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The displacement of a viscous fluid by an air bubble in the narrow gapbetween two parallel plates can readily drive complex interfacial patternformation known as viscous fingering. We focus on a modified system suggestedrecently by [1], in which the onset of the fingering instability is delayed byintroducing a time-dependent (power-law) plate separation. We perform acomplete linear stability analysis of a depth-averaged theoretical model toshow that the plate separation delays the onset of non-axisymmetricinstabilities, in qualitative agreement with the predictions obtained from asimplified analysis by [1]. We then employ direct numerical simulations to showthat in the parameter regime where the axisymmetrically expanding air bubble isunstable to nonaxisymmetric perturbations, the interface can evolve in aself-similar fashion such that the interface shape at a given time is simply arescaled version of the shape at an earlier time. These novel, self-similarsolutions are linearly stable but they only develop if the initially circularinterface is subjected to unimodal perturbations. Conversely, the applicationof non-unimodal perturbations (e.g. via the superposition of multiple linearlyunstable modes) leads to the development of complex, constantly evolving fingerpatterns similar to those that are typically observed in constant-widthHele-Shaw cells.  [1] Z. Zheng, H. Kim, and H. A. Stone, Controlling viscous fingering usingtime-dependent strategies, Phys. Rev. Lett. 115, 174501 (2015).

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