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

## TL;DR

This study investigates how time-varying cell depth in a Hele-Shaw cell affects viscous fingering, revealing delayed instability onset and the emergence of stable, self-similar interface patterns under specific perturbations.

## Contribution

It provides a comprehensive linear stability analysis and numerical simulations showing the effects of dynamic cell depth on pattern formation and stability in viscous fingering.

## Key findings

- Plate separation delays instability onset.
- Self-similar interface evolution occurs under unimodal perturbations.
- Complex finger patterns develop with non-unimodal perturbations.

## Abstract

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 [1], 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 non-axisymmetric instabilities, in qualitative agreement with the predictions obtained from a simplified analysis by [1]. 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 non-unimodal 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. [1] Z. Zheng, H. Kim, and H. A. Stone, Controlling viscous fingering using time-dependent strategies, Phys. Rev. Lett. 115, 174501 (2015).