to mathematical modeling of soap bubble and smoke in air (2015)
  • Author(s): Hajime Koba
  • Organizer: The 40th Sapporo Symposium on Partial Differential Equations
  • Place of Presentation: 北海道大学
  • Year and Date: 2015-08-20
  • Int'l Joint Research / Invited