Modelling finger propagation in elasto-rigid channels

TL;DR

This paper presents a theoretical model of finger propagation in elastic Hele-Shaw channels, capturing complex interface behaviors and instabilities, with validation against experimental data.

Contribution

It introduces a depth-averaged model for two-phase fluid-structure interaction in elastic channels, revealing diverse propagation modes and instabilities, and aligns well with experimental observations.

Findings

Model captures steady and unsteady finger propagation modes.

Increased initial collapse leads to instabilities like tip-splitting.

Good quantitative agreement with experimental data when including liquid film effects.

Abstract

We conduct a theoretical study of a two-phase-fluid-structure interaction problem in which air is driven at constant volume flux into a liquid-filled Hele-Shaw channel whose upper boundary is an elastic sheet. A depth-averaged model in the frame of reference of the advancing air-liquid interface is used to investigate the steady and unsteady interface propagation modes via numerical simulation. In slightly collapsed channels, the steadily-propagating interface adopts a shape that is similar to the classic Saffman--Taylor finger in rigid Hele-Shaw cells. As the level of initial collapse increases the induced gradients in channel depth alter the morphology of the propagating finger and promote a variety of instabilities from tip-splitting to small-scale fingering on the curved interface, in qualitative agreement with experiments. The model has a complex solution structure with a wide range of stable and unstable, steady and time-periodic modes, many of which have similar driving pressures. We find good quantitative agreement between our model and the experimental data of Duclou\'{e} et al. (J. Fluid Mech. vol. 819, 2017, p 121) for the finger width, sheet profile and finger pressure, provided that corrections to account for the presence of liquid films on the upper and lower walls of the channel are included in the model.