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

Liam C. Morrow1, Nicolas De Cock2, and Scott W. McCue3,*

  • 1Department of Engineering Science, University of Oxford, Oxford OX13PJ, United Kingdom
  • 2TERRA Research and Teaching Centre, Gembloux Agro Bio-Tech, University of Liege, 5030 Gembloux, Belgium
  • 3School of Mathematical Sciences, Queensland University of Technology, Brisbane QLD 4001, Australia

  • *scott.mccue@qut.edu.au
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Phys. Rev. Fluids 8, 014001 – Published 10 January, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.014001

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 interface between the fluid and air. Here we consider a more realistic model, which assumes the viscous fluid is finite, covering a doubly connected two-dimensional region bounded by two such interfaces. For the case in which the flow is driven by a prescribed 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 interfaces can be unstable and develop fingering patterns, depending on the rotation rate and the volume of fluid present.

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Phys. Rev. Fluids 8, 014001– Published 10 January, 2023
  • Received 24 June 2022
  • Accepted 23 November 2022
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