AI 中文总结
本综述探讨软物质系统中的有限时间奇点，分析其自相似动力学、小尺度截断机制及复杂流体等的影响，揭示连续介质理论的失效边界与工程应用关联。
AI 中文摘要
当液丝断流时，其颈部会变薄并分离为两个互不相连的区域。利用连续介质力学，我们可以预测颈部在有限时间内达到零半径，同时其曲率会无限增长。消失的颈部与发散的曲率共同构成了有限时间奇点。然而，真实流体并不会达到这些数学极限，因为一旦颈部变得足够小，分子或材料物理会接管主导作用。在软物质中，只要在 vanishing length scales（消失的长度尺度）下使用平滑的连续介质描述，就会出现类似的奇点。本综述探讨收缩区域遗忘了什么、保留了什么，以及哪种材料长度、时间或应力会切断表观发散。奇点附近的动力学通常具有自相似性，当用收缩的局部长度进行缩放时，不同时间的轮廓会坍缩为单一形状。有时这种坍缩具有足够的普适性，使得周围的几何结构和驱动力不再决定局部动力学。尽管如此，测量得到的输出仍可能取决于周围流动对收缩区域的供给方式，以及最终替代理想发散的小尺度物理。复杂流体和活性物质会将自身的时间尺度带入收缩区域，从而改变这种局部平衡。除界面外，当局部化对象是应力集中、几何缺陷或序缺陷而非运动表面时，同样的逻辑适用。奇点的重要性在于，它们揭示了连续介质理论何时不再是相关描述，以及小尺度截断如何决定印刷、涂层、气溶胶和可拉伸固体中关键的输出结果。
英文摘要
When a liquid thread pinches off, its neck thins as it separates into two unconnected regions. Using continuum mechanics, we can predict that the neck reaches zero radius in finite time while its curvature grows without bound. Together, the vanishing neck and diverging curvature form a finite-time singularity. However, a real fluid does not realise these mathematical limits as molecular or material physics takes over once the neck becomes sufficiently small. Similar singularities arise throughout soft matter whenever a smooth continuum description is used at vanishing length scales. This review asks what the shrinking region forgets, what it retains, and which material length, time, or stress cuts off the apparent divergence. The dynamics near a singularity often become self-similar, with profiles at different times collapsing onto one shape when rescaled by the shrinking local length. Sometimes that collapse is universal enough that the surrounding geometry and forcing no longer determine the local dynamics. Nonetheless, the measured output could still depend on how the shrinking region is fed by the surrounding flow and on the small-scale physics that finally replaces the ideal divergence. Complex fluids and active matter change the same local balance by bringing their own timescales into the shrinking region. Beyond interfaces, the same logic applies when the localised object is a stress concentration or a defect in geometry or order rather than a moving surface. Singularities matter because they show where continuum theory stops being the relevant description and how the small-scale cutoff sets the outputs that count in printing, coating, aerosols, and stretchable solids.
