Dynamics of Topological Transitions in Soap Films Spanning Deformable Contours

极小曲面的研究可以追溯到欧拉和拉格朗日的工作。'高原的问题'，即证明存在一个极小的表面跨越一个给定的轮廓，是在1930年代解决，随后的数学工作主要集中在静力学，涉及，例如，证明存在这样的表面规定的拓扑结构在更高的维度和分类的周期性极小表面。除了少数例外，很少有人关注将一个表面转换到另一个表面的过渡。在数学科学的其他领域，拓扑转变已经被广泛研究;从分裂液滴到重新连接太阳磁场线，这样的转变在自然界中比比皆是，并且通常与快速演化到新状态的奇异结构有关。在流体力学领域，长期以来一直强调粘性流动中的界面塌陷，以及流体和肥皂膜运动以及膜连接网络的更无粘性问题。令人惊讶的是，这些技术在其他情况下如此成功，但尚未被应用于理解将一个最小表面带到另一个最小表面的动力学过程，数学家柯朗在1940年的一篇优雅的文章中提出了一些关于最小面积表面的基本问题，这些表面可以用肥皂膜跨越各种形状的金属丝框架来可视化。他指出，当框架是一个双循环，它可以支持一个莫比乌斯带，一个片面的表面形式的电影。拉开和解开线圈会导致不稳定性，其中薄膜会跳到双面解决方案。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。虽然柯朗集中在这些系统中的关键静态问题，但没有考虑伴随这种转变的动力学过程;这并不奇怪，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，其工具仅在过去二十年中才开发出来。本提案中描述的研究将建立在我们最近对边界变形驱动的单面到双面肥皂膜过渡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，这种转变是与一个奇点，在该连接数之间的平台边界的电影和边界跳跃从2到零。此外，奇异性总是发生在膜边界处，并且在崩溃动力学之前，崩溃动力学显示出从粘性主导运动到惯性状态的明显交叉，并最终导致高原边界的重新连接。奇异后留下的静态最小表面表现出一个局部区域的高扭曲的重新排列的高原边界，这是一个奇异摄动现象的规模上的线半径。本研究将使用分析，数值和实验方法相结合，以更深入地了解这个和其他拓扑转换涉及最小曲面的相互转换，与分类这些过渡和他们的奇点的最终目标。使用莫比乌斯带问题作为范例，我们将开发一个合适的自由边界理论，结合空气惯性和高原边界耗散来研究崩溃动力学，特别注意高原边界在过渡点处所经历的扭曲奇异性。这些理论描述的发展将与实验研究密切联系，其中材料特性（粘度，表面张力，边界几何形状）的变化将用于提供洞察力和验证。将进一步发展基于直纹表面的数学模型，以阐明已被证明的各种拓扑连接。

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.