Experimental and theoretical work reported here on the collapse of catenoidal soap films of various viscosities reveal the existence of a robust geometric feature that appears not to have been analyzed previously; prior to the ultimate pinchoff event on the central axis, which is associated with the formation of a well-studied local double-cone structure folded back on itself, the film transiently consists of two acute-angle cones connected to the supporting rings, joined by a central quasicylindrical region. As the cylindrical region becomes unstable and pinches, the opening angle of those cones is found to be universal, independent of film viscosity. Moreover, that same opening angle at pinching is found when the transition occurs in a hemicatenoid bounded by a surface. The approach to the conical structure is found to obey classical Keller-Miksis scaling of the minimum radius as a function of time, down to very small but finite radii. While there is a large body of work on the detailed structure of the singularities associated with ultimate pinchoff events, these large-scale features have not been addressed. Here we study these geometrical aspects of film collapse by several distinct approaches, including a systematic analysis of the linear and weakly nonlinear dynamics in the neighborhood of the saddle node bifurcation leading to collapse, both within mean curvature flow and the physically realistic Euler flow associated with the incompressible dynamics of the surrounding air. These analyses are used to show how much of the geometry of collapsing catenoids is accurately captured by a few active modes triggered by boundary deformation. A separate analysis based on a mathematical sequence of shapes progressing from the critical catenoid towards the Goldschmidt solution is shown to predict accurately the cone angle at pinching. We suggest that the approach to the conical structures can be viewed as passage close to an unstable fixed point of conical similarity solutions. The overall analysis provides the basis for the systematic study of more complex problems of surface instabilities triggered by deformations of the supporting boundaries.

中文翻译：

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边界变形引起的悬链皂膜坍塌几何

此处报告的关于各种粘度的悬链皂膜坍塌的实验和理论工作揭示了以前似乎没有分析过的稳健几何特征的存在；在中心轴上的最终夹断事件之前，该事件与经过充分研究的局部双锥结构的形成有关，它自身向后折叠，薄膜暂时由连接到支撑环的两个锐角锥体组成，通过中央准圆柱区。随着圆柱形区域变得不稳定和收缩，发现这些锥体的张角是通用的，与薄膜粘度无关。此外，当过渡发生在以表面为界的半连体中时，发现了相同的收缩张角。发现锥形结构的方法遵循最小半径作为时间函数的经典 Keller-Miksis 缩放，直到非常小但有限的半径。虽然有大量关于与最终夹断事件相关的奇点的详细结构的工作，但尚未解决这些大规模特征。在这里，我们通过几种不同的方法研究了薄膜坍塌的这些几何方面，包括对导致坍塌的鞍节点分岔附近的线性和弱非线性动力学的系统分析，包括平均曲率流和与周围空气的不可压缩动力学。这些分析用于显示由边界变形触发的一些活动模式准确捕获了多少塌陷悬链线的几何形状。显示了基于从临界悬链线到 Goldschmidt 解的形状数学序列的单独分析，以准确预测收缩时的锥角。我们建议可以将锥形结构的方法视为接近锥形相似解的不稳定固定点的通道。整体分析为系统研究由支撑边界变形引发的更复杂的表面不稳定性问题提供了基础。显示了基于从临界悬链线到 Goldschmidt 解的形状数学序列的单独分析，以准确预测收缩时的锥角。我们建议可以将锥形结构的方法视为接近锥形相似解的不稳定固定点的通道。整体分析为系统研究由支撑边界变形引发的更复杂的表面不稳定性问题提供了基础。显示了基于从临界悬链线到 Goldschmidt 解的形状数学序列的单独分析，以准确预测收缩时的锥角。我们建议可以将锥形结构的方法视为接近锥形相似解的不稳定固定点的通道。整体分析为系统研究由支撑边界变形引发的更复杂的表面不稳定性问题提供了基础。