Viscous fingering patterns for Hele--Shaw flow in a doubly connected geometry driven by a pressure differential or rotation

TL;DR

This paper models viscous fingering in a finite, doubly connected Hele-Shaw cell driven by pressure differences or rotation, using numerical simulations that align well with experimental observations.

Contribution

It introduces a numerical level set method for simulating viscous fingering in finite, doubly connected Hele-Shaw geometries, extending traditional models to more realistic scenarios.

Findings

Numerical scheme reproduces experimental finger patterns.

Fingering instability depends on pressure and rotation rate.

Both interfaces can become unstable under certain conditions.

Abstract

Traditional mathematical models of Hele--Shaw flow consider the injection (or withdrawal) of an air bubble into (or from) an infinite body of viscous fluid. The most commonly studied feature of such a model is how the Saffman-Taylor instability drives viscous fingering patterns at the fluid/air interface. Here we consider a more realistic model, which assumes the viscous fluid is finite, covering a doubly connected two-dimensional region bounded by two fluid/air interfaces. For the case in which the flow is driven by a pressure difference across the two interfaces, we explore this model numerically, highlighting the development of viscous fingering patterns on the interface with the higher pressure. Our numerical scheme is based on the level set method, where each interface is represented as a separate level set function. We show that the scheme is able to reproduce the characteristic finger patterns observed experimentally up to the point at which one of the interfaces bursts through the other. The simulations are shown to compare well with experimental results. Further, we consider a model for the problem in which an annular body of fluid is evolving in a rotating Hele--Shaw cell. In this case, our simulations explore how either one or both of the interfaces can be unstable and develop fingering patterns, depending on the rotation rate and the volume of fluid present.