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

Morrow, Liam C.; Cock, Nicolas De; McCue, Scott W.

doi:10.1103/physrevfluids.8.014001

Cited by 3 publications

(5 citation statements)

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“…Our linear stability analysis here for a finite viscous fluid evolving in a Hele-Shaw channel is analogous to that presented for a radial geometry with two interfaces (Morrow et al 2023). Generalisations and alterations to include different viscous fluids on either side of the interfaces have been conducted for both linear and weakly nonlinear frameworks (Gin & Daripa 2015;Anjos & Li 2020).…”

Section: Linear Stabilitymentioning

confidence: 99%

“…To validate solutions of the thin-filament model (3.19) it is valuable to compare it with solutions of the full two-interface system (2.1), which is a more challenging numerical problem. We solve (2.1) using the numerical framework proposed by Morrow et al (2021Morrow et al ( , 2023, which we briefly summarise here. The framework is based on the level-set method, in which we represent each interface, f L and f U , as the zero level set of the associated level-set functions ψ L and ψ U .…”

Section: Validation Against Solution Of the Two-interface Problemmentioning

confidence: 99%

“…We solve (2.1) using the numerical framework proposed by Morrow et al. (2021, 2023), which we briefly summarise here. The framework is based on the level-set method, in which we represent each interface, and , as the zero level set of the associated level-set functions and .…”

Section: Numerical Simulation and Comparisonmentioning

confidence: 99%

“…The two-interface Hele-Shaw model in the limit that the interfaces are close is of great importance as, even if the fluid region is not initially thin, the effect of the Saffman-Taylor instability will result in a thin fluid region or filament developing (see, for example, the experimental results in Ward & White (2011), Morrow, De Cock & McCue (2023) and Cardoso & Woods (1995), although the latter study focuses on a much less viscous thin annular region in a more viscous ambient fluid). This formation of a thin filament precedes the fluid 'bursting', at which point the two inviscid regions meet and the pressure rapidly equalises.…”

Section: Introductionmentioning

confidence: 99%

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Two-interface and thin-filament approximation in Hele-Shaw channel flow

Dallaston,

Jackson,

Morrow

et al. 2024

J. Fluid Mech.

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When a viscous fluid partially fills a Hele-Shaw channel, and is pushed by a pressure difference, the fluid interface is unstable due to the Saffman–Taylor instability. We consider the evolution of a fluid region of finite extent, bounded between two interfaces, in the limit that the interfaces are close, that is, when the fluid region is a thin liquid filament separating two gases of different pressure. In this limit, we derive a second-order ‘thin-filament’ model that describes the normal velocity of the filament centreline, and evolution of the filament thickness, as functions of the thickness, centreline curvature and their derivatives. We show that the second-order terms in this model, that include the effect of transverse flow along the filament, are necessary to regularise the instability. Numerical simulation of the thin-filament model is shown to be in accordance with level-set computations of the complete two-interface model. Solutions ultimately evolve to form a bubble of rapidly increasing radius and decreasing thickness.

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Section: Linear Stabilitymentioning

confidence: 99%

Section: Validation Against Solution Of the Two-interface Problemmentioning

confidence: 99%

Section: Numerical Simulation and Comparisonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Two-interface and thin-filament approximation in Hele-Shaw channel flow

Dallaston,

Jackson,

Morrow

et al. 2024

J. Fluid Mech.

Self Cite

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“…Our study also provides an unusually thorough comparison between simulations and experiments of viscous fingering. Previous studies have made quantitative comparisons between experimental results and simulations over a relatively small portion of the parameter space [17,18,[41][42][43]. The present study is the most extensive comparison of its type, featuring quantitative examination of the fingering patterns, the time-dependent injection rate, gas pressure, and the onset and development of the instability.…”

Section: Introductionmentioning

confidence: 99%

Compression-driven viscous fingering in a radial Hele-Shaw cell

Cuttle¹,

Morrow²,

MacMinn³

2023

Preprint

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The displacement of a viscous liquid by a gas within a Hele-Shaw cell is a classical problem.The gas-liquid interface is hydrodynamically unstable, forming striking finger-like patterns that have been the subject of intense focus. Generally, both the gas and liquid phases are taken to be incompressible, with the capillary number being the key parameter that determines the severity of the instability. Here, we consider a radially outward displacement driven by the steady compression of a gas reservoir. The associated gas injection rate is then unsteady due to the compressibility of the gas. We identify a second nondimensional parameter, the compressibility number, that plays a strong role in the development of the fingering pattern. We use an axisymmetric model to study the impact of compressibility number on the unsteady evolution of injection rate and gas pressure. We use linear stability analysis to show that increasing the compressibility number delays the onset of finger development relative to the corresponding incompressible case. Finally, we present and compare a series of experiments and fully nonlinear simulations over a broad range of capillary and compressibility numbers. These results show that increasing the compressibility number systematically decreases the severity of the fingering pattern at high capillary number. Our results provide an unprecedented comparison of experiments with simulations for viscous fingering, a comprehensive understanding of the role of compressibility in unstable gas-liquid displacement flows, and insight into a new mechanism for controlling the development of fingering patterns.

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