Dynamics of Topological Transitions in Soap Films Spanning Deformable Contours

对极小曲面的研究可以追溯到欧拉和拉格朗日的工作。20世纪30年代，证明跨越给定轮廓的极小曲面的存在的问题--高原问题--得到了解决，随后的数学工作主要集中在静力学上，例如，涉及高维指定拓扑的这种曲面的存在的证明，以及周期极小曲面的分类。除了少数例外，很少有人注意到从一个表面到另一个表面的转变。在数学科学的其他领域，人们已经对拓扑转变进行了广泛的研究；从分裂液滴到重新连接太阳磁力线，这样的转变在自然界中随处可见，而且经常与迅速演化到新状态的奇异结构联系在一起。在流体力学领域，长期以来一直强调粘性流动中的界面坍塌，以及流体和肥皂膜运动以及膜连接网络等更无粘性的问题。令人惊讶的是，这些在其他情况下如此成功的技术，还没有被应用于理解将一个极小曲面连接到另一个极小曲面的动力学过程。在1940年的一篇优雅的文章中，数学家库兰特列出了一些关于最小面积曲面的基本问题，这些问题可以用跨越各种形状的线框的肥皂膜来可视化。他指出，当框架是一个双环时，它可以支撑莫比乌斯带形式的薄膜，即单面表面。拆开并解开环路会导致胶片不稳定，从而使胶片跳跃到双面解决方案。这构成了已知的最简单的拓扑转换，它将单侧曲面转换为双侧曲面。虽然Courant专注于这些系统中的关键静态问题，但没有考虑到伴随这种转变的动态过程；这并不令人惊讶，因为它们构成了自由边界动力学领域中非常现代的一类问题，而工具只是在过去20年才开发出来的。这项研究将建立在我们最近关于由边界变形驱动的单面肥皂膜向双面肥皂膜转变的理论和实验研究的基础上，其中发现了丰富而复杂的动力学。我们发现，这种转变与一个奇点有关，在该奇点处，薄膜的平台边界和边界之间的连接数从2跃升到0。此外，这种奇异性总是发生在薄膜边界上，在此之前发生了崩塌动力学，表现出明显的从粘性为主的运动到惯性区域的交叉，并最终导致高原边界的重新连接。奇点后的静态极小表面在重新排列的Platform边界上出现了一个局域化的高扭度区域，这是丝半径尺度上的奇异摄动现象。这项研究将使用分析、数值和实验相结合的方法来更深入地理解这种和其他涉及极小曲面相互转换的拓扑转变，最终目的是对这些转变及其奇点进行分类。以莫比乌斯条形问题为范例，我们将发展一种结合空气惯性和高原边界耗散的合适的自由边界理论来研究崩塌动力学，特别关注高原边界在转折点经历的扭转奇异性。这些理论描述的开发将与实验研究密切联系，其中将使用材料性质(粘度、表面张力、边界几何)的变化来提供洞察和验证。基于直纹面的数学模型将进一步发展，以阐明已经猜测的各种拓扑联系。

The study of minimal surfaces dates back to the work of Euler and Lagrange. 'Plateau's problem,' that of proving the existence of a minimal surface spanning a given contour, was solved in the 1930s, and subsequent mathematical work has focused chiefly on statics, involving, for example, proofs of the existence of such surfaces of prescribed topology in higher dimensions and classification of periodic minimal surfaces. With few exceptions, little attention has been paid to transitions that take one surface to another. Elsewhere in the mathematical sciences topological transitions have been studied extensively; from splitting fluid drops to reconnecting solar magnetic field lines, such transitions abound in nature, and are often associated with singular structures that evolve rapidly to a new state. In the field of Fluid Mechanics there has been a longstanding emphasis on interface collapse in viscous flows, and on the more inviscid problems of fluid and soap film motion and networks of film junctions. Surprisingly, these techniques, so successful in other contexts, have yet to be applied to understand the dynamical processes that take one minimal surface to another.In an elegant article in 1940, the mathematician Courant laid out a number of fundamental questions about minimal-area surfaces that could be visualized with soap films spanning wire frames of various shapes. He noted that when the frame is a double loop it can support a film in the form of a Mobius strip, a one-sided surface. Pulling apart and untwisting the loops leads to an instability where the film jumps to a two-sided solution. This constitutes the simplest known topological transition which converts a one-sided surface to a two-sided one. While Courant concentrated on the key static issues in these systems, the dynamical processes that accompany such transitions were not considered; this is not surprising because they constitute a very modern class of problems in the field of free-boundary dynamics, the tools for which have been developed only over the last two decades. The research described in this proposal will build upon our recent theoretical and experimental studies of the transition from one- to two-sided soap films driven by boundary deformation, in which a rich and complex dynamics was discovered. We found that this transition is associated with a singularity at which the linking number between the Plateau border of the film and the boundary jumps from two to zero. Moreover, that singularity occurs always at the film boundary and is preceded by collapse dynamics that displays an apparent crossover from viscous-dominated motion to an inertial regime, and ultimately leads to reconnection of the Plateau Border. The static minimal surface left after the singularity exhibits a localized region of high twist of the rearranged Plateau border which is a singular perturbation phenomenon on the scale of the wire radius. This research will use a combination of analytical, numerical, and experimental methods to understand more deeply this and other topological transitions involving interconversion of minimal surfaces, with the ultimate goal of classifying these transitions and their singularities. Using the Mobius strip problem as a paradigm, we will develop a suitable free-boundary theory that incorporates air inertia and Plateau border dissipation to study the collapse dynamics, with particular attention to the twist singularity experienced by the Plateau border at the transition point. Development of these theoretical descriptions will be done in close contact with experimental studies in which variations in material properties (viscosity, surface tension, boundary geometry) will be used to provide insight and verification. Mathematical models based on ruled surfaces will be further developed to elucidate various topological connections that have been conjectured.