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Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell

Published online by Cambridge University Press:  18 January 2021

Meng Zhao Open the ORCID record for Meng Zhao [Opens in a new window] ,
Zahra Niroobakhsh ,
John Lowengrub Open the ORCID record for John Lowengrub [Opens in a new window]  and
Shuwang Li
Meng Zhao
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA92617, USA
Zahra Niroobakhsh
Affiliation:
Department of Civil and Mechanical Engineering, University of Missouri-Kansas City, Kansas City, MO64110, USA
John Lowengrub*
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA92617, USA Department of Biomedical Engineering, Center for Complex Biological Systems University of California at Irvine, Irvine, CA92617, USA
Shuwang Li*
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL60616, USA
*
†Email addresses for correspondence: lowengrb@math.uci.edu, sli@math.iit.edu
†Email addresses for correspondence: lowengrb@math.uci.edu, sli@math.iit.edu
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Abstract

The flow in a Hele-Shaw cell with a time-increasing gap poses a unique shrinking interface problem. When the upper plate of the cell is lifted perpendicularly at a prescribed speed, the exterior less viscous fluid penetrates the interior more viscous fluid, which generates complex, time-dependent interfacial patterns through the Saffman–Taylor instability. The pattern formation process sensitively depends on the lifting speed and is still not fully understood. For some lifting speeds, such as linear or exponential speed, the instability is transient and the interface eventually shrinks as a circle. However, linear stability analysis suggests there exist shape invariant shrinking patterns if the gap $b(t)$ is increased more rapidly: $b(t)=\left (1-({7}/{2})\tau \mathcal {C} t\right )^{-{2}/{7}}$, where $\tau$ is the surface tension and $\mathcal {C}$ is a function of the interface perturbation mode $k$. Here, we use a spectrally accurate boundary integral method together with an efficient time adaptive rescaling scheme, which for the first time makes it possible to explore the nonlinear limiting dynamical behaviour of a vanishing interface. When the gap is increased at a constant rate, our numerical results quantitatively agree with experimental observations (Nase et al., Phys. Fluids, vol. 23, 2011, 123101). When we use the shape invariant gap $b(t)$, our nonlinear results reveal the existence of $k$-fold dominant, one-dimensional, web-like networks, where the fractal dimension is reduced to almost unity at late times. We conclude by constructing a morphology diagram for pattern selection that relates the dominant mode $k$ of the vanishing interface and the control parameter $\mathcal {C}$.

JFM classification

Interfacial Flows (free surface): Fingering instability Nonlinear Dynamical Systems: Pattern formation Low-Reynolds-number flows: Boundary integral methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A41
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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