Mathematical Sciences: Pattern Formation Relevant to Turing and Morphological Instabilities: Comparison of Theory with Experiment

数学科学：与图灵和形态不稳定性相关的模式形成：理论与实验的比较

• 批准号: 9531797
• 负责人: David Wollkind
• 金额: $ 6万
• 依托单位: Washington State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531797&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Pattern; Formation; Relevant

9531797 Wollkind The development of one- and two-dimensional Turing patterns characteristic of the chlorite-iodide-malonic acid/indicator reaction occurring in an open gel continuously fed unstirred reactor is to be investigated by means of various weakly nonlinear stability analyses applied to the appropriately scaled govern- ing chlorine dioxide-iodine-malonic acid/indicator model system. It is then proposed that the resulting theoretical predictions deduced from these pattern formation studies be compared both with the recent experimental evidence rele- vant to the diffusive instabilities under examination which consist of stripes, rhombic arrays of rectangles, and hexagonal arrays of spots or nets, and with similar equilibrium structures characteristic of a diffusion system used to model interfacial morphologies observed during alloy solidification. The ra- tionale for these comparisons is to explain more fully the transition to such stationary symmetry breaking spatial Turing structures when the malonic acid reservoir concentration varies in the first case and to examine the unifying as well as divergent aspects of the model systems in the second. Finally the nonequilibrium Turing and morphological instabilities of chemical turbulence and cellular dendrites, respectively, are to be investigated by application of a numerical bifurcation code to the amplitude equations of nonlinear stability theory in the appropriate parameter ranges for these two phenomena. The antici- pated results of this research have the potential to contribute to the under- standing of pattern formation in a wide variety of application areas. %%% The long-term goal of this project is to develop the simplest reasonable natu- ral science models which preserve the essential features of pattern formation while still being consistent with observation. The candidates for such mathe- matical modeling are those structures generated during chemical reactions in gels and during the solidif ication of two-component mixtures, respectively. Quantification of that sort would allow an experimentalist to obtain a desired outcome without wasting the time and money caused by repeated unsuccessful trials. The relevant application areas for this work range from combustion theory and other energy related interactions to materials processing involving semi-conductor production and steel manufacturing. ***

本文采用各种弱非线性稳定性分析方法，研究了在开放的凝胶连续进料无搅拌反应器中发生的氯-碘-丙二酸/指示剂反应的一维和二维图灵图特征的发展。然后，建议将从这些模式形成研究中推断出的理论预测与最近与所研究的扩散不稳定性相关的实验证据进行比较，这些不稳定性包括条纹，矩形的菱形阵列和六边形的点或网阵列，以及用于模拟合金凝固过程中观察到的界面形态的扩散系统的相似平衡结构特征。这些比较的合理性是为了更充分地解释当丙二酸储层浓度在第一种情况下发生变化时，向这种静态对称破断空间图灵结构的过渡，并在第二种情况下检查模型系统的统一和分歧方面。最后，在适当的参数范围内，对非线性稳定性理论的振幅方程应用数值分岔程序，分别研究了化学湍流和细胞树突的非平衡图灵和形态不稳定性。本研究的预期结果有可能有助于在广泛的应用领域对模式形成的理解。该项目的长期目标是建立最简单合理的自然科学模型，该模型既保留了图案形成的基本特征，又与观测结果保持一致。这种数学建模的候选对象分别是在凝胶化学反应和双组分混合物凝固过程中产生的结构。这样的量化可以让实验学家在不浪费时间和金钱的情况下获得理想的结果，而这些时间和金钱是由多次失败的试验造成的。这项工作的相关应用领域包括燃烧理论和其他与能源相关的相互作用，以及涉及半导体生产和钢铁制造的材料加工。＊＊＊