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

9531797沃尔金德研究了在开式凝胶连续进料非搅拌反应器中发生的亚氯酸盐-碘-丙二酸/指示剂反应的一维和二维图灵图的发展特征，并将各种弱非线性稳定性分析应用于适当规模的二氧化氯-碘-丙二酸/指示剂模型体系。然后建议将从这些图案形成研究中得到的理论预测与最近的实验证据进行比较，这些证据与所研究的扩散不稳定性有关，这些扩散不稳定性由条纹、菱形矩形阵列和六角形点阵或网点组成，并且具有与用于模拟合金凝固过程中观察到的界面形态的扩散系统相似的平衡结构特征。这些比较的目的是为了更全面地解释在第一种情况下丙二酸库浓度变化时向这种静止对称破缺空间图灵结构的转变，并在第二种情况下检验模型体系的统一和发散方面。最后，在适当的参数范围内，对化学湍流和胞状树枝晶的非平衡图灵不稳定性和胞状树枝晶的形态不稳定性分别进行了研究。这项研究的反不利结果有可能有助于理解在广泛的应用领域中的图案形成。这个项目的长期目标是开发出最简单、合理的自然科学模型，既能保留格局形成的基本特征，又能与观测结果保持一致。这种数学模型的候选结构分别是在凝胶中发生化学反应和在双组分混合物凝固过程中产生的结构。这种量化将允许实验者获得预期的结果，而不会浪费反复失败的试验所造成的时间和金钱。这项工作的相关应用领域从燃烧理论和其他与能源有关的相互作用，到涉及半导体生产和钢铁制造的材料加工。***

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. ***