Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell

TL;DR

This paper investigates the nonlinear dynamics of shrinking interfaces in a Hele-Shaw cell with a time-increasing gap, revealing shape-invariant patterns and a morphology diagram through advanced numerical methods and stability analysis.

Contribution

It introduces a spectrally accurate boundary integral method with adaptive rescaling to explore nonlinear interface behavior and pattern formation in Hele-Shaw flows with variable gap growth.

Findings

Quantitative agreement with experimental observations for constant gap increase.

Discovery of shape-invariant shrinking patterns for specific gap growth functions.

Development of a morphology diagram linking dominant mode and control parameters.

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: $\displaystyle b(t)=\left(1-\frac{7}{2}\tau \mathcal{C} t\right)^{-{2}/{7}}$, 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 behavior 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, pp. 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 one 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 $\mathcal{C}$.