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]).

在这个项目中，我们关注以下现象。1）在自我传播高温合成中的光敏性Belousov-Zhabotinsky反应和螺旋波中螺旋波的模式形成。2）细菌细胞种群自组织的菌落模式。33）由Hele-Shaw细胞的螺旋细胞构成的螺旋式分布和泡沫的实验在Ply trimement Intriontion中的分布序列。 5）裂缝力学中的裂纹路径的预测和分析。6）晶体生长问题中的几何模型。我们获得了以下结果。在1）和2）2）模式形成的机理由反应 - 扩散方程式描述，并使用单数限制和bifurcation的理论（3] [1] [1]，[1] [1] [5]）。在3）3）修改后的达西定律是通过考虑剪切粉的索引来得出的，并很好地预测了手指速度（[6]，[7]）。在4）中描述了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。支撑开始分为两个不相交组的足够条件（[[8]，[9]）。在5）中，一些公式连接了平滑裂纹的方向和曲率。这些公式的有效性在简单的示例中显示（[10]，[11]）。在6）基于具有非不同能量的梯度系统中，提出了刻面晶体生长和晶界的模型。获得了合理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决方案如何发展（[12]，[13]，[14]）。