Stable and unstable singularities in the unforced Hele‐Shaw cell
Stable and unstable singularities in the unforced Hele‐Shaw cell
复制标题
非受迫 Hele-Shaw 晶胞中的稳定和不稳定奇点
DOI：
10.1063/1.868915
复制
发表时间：
1996
影响因子：
4.6
通讯作者：
M. Brenner
中科院分区：
文献类型：
--
作者：
R. Almgren;A. Bertozzi;M. Brenner
我们研究了非受迫 Hele-Shaw 系统润滑模型中奇点的形成，描述了限制在两个狭窄间隔玻璃板之间的两个液滴的破裂。通过改变初始数据，我们展示了四种不同的场景：（1）液滴在有限时间内破裂，两个夹点彼此靠近并在奇异时间合并； (2) 液滴在有限时间内破裂，两个不对称夹点相互远离； (3) 液滴在有限时间内破裂，具有单个对称夹点； (4) 液滴松弛至稳定的平衡形状，而没有有限时间的破裂。三个奇异场景中的每一个都具有具有不同标度法则的自相似结构；第一种情况之前在其他 Hele-Shaw 研究中尚未观察到。我们证明了第二和第三种情况的不稳定性，其中解决方案在厚度上改变其行为，该厚度可以是任意小的，具体取决于初始条件。这些交通...
We study singularity formation in the lubrication model for the unforced Hele‐Shaw system, describing the breaking in two of a fluid droplet confined between two narrowly spaced glass plates. By varying the initial data, we exhibit four different scenarios: (1) the droplet breaks in finite time, with two pinch points moving toward each other and merging at the singular time; (2) the droplet breaks in finite time, with two asymmetric pinch points propagating away from each other; (3) the droplet breaks in finite time, with a single symmetric pinch point; or (4) the droplet relaxes to a stable equilibrium shape without a finite time breakup. Each of the three singular scenarios has a self‐similar structure with different scaling laws; the first scenario has not been observed before in other Hele‐Shaw studies. We demonstrate instabilities of the second and third scenarios, in which the solution changes its behavior at a thickness that can be arbitrarily small depending on the initial condition. These transit...