A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele–Shaw cell

Li, Shuwang; Lowengrub, John; Leo, Perry H.

doi:10.1016/j.jcp.2006.12.023

Cited by 73 publications

(67 citation statements)

References 42 publications

(82 reference statements)

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“…In the absence of regularisations such as surface tension, the interface generically develops cusps (curvature singularities) in finite time [9,1], while surface tension suppresses these singularities, allowing the solution to continue for all time; the interface develops long fingers that leave behind 'fjords' of viscous fluid [12]. Surface tension in this model serves the same purpose as finite viscosity in ill-posed ideal flow problems exhibiting Rayleigh-Taylor or KelvinHelmholtz instabilities, which otherwise develop curvature singularities [7, e.g.].…”

Section: Expanding Bubblesmentioning

confidence: 99%

An accurate numerical scheme for the contraction of a bubble in a Hele--Shaw cell

Dallaston

McCue

2013

ANZIAMJ

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We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

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Section: Expanding Bubblesmentioning

confidence: 99%

An accurate numerical scheme for the contraction of a bubble in a Hele--Shaw cell

Dallaston

McCue

2013

ANZIAMJ

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“…The kernel in (4) has a limit for s → t which can be computed analytically and used for the diagonal entries of the system matrix in the discretization. This is standard and done in [2,3,9]. In the discretization of the post-processor (5) one can use the alternate point trapezoidal rule [13] for the Cauchy principal value integral and Fourier approximation and FFT for the differentiation with respect to arc length.…”

Section: Classic Spectrally Accurate Nyström Schemesmentioning

confidence: 99%

“…The 1993 GGM paper [3] on solving Laplace's equation and computing the Dirichlet-Neumann map on domains exterior to M closed contours has fueled a rapid development in computational multicomponent fluid flow and multiphase materials science [2,8,9,14]. There has also been some recent basic algorithmic development.…”

Section: Introductionmentioning

confidence: 99%

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Faster convergence and higher accuracy for the Dirichlet–Neumann map

Helsing

2009

Journal of Computational Physics

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New techniques allow for more efficient boundary integral algorithms to compute the Dirichlet-Neumann map for Laplace's equation in two-dimensional exterior domains. Novelties include a new post-processor which reduces the need for discretization points with 50 per cent, a new integral equation which reduces the error for resolved geometries with a factor equal to the system size, systematic use of regularization which reduces the error even further, and adaptive mesh generation based on kernel resolution.

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“…Therefore, most numerical approaches consider only the external fluid, with an injected fluid of negligible viscosity, resulting in a single-phase model [3,4].…”

Section: Introductionmentioning

confidence: 99%

A direct boundary element approach for the numerical simulation of finite mobility ratio immiscible displacement in a Hele-Shaw cell

Jackson¹,

Stevens²,

Power³

et al. 2014

WIT Transactions on Modelling and Simulation

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In this work, the interaction between two immiscible fluids with a finite mobility is investigated numerically in a Hele-Shaw cell, simulating conditions found during the injection and lateral spreading of supercritical CO 2 in a deep subsurface aquifer.A two-phase numerical method is presented that uses a direct boundary element approach to compute the normal velocity at the interface between two fluids in a 2-D Hele-Shaw cell, through the evaluation of a hypersingular integral. The resulting second kind Fredholm equation is solved numerically using a truncated convergent Neumann series.Utilising cubic B-Spline surface geometry and function interpolation, the numerical scheme exhibits 6th order spatial convergence and a computational cost that scales with O(N 2 ). This allows the long term non-linear dynamics of a growing CO 2 -brine interface to be explored accurately and efficiently, revealing large differences with previous single-phase models and interface capturing techniques.

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A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele–Shaw cell

Cited by 73 publications

References 42 publications

An accurate numerical scheme for the contraction of a bubble in a Hele--Shaw cell

An accurate numerical scheme for the contraction of a bubble in a Hele--Shaw cell

Faster convergence and higher accuracy for the Dirichlet–Neumann map

A direct boundary element approach for the numerical simulation of finite mobility ratio immiscible displacement in a Hele-Shaw cell

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