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A REVIEW OF ONE-PHASE HELE-SHAW FLOWS AND A LEVEL-SET METHOD FOR NONSTANDARD CONFIGURATIONS

Part of: Miscellaneous topics - Partial differential equations Partial differential equations, boundary value problems Incompressible viscous fluids Geometric function theory

Published online by Cambridge University Press:  23 September 2021

LIAM C. MORROW Open the ORCID record for LIAM C. MORROW [Opens in a new window] ,
TIMOTHY J. MORONEY Open the ORCID record for TIMOTHY J. MORONEY [Opens in a new window] ,
MICHAEL C. DALLASTON Open the ORCID record for MICHAEL C. DALLASTON [Opens in a new window]  and
SCOTT W. MCCUE Open the ORCID record for SCOTT W. MCCUE [Opens in a new window]
LIAM C. MORROW
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK; e-mail: liam.morrow@eng.ox.ac.uk School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia; e-mail: t.moroney@qut.edu.au, michael.dallaston@qut.edu.au
TIMOTHY J. MORONEY
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia; e-mail: t.moroney@qut.edu.au, michael.dallaston@qut.edu.au
MICHAEL C. DALLASTON
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia; e-mail: t.moroney@qut.edu.au, michael.dallaston@qut.edu.au
SCOTT W. MCCUE*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4001, Australia; e-mail: t.moroney@qut.edu.au, michael.dallaston@qut.edu.au
*
e-mail: scott.mccue@qut.edu.au
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Abstract

The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.

Keywords

Hele-Shaw flowSaffman–Taylor instabilityviscous fingering patternsmoving boundary problemconformal mappinglevel-set method

MSC classification

Primary: 76D27: Other free-boundary flows; Hele-Shaw flows
Secondary: 35R37: Moving boundary problems 65N06: Finite difference methods 30C35: General theory of conformal mappings
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 3 , July 2021 , pp. 269 - 307
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Copyright
© Australian Mathematical Society 2021

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Footnotes

*

This is a contribution to the series of invited papers by past Tuck medallists (Editorial, Issue 62(1)). Scott W. McCue was awarded the 2019 Tuck medal.

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