Thin filaments in Hele-Shaw cells

TL;DR

This paper analyzes the stability of fluid filaments in Hele-Shaw cells using a filament model, identifying critical radii for growth and stability, and describing nonlinear growth behaviors.

Contribution

It introduces a stability analysis of fluid filaments in Hele-Shaw cells, including the concept of a 'pinned circle' solution and its finite-time blow-up behavior.

Findings

Thin filaments grow if initial radius exceeds a critical value.

All modes are stable below twice the critical radius.

Large radii lead to unstable modes and nonlinear growth.

Abstract

Using a recently derived filament model, the stability of fluid filaments in Hele-Shaw cells, driven by a constant pressure gradient, is studied. It is found that thin circular filaments grow if their initial radius exceeds a dimensionless critical radius. Further, linear stability of the axisymmetric solution reveals that all modes are stable for twice this critical radius, with modes becoming unstable for larger radii. A translating circular solution is found for asymptotically large radius, termed a `pinned circle'. These are thought to describe the observed non-linear growth of filament into circular-like solutions. The solutions exhibit a finite-time blow up in the pinned circle radius, attributed to the circle shedding mass as it grows. This report presents the results of a project undertaken by the first author at the Matrix Workshop Instabilities in Porous Media, April 3-23, 2024, under the supervision of the other authors.