Computation of a Shrinking Interface in a Hele-Shaw Cell

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, where τ is the surface tension and 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 C.

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

“…Indeed, in Zhao et al. (2018), it was shown that linear theory underpredicts the amplitudes of growing modes in the lifting plate problem.…”

“…(1997) and Zhao et al. (2018). We consider the interface to be a slightly perturbed circle, where , the perturbation mode is an integer, is the polar angle and is the amplitude of the perturbation.…”

“…To overcome these numerical issues, we have developed a rescaled boundary integral scheme (Zhao et al. 2018); the method is briefly described here. The idea is to map the original time and space into new coordinates such that the interface can evolve at an arbitrary speed in the new rescaled frame (see also Li et al.…”

“…Further details of the numerical method, including convergence studies, can be found in Zhao et al. (2018).…”

“…Here, we use a recently developed spectrally accurate boundary integral method with space–time rescaling (Zhao et al. 2018) that enables the accurate simulation of nonlinear interface dynamics in the fast shrinking regime for the first time. In particular, the interface is mapped back to its initial size, and time is rescaled such that the speed of the interface in the new frame is prescribed and is slower than the dynamics in the original frame.…”