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Experimental, numerical and mathematical approach to the complexity of the phenomena of interfaces

界面现象复杂性的实验、数值和数学方法

基本信息

批准号：
11440035
负责人：
TOMOEDA Kenji
金额：
$ 7.49万
依托单位：
Osaka Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B).
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

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 mechanics6) 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 ([1], [2] , [3] , [4] , [5]).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 ([6], [7]).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 ([8], [9]).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 ([10], [11]).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 ([12], [13], [14]).

本项目主要研究了以下现象：1）光敏Belousov-Zhabotinsky反应中螺旋波的形成和自传播高温合成中螺旋波的形成; 2）细菌细胞群的自组织菌落图案; 3）Hele-Shaw细胞实验中聚合物溶液中粘性指进和气泡运动; 4）聚合物溶液中气泡运动的研究。支持多孔介质流动中扩散与吸收相互作用引起的分裂现象。5）断裂力学中裂纹路径的预测与分析。6）晶体生长问题中的几何模型。我们得到了以下结果。在1）和2）用反应扩散方程描述了斑图的形成机理，并利用奇异极限方法和分岔理论对斑图的形成机理进行了分析（[1]，[2]，[3]，[4]，[5]）.（3）引入剪切稀化指数，推导出修正的达西定律，并对指状体速度进行了较好的预测（[6]，[7]）.在4）中，描述这种现象的数学模型被写成具有强吸收的非线性扩散方程。得到了支座开始分裂为两个不相交集的充分条件（[8]，[9]）。5）导出了光滑裂纹方向与曲率的关系式。这些公式的有效性在简单的例子中得到了证明（[10]，[11]）。6）基于不可微能量梯度系统，提出了小面晶体生长和晶界生长的模型。得到了证明和分析这些模型的数学基础，理论和数值方法显示了这些模型的解是如何演变的（[12]，[13]，[14]）。