Self-similar and disordered front propagation in a radial Hele-Shaw channel with time-varying cell depth.

The displacement of a viscous fluid by an air bubble in the narrow gap between two parallel plates can readily drive complex interfacial pattern formation known as viscous fingering. We focus on a modified system suggested recently by Zheng et al. [Phys. Rev. Lett. 115, 174501 (2015)], in which the onset of the fingering instability is delayed by introducing a time-dependent (power-law) plate separation. We perform a complete linear stability analysis of a depth-averaged theoretical model to show that the plate separation delays the onset of nonaxisymmetric instabilities, in qualitative agreement with the predictions obtained from a simplified analysis by Zheng et al. We then employ direct numerical simulations to show that in the parameter regime where the axisymmetrically expanding air bubble is unstable to nonaxisymmetric perturbations, the interface can evolve in a self-similar fashion such that the interface shape at a given time is simply a rescaled version of the shape at an earlier time. These novel, self-similar solutions are linearly stable but they only develop if the initially circular interface is subjected to unimodal perturbations. Conversely, the application of nonunimodal perturbations (e.g., via the superposition of multiple linearly unstable modes) leads to the development of complex, constantly evolving finger patterns similar to those that are typically observed in constant-width Hele-Shaw cells.