Institut Jean Le Rond ∂'Alembert, Sorbonne Université, Centre National de la Recherche Scientifique, UMR 7190, F-75005 Paris, France.
LadHyX, CNRS, Ecole Polytechnique, UMR 7646, 91128 Palaiseau, France.
Proc Natl Acad Sci U S A. 2018 Dec 4;115(49):12371-12376. doi: 10.1073/pnas.1814242115. Epub 2018 Nov 19.
Many differential equations involved in natural sciences show singular behaviors; i.e., quantities in the model diverge as the solution goes to zero. Nonetheless, the evolution of the singularity can be captured with self-similar solutions, several of which may exist for a given system. How to characterize the transition from one self-similar regime to another remains an open question. By studying the classic example of the pinch-off of a viscous liquid thread, we show experimentally that the geometry of the system and external perturbations play an essential role in the transition from a symmetric to an asymmetric solution. Moreover, this transient regime undergoes unexpected log-scale oscillations that delay dramatically the onset of the final self-similar solution. This result sheds light on the strong impact external constraints can have on predictions established to explain the formation of satellite droplets or on the rheological tests applied on a fluid, for example.
许多自然科学领域中的微分方程表现出奇异行为；即，模型中的量随着解趋近于零而发散。然而，奇点的演化可以用自相似解来捕捉，对于给定的系统可能存在几个自相似解。如何描述从一个自相似区域到另一个自相似区域的转变仍然是一个悬而未决的问题。通过研究粘性液体线的挤压这一经典范例，我们实验表明，系统的几何形状和外部扰动在外延的过渡中扮演了重要的角色，从而从对称解到非对称解。此外，这种瞬态过程会经历意想不到的对数尺度振荡，从而显著延迟最终自相似解的开始。这一结果表明，外部约束可以对解释卫星液滴形成或应用于流体的流变测试等方面的预测产生强烈的影响。
