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Numerical simulations and universal saturation profiles for viscous fingering patterns in Hele-Shaw flow

Írio M. Coutinho1,*,†, Liam C. Morrow2, and Scott W. McCue3,‡

  • *Contact author: iriomenezes@esp.puc-rio.br
  • †Present address: Departamento de EngenhariaMecânica, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro 22451-900, Brazil.
  • ‡Contact author: scott.mccue@qut.edu.au
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Phys. Rev. Fluids 11, 084003 – Published 13 August, 2026

DOI: https://doi.org/10.1103/gd5k-txp3

Abstract

Hele-Shaw flows with an interface are known to give rise to complex pattern formation, whereby the Saffman-Taylor instability triggers a viscous fingering process accompanied by tip splitting and branching. The most popular of these experiments involves a radial configuration with a less viscous fluid injected into a more viscous fluid. In an attempt to characterize the resulting complexity in such an experiment, T. H. Beeson-Jones et al. [Sci. Rep. 9, 7780 (2019)] have proposed a type of simple empirical model that aims to predict the saturation profile of the invading fingers as a function of a radial coordinate. We revisit the proposed saturation model and test its validity over a broad parameter range using fully nonlinear numerical simulations computed with a level-set method. We find that the saturation model is very effective at predicting some near-universal properties of the viscous fingering patterns for one-phase flows, where the invading fluid is neglected, with a sufficiently small surface tension parameter. For larger values of this parameter and for two-phase flows, there are discrepancies between the model and our observations. We explain these differences by studying the morphology of the advancing fingers, including pinching at the base and the rate of tip splitting. Overall, our study shows that the Beeson-Jones–Woods saturation model serves as a valid description of diffusion-limited aggregation–like patterns but is not universal over two-phase flows, where surface tension and viscosity ratio substantially alter finger morphology and the resulting saturation profile.

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Phys. Rev. Fluids 11, 084003– Published 13 August, 2026
  • Received 24 March 2026
  • Accepted 24 July 2026
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