This work is motivated by the recent experiments of two reacting fluids in a Hele-Shaw cell [Phys. Rev. E 76 (1) (2007) 016202] and associated linear stability analysis of a curvature weakening model [SIAM J. Appl. Math 72 (3) (2012) 842-856]. Unlike the classical Hele-Shaw problem posed for moving interfaces with surface tension, the curvature weakening model is concerned with a newly-produced gel-like phase that stiffens the interface, thus the interface is modeled as an elastic membrane with curvature dependent rigidity that reflects geometrically induced breaking of intermolecular bonds. Here we are interested in exploring the long-time interface dynamics in the nonlinear regime. We perform simulations using a spectrally accurate boundary integral method, together with a rescaling scheme to dramatically speed up the intrinsically slow evolution of the interface. We find curvature weakening inhibits tip-splitting and promotes side-branching morphology. At long times, numerical results reveal that there exist nonlinear, stable, self-similarly evolving morphologies.

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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.

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