Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell
Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell
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Hele-Shaw池中收缩界面的非线性极限动力学
DOI：
10.1017/jfm.2020.983
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发表时间：
2020-05
影响因子：
3.7
通讯作者：
Meng Zhao;Zahra Niroobakhsh;J. Lowengrub;Shuwang Li
中科院分区：
文献类型：
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作者：
Meng Zhao;Zahra Niroobakhsh;J. Lowengrub;Shuwang Li
Hele-Shaw盒中间隙随时间增加的流动构成了一个独特的收缩界面问题。当电池的上板以规定的速度垂直提升时，外部粘性较小的流体渗透到内部粘性较大的流体中，这通过Saffman-Taylor不稳定性产生复杂的、随时间变化的界面图案。图案形成过程敏感地取决于提升速度，并且仍然没有完全理解。对于某些提升速度，如线性或指数速度，不稳定性是短暂的，界面最终收缩为圆形。然而，线性稳定性分析表明，如果差距$B（t）$增加得更快，则存在形状不变的收缩模式：$B（t）=\left（1-（{7}/{2}）\tau \mathcal {C} t\right）^{-{2}/{7}}$，其中$\tau$是表面张力，$\mathcal {C}$是界面扰动模式$k$的函数。在这里，我们使用一个频谱精确的边界积分方法与一个有效的时间自适应重新缩放计划，这是第一次，使人们有可能探索的非线性限制动力学行为的消失接口。当差距以恒定速率增加时，我们的数值计算结果与实验观察结果定量一致（Nase等人，Phys. Fluids，第23卷，2011，123101）。当我们使用形状不变的间隙$B（t）$时，我们的非线性结果揭示了存在k$倍占主导地位的一维网状网络，其中分形维数在后期降低到几乎为1。我们的结论是通过构建一个形态图的模式选择，涉及的主导模式$k$的消失接口和控制参数$\mathcal {C}$。
Abstract The flow in a Hele-Shaw cell with a time-increasing gap poses a unique shrinking interface problem. When the upper plate of the cell is lifted perpendicularly at a prescribed speed, the exterior less viscous fluid penetrates the interior more viscous fluid, which generates complex, time-dependent interfacial patterns through the Saffman–Taylor instability. The pattern formation process sensitively depends on the lifting speed and is still not fully understood. For some lifting speeds, such as linear or exponential speed, the instability is transient and the interface eventually shrinks as a circle. However, linear stability analysis suggests there exist shape invariant shrinking patterns if the gap $b(t)$ is increased more rapidly: $b(t)=\left (1-({7}/{2})\tau \mathcal {C} t\right )^{-{2}/{7}}$, where $\tau$ is the surface tension and $\mathcal {C}$ is a function of the interface perturbation mode $k$. Here, we use a spectrally accurate boundary integral method together with an efficient time adaptive rescaling scheme, which for the first time makes it possible to explore the nonlinear limiting dynamical behaviour of a vanishing interface. When the gap is increased at a constant rate, our numerical results quantitatively agree with experimental observations (Nase et al., Phys. Fluids, vol. 23, 2011, 123101). When we use the shape invariant gap $b(t)$, our nonlinear results reveal the existence of $k$-fold dominant, one-dimensional, web-like networks, where the fractal dimension is reduced to almost unity at late times. We conclude by constructing a morphology diagram for pattern selection that relates the dominant mode $k$ of the vanishing interface and the control parameter $\mathcal {C}$.