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

小行星9531797 本文用各种方法研究了在开放式凝胶连续进料无搅拌反应器中亚氯酸盐-碘化物-丙二酸/指示剂反应的一维和二维图灵模式特征。 弱非线性稳定性分析适用于适当规模的治理二氧化氯-碘-丙二酸/指示剂模型系统。然后 建议将从这些图案形成研究中推导出的理论预测结果与最近的实验证据进行比较，这些实验证据与受检查的扩散不稳定性有关，包括条纹，矩形的菱形阵列和点或网的六边形阵列，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库浓度变化时，在第一种情况下向这种稳态对称性破缺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发散方面。最后 非平衡图灵和形态学不稳定性的化学湍流和细胞枝晶，分别将研究应用 在适当的参数范围内，对这两种现象的非线性稳定性理论的振幅方程进行了数值分岔计算。本研究的预期结果有可能有助于理解各种应用领域中的图案形成。 %%% 该项目的长期目标是开发最简单合理的自然科学模型，这些模型保留了图案形成的基本特征，同时仍然与观察一致。这种数学的候选人- 数学建模是在化学反应过程中产生的那些结构， 凝胶和双组分混合物的固化过程中，分别。 这样的量化将允许实验者获得所需的 结果没有浪费时间和金钱造成的反复不成功 审判这项工作的相关应用领域从燃烧 理论和其他与材料加工有关的能量相互作用， 半导体生产和钢铁制造。 ***

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. ***