ITR: Free-Boundary Problems in Precipitative Growth

0219411 Goldstein这个项目解决了生长现象带来的数学和计算挑战，涉及流体流动、沉淀/结晶和蒸发中新的和困难的自由边界问题。它们出现在不同的环境中，如海底的热液喷口、陆地洞穴和电化学模式的形成，在这些环境中，人们发现通过降水形成了复杂的结构。从中空的苏打稻草钟乳石到奇特的弯曲螺旋石和石灰岩洞穴中起伏的片状窗帘，再到几十米高的海底硫化铁质烟囱。尽管进行了多年的实证研究，但这些引人入胜的增长形态在很大程度上无法用任何定量的方式来解释。我们将开发研究沉淀生长现象的计算方法，主要基于表面演化研究中的边界积分和微分几何方法。其中许多结构形成的巨大地质时间尺度创造了很少的实验结果，也几乎没有创造出实验室模拟结果。我们还将开发和探索能够在可访问的时间尺度上显示这种模式形成的实验模型系统。我们实验室发现的一个这样的系统，涉及通过电沉积进行管状生长，在这种系统中，宏观的管状沉积物在数十分钟内围绕从阴极释放出的氢气泡生长。理论方法将包括研究在大Peclet数下的基本电动力学和由于洛伦兹力引起的耗尽层修改，所有这些都是作为一种手段来理解表面生长的几何规律，这些几何规律可以产生在自然界中看到的丰富和迷人的形貌，以及研究这种规律的计算技术。

0219411GoldsteinThis project addresses mathematical and computational challenges posed by growth phenomena involving novel and difficult free-boundary problems in fluid flow, precipitation/crystallization and evaporation. These appear in contexts as diverse as hydrothermal vents in the ocean floor, terrestrial caves, and electrochemical pattern formation, where one finds growth of complex structures by precipitation. These range from hollow soda-straw stalactites to fantastic curved helictites and undulating sheet-like draperies in limestone caves, to iron-sulfide chimneys tens of meters high on the ocean floor. Despite years of empirical study, these fascinating growth morphologies are largely unexplained in any quantitative manner. We will develop computational methods to study precipitative growth phenomena, based primarily on boundary-integral and differential geometric methods in the study of surface evolution. The enormous geological time scales on which many of these structures form has created a paucity of experimental results, and few if any laboratory analogs have been created. We will also develop and explore experimentally model systems that can display such pattern formation on accessible time scales. One such system, discovered in our laboratory, involves tubular growth by electrodeposition, in which macroscopic tubular deposits grow in tens of minutes around bubbles of hydrogen gas evolving off a cathode. Theoretical approaches will include the study of the underlying electrokinetics at large Peclet number and depletion layer modifications due to Lorentz forces, all as a means of understanding the geometrical laws of surface growth that can produce the rich and fascinating morphologies seen in nature, and the computational techniques to study such laws.