Grant-in-Aid for Scientific Research (B).
|Allocation Type||Single-year Grants |
General mathematics (including Probability theory/Statistical mathematics)
|Research Institution||Osaka Institute of Technology |
TOMOEDA Kenji Osaka Institute of Technology, Faculty of Engineering, Professor, 工学部, 教授 (60033916)
IKEDA Tsutomu Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (50151296)
MIYOSHI Tetsuhiko Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (60040101)
MIMURA Masayasu Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50068128)
KAWAGUCHI Masami Mie University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (30093123)
GIGA Yoshikazu Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70144110)
河原田 秀夫 千葉大学, 工学部, 教授 (90010793)
|Project Period (FY)
1999 – 2000
Completed (Fiscal Year 2000)
|Budget Amount *help
¥11,700,000 (Direct Cost: ¥11,700,000)
Fiscal Year 2000: ¥4,300,000 (Direct Cost: ¥4,300,000)
Fiscal Year 1999: ¥7,400,000 (Direct Cost: ¥7,400,000)
|Keywords||Belousov-Zhabotinsky reaction / reaction-diffusion / Hele-Shaw cell / viscous fingering / support splitting phenomena / fracture mechanics / crystal growth / crystal line algorithm / ベルーソフ・ザボチンスキー反応 / ヘリカル進行波 / 樹枝状形状 / 浸透領域の変化 / 渦の相互作用 / 亀裂伸展 / ギンツブルグ・ランダウ方程式|
In this project we are concerned with the following phenomena.
1) Pattern formation of spiral waves in the photosensitive Belousov-Zhabotinsky reaction and helical waves in self-propagating high-temperature syntheses.
2) Self-organized colony patterns by a bacterial cell population.
3) Hele-Shaw cell experiments of viscous fingering and bubble motion in polymeric solutions.
4) Support splitting phenomena caused by the interaction between diffusion and absorption in porous medium flow.
5) Prediction and analysis for the crack path in fracture mechanics
6) Geometric models in crystal growth problems.
We have obtained the following results.
In 1) and 2) the mechanism of the pattern formation is described by the reaction-diffusion equations and are analyzed by using the method of the singular limit and the theory of bifurcation (,  ,  ,  , ).
In 3) The modified Darcy's law is derived by taking account of the index of shear-thinning, and gives a good prediction of the finger velocity (, ).
In 4) the mathematical models which describe such phenomena are written as the nonlinear diffusion equation with strong absorption. The sufficient conditions under which the support begins to split into two disjoint sets are obtained (, ).
In 5) some formulas connecting the direction and the curvature of the smooth cracks are derived. The validity of these formulas are shown in simple examples (, ).
In 6) Models of faceted crystal growth and of grain boundaries are proposed based on the gradient system with nondifferentiable energy. The mathematical basis for justifying and analyzing these models is obtained, and theoretical and numerical approaches show how the solutions of these models evolve (, , ).